Most Recent Proofs - Metamath Proof Explorer

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Most recent proofs These are the 100 (Unicode, GIF) or 1000 (Unicode, GIF) most recent proofs in the set.mm database for the Metamath Proof Explorer (and the Hilbert Space Explorer). The set.mm database is maintained on GitHub with master (stable) and develop (development) versions. This page was created from develop commit 212ee7d9, also available here: set.mm (43MB) or set.mm.bz2 (compressed, 13MB).

The original proofs of theorems with recently shortened proofs can often be found by appending "OLD" to the theorem name, for example 19.43OLD for 19.43. The "OLD" versions are usually deleted after a year.

Other links Email: Norm Megill. Mailing list: Metamath Google Group Updated 7-Dec-2021 . Contributing: How can I contribute to Metamath? Syndication: RSS feed (courtesy of Dan Getz) Related wikis: Ghilbert site; Ghilbert Google Group.

Recent news items (7-Aug-2021) Version 0.198 of the metamath program fixes a bug in "write source ... /rewrap" that prevented end-of-sentence punctuation from appearing in column 79, causing some rewrapped lines to be shorter than necessary. Because this affects about 2000 lines in set.mm, you should use version 0.198 or later for rewrapping before submitting to GitHub.

(7-May-2021) Mario Carneiro has written a Metamath verifier in Lean.

(5-May-2021) Marnix Klooster has written a Metamath verifier in Zig.

(24-Mar-2021) Metamath was mentioned in a couple of articles about OpenAI: Researchers find that large language models struggle with math and What Is GPT-F?.

(26-Dec-2020) Version 0.194 of the metamath program adds the keyword "htmlexturl" to the $t comment to specify external versions of theorem pages. This keyward has been added to set.mm, and you must update your local copy of set.mm for "verify markup" to pass with the new program version.

(19-Dec-2020) Aleksandr A. Adamov has translated the Wikipedia Metamath page into Russian.

(19-Nov-2020) Eric Schmidt's checkmm.cpp was used as a test case for C'est, "a non-standard version of the C++20 standard library, with enhanced support for compile-time evaluation." See C++20 Compile-time Metamath Proof Verification using C'est.

(10-Nov-2020) Filip Cernatescu has updated the XPuzzle (Android app) to version 1.2. XPuzzle is a puzzle with math formulas derived from the Metamath system. At the bottom of the web page is a link to the Google Play Store, where the app can be found.

(7-Nov-2020) Richard Penner created a cross-reference guide between Frege's logic notation and the notation used by set.mm.

(4-Sep-2020) Version 0.192 of the metamath program adds the qualifier '/extract' to 'write source'. See 'help write source' and also this Google Group post.

(23-Aug-2020) Version 0.188 of the metamath program adds keywords Conclusion, Fact, Introduction, Paragraph, Scolia, Scolion, Subsection, and Table to bibliographic references. See 'help write bibliography' for the complete current list.

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**Recent Additions to the Metamath Proof Explorer** Notes (last updated 7-Dec-2020 )
Date	Label	Description
Theorem

22-Jul-2026	sqrtqaa 47589	Square root of a rational number is algebraic. (Contributed by Ender Ting, 22-Jul-2026.)
⊢ (𝐴 ∈ ℚ → (√‘𝐴) ∈ 𝔸)

22-Jul-2026	sqrtnzqaa 47588	Square root of a nonzero rational is algebraic. (Contributed by Ender Ting, 22-Jul-2026.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (√‘𝐴) ∈ 𝔸)

21-Jul-2026	sqrtnnaa 47587	Square root of a natural number is algebraic. (Contributed by Ender Ting, 21-Jul-2026.)
⊢ (𝐴 ∈ ℕ → (√‘𝐴) ∈ 𝔸)

21-Jul-2026	nmullid 36679	Identity law for natural multiplication. (Contributed by Scott Fenton, 21-Jul-2026.)
⊢ (𝐴 ∈ On → (1_o ·_no 𝐴) = 𝐴)

21-Jul-2026	nmulrid 36678	Identity law for natural multiplication. (Contributed by Scott Fenton, 21-Jul-2026.)
⊢ (𝐴 ∈ On → (𝐴 ·_no 1_o) = 𝐴)

21-Jul-2026	naddle 36677	Condition for bounding natural addition above. (Contributed by Scott Fenton, 21-Jul-2026.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +no 𝐵) ⊆ 𝐶 ↔ (∀𝑎 ∈ 𝐴 (𝑎 +no 𝐵) ∈ 𝐶 ∧ ∀𝑏 ∈ 𝐵 (𝐴 +no 𝑏) ∈ 𝐶)))

21-Jul-2026	ltnadd 36676	Condition for bounding a natural sum below. (Contributed by Scott Fenton, 21-Jul-2026.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ (𝐵 +no 𝐶) ↔ (∃𝑏 ∈ 𝐵 𝐴 ⊆ (𝑏 +no 𝐶) ∨ ∃𝑐 ∈ 𝐶 𝐴 ⊆ (𝐵 +no 𝑐))))

21-Jul-2026	aaliou3r 26496	The sum presented above is convergent, which means that the "Liouville number" is indeed real. (Contributed by Ender Ting, 21-Jul-2026.)
⊢ Σ𝑘 ∈ ℕ (2↑-(!‘𝑘)) ∈ ℝ

21-Jul-2026	1aa 26468	One is algebraic. (Contributed by Ender Ting, 21-Jul-2026.)
⊢ 1 ∈ 𝔸

21-Jul-2026	0aa 26467	Zero is algebraic. (Contributed by Ender Ting, 21-Jul-2026.)
⊢ 0 ∈ 𝔸

20-Jul-2026	ralsanmo 50572	An "all some" statement restricted to a class, conjoined with the claim that at most one 𝑥 in 𝐴 satisfies its antecedent, is equivalent to the universal part conjoined with the claim that exactly one 𝑥 in 𝐴 satisfies the antecedent. This is the restricted counterpart of alsanmo 50571. (Contributed by Peter Mazsa and David A. Wheeler, 20-Jul-2026.)
⊢ ((∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ∧ ∃*𝑥 ∈ 𝐴 𝜑) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃!𝑥 ∈ 𝐴 𝜑))

20-Jul-2026	alsanmo 50571	An "all some" statement conjoined with the claim that at most one 𝑥 satisfies its antecedent is equivalent to the universal part conjoined with the claim that exactly one 𝑥 satisfies the antecedent. The "all some" quantifier supplies the existence of such an 𝑥 and ∃*𝑥𝜑 supplies the at-most-one part, so together they yield ∃!𝑥𝜑. (Contributed by Peter Mazsa and David A. Wheeler, 20-Jul-2026.)
⊢ ((∀∃𝑥(𝜑 → 𝜓) ∧ ∃*𝑥𝜑) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃!𝑥𝜑))

20-Jul-2026	rexrals 50570	If a member of 𝐴 satisfying the antecedent exists, then a restricted "all some" statement reduces to its universal part. This is the restricted counterpart of rexals 50576. (Contributed by Peter Mazsa and David A. Wheeler, 20-Jul-2026.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 → (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)))

20-Jul-2026	ralrals 50569	If the universal part of a restricted "all some" statement holds, then the statement reduces to the existence of a member of 𝐴 satisfying its antecedent. This is the restricted counterpart of ralals 50575. (Contributed by Peter Mazsa and David A. Wheeler, 20-Jul-2026.)
⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∃𝑥 ∈ 𝐴 𝜑))

20-Jul-2026	tgaltai 29198	Parallelism implies alternate angles congruence: if a line (𝑋𝐿𝑍) crosses two parallel lines (𝑋𝐿𝑌) and (𝑍𝐿𝑊), then the alternate angles are congruent. First direction of Theorem 12.21 of [Schwabhauser] p. 126. This is also Proposition 1.29 of Euclid's Elements . (Contributed by Thierry Arnoux, 20-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ ∥ = (parlnG‘𝐺) & ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑍))) ∧ ∃𝑡 ∈ (𝑋𝐿𝑍)𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺 ∈ TarskiG_E) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝑊 ∈ 𝑃) & ⊢ (𝜑 → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊)) & ⊢ (𝜑 → 𝑌𝑂𝑊) & ⊢ (𝜑 → 𝑋 ≠ 𝑍) ⇒ ⊢ (𝜑 → ⟨“𝑌𝑋𝑍”⟩(cgrA‘𝐺)⟨“𝑊𝑍𝑋”⟩)

20-Jul-2026	quadcgrprlng 29197	Nontrivial quadrilaterals with congruent and parallel opposite sides are parallelograms. Theorem 12.20 of [Schwabhauser] p. 126. (Contributed by Thierry Arnoux, 20-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ ∥ = (parlnG‘𝐺) & ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑍))) ∧ ∃𝑡 ∈ (𝑋𝐿𝑍)𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺 ∈ TarskiG_E) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝑊 ∈ 𝑃) & ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) & ⊢ (𝜑 → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊)) & ⊢ (𝜑 → (𝑋 − 𝑌) = (𝑍 − 𝑊)) & ⊢ (𝜑 → 𝑌𝑂𝑊) ⇒ ⊢ (𝜑 → ((𝑌𝐿𝑍) ∥ (𝑊𝐿𝑋) ∧ (𝑌 − 𝑍) = (𝑊 − 𝑋)))

20-Jul-2026	prlngsymquadopp 29196	In parallelograms, opposing vertices are on opposite sides of the diagonal. Second part of Theorem 12.19 of [Schwabhauser] p. 126. (Contributed by Thierry Arnoux, 20-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ ∥ = (parlnG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺 ∈ TarskiG_E) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝑊 ∈ 𝑃) & ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) & ⊢ (𝜑 → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊)) & ⊢ (𝜑 → (𝑌𝐿𝑍) ∥ (𝑊𝐿𝑋)) & ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑍))) ∧ ∃𝑡 ∈ (𝑋𝐿𝑍)𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ 𝐼 = (Itv‘𝐺) ⇒ ⊢ (𝜑 → 𝑊𝑂𝑌)

20-Jul-2026	prlngsymquad 29195	All parallelograms are symmetric quadrilaterals. First part of Theorem 12.19 of [Schwabhauser] p. 126. (Contributed by Thierry Arnoux, 20-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ ∥ = (parlnG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺 ∈ TarskiG_E) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝑊 ∈ 𝑃) & ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) & ⊢ (𝜑 → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊)) & ⊢ (𝜑 → (𝑌𝐿𝑍) ∥ (𝑊𝐿𝑋)) ⇒ ⊢ (𝜑 → ((𝑋 − 𝑌) = (𝑍 − 𝑊) ∧ (𝑌 − 𝑍) = (𝑊 − 𝑋)))

20-Jul-2026	prlngsymquadlem 29194	Lemma for prlngsymquad 29195. (Contributed by Thierry Arnoux, 20-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ ∥ = (parlnG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺 ∈ TarskiG_E) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝑊 ∈ 𝑃) & ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) & ⊢ (𝜑 → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊)) & ⊢ (𝜑 → (𝑌𝐿𝑍) ∥ (𝑊𝐿𝑋)) & ⊢ 𝑇 = (((pInvG‘𝐺)‘(𝑋(midG‘𝐺)𝑍))‘𝑌) ⇒ ⊢ (𝜑 → 𝑇 = 𝑊)

20-Jul-2026	symquadprlng 29193	Symmetrical quadrilaterals are parallelograms. Theorem 12.18 of [Schwabhauser] p. 126. (Contributed by Thierry Arnoux, 20-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ ∥ = (parlnG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺 ∈ TarskiG_E) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝑊 ∈ 𝑃) & ⊢ (𝜑 → (𝑋 − 𝑌) = (𝑍 − 𝑊)) & ⊢ (𝜑 → (𝑌 − 𝑍) = (𝑊 − 𝑋)) & ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) & ⊢ (𝜑 → 𝑌 ≠ 𝑊) & ⊢ (𝜑 → 𝑇 ∈ (𝑋𝐿𝑍)) & ⊢ (𝜑 → 𝑇 ∈ (𝑌𝐿𝑊)) ⇒ ⊢ (𝜑 → ((𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊) ∧ (𝑌𝐿𝑍) ∥ (𝑊𝐿𝑋)))

20-Jul-2026	prlngeq 29188	Playfair's axiom, written as an equality: if two different lines are parallel to a given line at a given point, they are equal. (Contributed by Thierry Arnoux, 20-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ ∥ = (parlnG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺 ∈ TarskiG_E) & ⊢ (𝜑 → 𝐴 ∥ 𝐵) & ⊢ (𝜑 → 𝐴 ∥ 𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ∈ 𝐶) ⇒ ⊢ (𝜑 → 𝐵 = 𝐶)

20-Jul-2026	symquadmid 29089	In a symmetrical quadrilateral, the midpoints of the diagonals coincide. Corollary of Lemma 7.21 of [Schwabhauser] p. 52. (Contributed by Thierry Arnoux, 20-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑀 = (midG‘𝐺) & ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑍)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑍))) ∧ ∃𝑡 ∈ (𝑋𝐿𝑍)𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝑊 ∈ 𝑃) & ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) & ⊢ (𝜑 → 𝑌 ≠ 𝑊) & ⊢ (𝜑 → (𝑋 − 𝑌) = (𝑍 − 𝑊)) & ⊢ (𝜑 → (𝑌 − 𝑍) = (𝑊 − 𝑋)) & ⊢ (𝜑 → 𝑌𝑂𝑊) ⇒ ⊢ (𝜑 → (𝑋𝑀𝑍) = (𝑌𝑀𝑊))

20-Jul-2026	hlopp 29035	If two points 𝑋 and 𝑌 lie on opposite sides of a line 𝐴, then given a point 𝑍 on 𝐴, any point 𝑊 on the line (𝑋𝐿𝑍) opposite to 𝑌 lies on the half line (𝑍𝑋) (Contributed by Thierry Arnoux, 20-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐴) ∧ 𝑏 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑡 ∈ 𝐴 𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑋𝑂𝑌) & ⊢ (𝜑 → 𝑍 ∈ 𝐴) & ⊢ (𝜑 → 𝑊𝑂𝑌) & ⊢ (𝜑 → 𝑊 ∈ (𝑋𝐿𝑍)) ⇒ ⊢ (𝜑 → 𝑊(𝐾‘𝑍)𝑋)

20-Jul-2026	symquadprlnglem 28951	Lemma for symquadprlnglem 28951. (Contributed by Thierry Arnoux, 20-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝑊 ∈ 𝑃) & ⊢ (𝜑 → (𝑋 − 𝑌) = (𝑍 − 𝑊)) & ⊢ (𝜑 → (𝑌 − 𝑍) = (𝑊 − 𝑋)) & ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) & ⊢ (𝜑 → 𝑌 ≠ 𝑊) & ⊢ (𝜑 → 𝑇 ∈ (𝑋𝐿𝑍)) & ⊢ (𝜑 → 𝑇 ∈ (𝑌𝐿𝑊)) ⇒ ⊢ (𝜑 → ¬ (𝑊 ∈ (𝑍𝐿𝑌) ∨ 𝑍 = 𝑌))

19-Jul-2026	disjdifg 4433	A class does not intersect a relative complement of a superclass. (Contributed by NM, 24-Mar-1998.) Generalize from disjdif 4434. (Revised by BJ, 19-Jul-2026.)
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ (𝐶 ∖ 𝐵)) = ∅)

19-Jul-2026	sseq0b 4361	The only subclass of the empty class is itself. (Contributed by NM, 7-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) Strengthen sseq0 4362 to a biconditional. (Revised by BJ, 19-Jul-2026.)
⊢ (𝐴 = ∅ → (𝐵 ⊆ 𝐴 ↔ 𝐵 = ∅))

18-Jul-2026	sepab 5304	Separation Scheme (Aussonderung) in terms of a class abstraction. Prefer using the more natural statement rabexg 5309. (Contributed by NM, 8-Jun-1994.) Put in closed form. (Revised by BJ, 18-Jul-2026.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V)

18-Jul-2026	ssex 5292	A subclass of a set is a set. Exercise 3 of [TakeutiZaring] p. 22. This is one way to express the Axiom of Separation ax-sep 5258 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.) (Proof shortened by BJ, 18-Jul-2026.)
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)

18-Jul-2026	ssexg 5291	A subclass of a set is a set. Exercise 3 of [TakeutiZaring] p. 22 (generalized). (Contributed by NM, 14-Aug-1994.) (Proof shortened by BJ, 18-Jul-2026.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ V)

18-Jul-2026	inssdif0 4330	Intersection, subclass, and difference relationship. (Contributed by NM, 27-Oct-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Sep-2014.) (Proof shortened by BJ, 18-Jul-2026.)
⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐶 ↔ (𝐴 ∩ (𝐵 ∖ 𝐶)) = ∅)

16-Jul-2026	nmulle 36675	A condition for bounding a natural product above. Converse of ltnmul 36674. (Contributed by Scott Fenton, 16-Jul-2026.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ·_no 𝐵) ⊆ 𝐶 ↔ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 ((𝑎 ·_no 𝐵) +no (𝐴 ·_no 𝑏)) ∈ (𝐶 +no (𝑎 ·_no 𝑏))))

15-Jul-2026	2alsraln0id 50579	Nested general "all some" quantifiers with class membership as their antecedents, for the same class 𝐴: 𝜑 holds for every 𝑥 and every 𝑦 in 𝐴, and 𝐴 is not empty. (Contributed by Peter Mazsa, 28-May-2019.) (Revised by David A. Wheeler, 15-Jul-2026.)
⊢ (∀∃𝑥(𝑥 ∈ 𝐴 → ∀∃𝑦(𝑦 ∈ 𝐴 → 𝜑)) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ∧ 𝐴 ≠ ∅))

15-Jul-2026	2alsraln0 50578	Nested general "all some" quantifiers with class membership as their antecedents: 𝜑 holds for every 𝑥 in 𝐴 and every 𝑦 in 𝐵, and both 𝐴 and 𝐵 are not empty. (Contributed by Peter Mazsa, 28-May-2019.) (Revised by David A. Wheeler, 15-Jul-2026.)
⊢ (∀∃𝑥(𝑥 ∈ 𝐴 → ∀∃𝑦(𝑦 ∈ 𝐵 → 𝜑)) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ (𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅)))

15-Jul-2026	n0als 50577	If 𝐴 is not empty, then the general "all some" quantifier with class membership as its antecedent reduces to the assertion that 𝜑 holds for every 𝑥 in 𝐴. (Contributed by Peter Mazsa, 19-Dec-2018.) (Revised by David A. Wheeler, 15-Jul-2026.)
⊢ (𝐴 ≠ ∅ → (∀∃𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥 ∈ 𝐴 𝜑))

15-Jul-2026	rexals 50576	If some 𝑥 in 𝐴 satisfies 𝜑, then the general "all some" quantifier with class membership as its antecedent reduces to the assertion that 𝜑 holds for every 𝑥 in 𝐴. See rexrals 50570 for the restricted counterpart. (Contributed by Peter Mazsa, 19-Dec-2018.) (Revised by David A. Wheeler, 15-Jul-2026.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 → (∀∃𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥 ∈ 𝐴 𝜑))

15-Jul-2026	ralals 50575	If 𝜑 holds for every 𝑥 in 𝐴, then the general "all some" quantifier with class membership as its antecedent reduces to the assertion that some 𝑥 in 𝐴 satisfies 𝜑. See ralrals 50569 for the restricted counterpart. (Contributed by Peter Mazsa, 19-Dec-2018.) (Revised by David A. Wheeler, 15-Jul-2026.)
⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∀∃𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∃𝑥 ∈ 𝐴 𝜑))

15-Jul-2026	alsraln0 50574	The general "all some" quantifier with class membership as its antecedent holds if and only if 𝜑 holds for every 𝑥 in 𝐴 and 𝐴 is not empty. (Contributed by Peter Mazsa, 28-Nov-2018.) (Revised by David A. Wheeler, 15-Jul-2026.)
⊢ (∀∃𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ 𝐴 ≠ ∅))

15-Jul-2026	alsralrex 50573	The general "all some" quantifier with class membership as its antecedent holds if and only if 𝜑 holds for every 𝑥 in 𝐴 and some 𝑥 in 𝐴 satisfies 𝜑. (Contributed by Peter Mazsa, 27-Nov-2018.) (Revised by David A. Wheeler, 15-Jul-2026.)
⊢ (∀∃𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜑))

15-Jul-2026	ltnmul 36674	Characterize less-than a natural product. (Contributed by Scott Fenton, 15-Jul-2026.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ (𝐵 ·_no 𝐶) ↔ ∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 (𝐴 +no (𝑏 ·_no 𝑐)) ⊆ ((𝑏 ·_no 𝐶) +no (𝐵 ·_no 𝑐))))

15-Jul-2026	nmulel1 36673	Natural multiplication by a non-zero number preserves less-than. (Contributed by Scott Fenton, 15-Jul-2026.)
⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ≠ ∅)) → (𝐶 ·_no 𝐴) ∈ (𝐶 ·_no 𝐵))

15-Jul-2026	nmulss1 36672	Natural multiplication preserves less-than or equal. (Contributed by Scott Fenton, 15-Jul-2026.)
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝐴 ⊆ 𝐵) → (𝐶 ·_no 𝐴) ⊆ (𝐶 ·_no 𝐵))

15-Jul-2026	nmuladdss 36671	Ordering relationship for natural ordinal operations. (Contributed by Scott Fenton, 15-Jul-2026.)
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵)) → ((𝐶 ·_no 𝐵) +no (𝐴 ·_no 𝐷)) ⊆ ((𝐴 ·_no 𝐵) +no (𝐶 ·_no 𝐷)))

15-Jul-2026	nmuladdel 36670	Ordering relationship for natural ordinal operations. (Contributed by Scott Fenton, 15-Jul-2026.)
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → ((𝐶 ·_no 𝐵) +no (𝐴 ·_no 𝐷)) ∈ ((𝐴 ·_no 𝐵) +no (𝐶 ·_no 𝐷)))

14-Jul-2026	bj-inex1gALT 37541	Proof of inex1g 5289 from sepg 5260 to then allow proving inex1 5287 from it. That does not reduce the combined proof size of inex1 5287 and inex1g 5289. (Contributed by BJ, 14-Jul-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V)

14-Jul-2026	uniex2 7737	The Axiom of Union using the standard abbreviation for union. Given any set 𝑥, its union 𝑦 exists. (Contributed by NM, 4-Jun-2006.) (Proof shortened by BJ, 14-Jul-2026.)
⊢ ∃𝑦 𝑦 = ∪ 𝑥

14-Jul-2026	sepgi 5261	Inference associated with sepg 5260. The requirement that 𝑦 not occur in 𝜑 is necessary, as notsep 5336 shows. (Contributed by NM, 21-Jun-1993.) (Revised by BJ, 14-Jul-2026.)
⊢ 𝐴 ∈ V ⇒ ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))

13-Jul-2026	idomcanr 49096	Cancellation law for domains. (Contributed by Jeff Madsen, 6-Jan-2011.) (Revised by AV, 13-Jul-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (._r‘𝑅) & ⊢ 0 = (0_g‘𝑅) ⇒ ⊢ (((𝑅 ∈ IDomn ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ 𝑍 ≠ 0 ) → ((𝑋 · 𝑍) = (𝑌 · 𝑍) → 𝑋 = 𝑌))

13-Jul-2026	idomcanl 49095	Cancellation law for domains. (Contributed by Jeff Madsen, 6-Jan-2011.) (Revised by AV, 13-Jul-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (._r‘𝑅) & ⊢ 0 = (0_g‘𝑅) ⇒ ⊢ (((𝑅 ∈ IDomn ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ 𝑋 ≠ 0 ) → ((𝑋 · 𝑌) = (𝑋 · 𝑍) → 𝑌 = 𝑍))

13-Jul-2026	prlngmid2 29192	If the midpoints of two segments (𝑋𝐼𝑍) and (𝑌𝐼𝑊) coincide, the points 𝑋, 𝑌, 𝑍 and 𝑊 form a parallelogram, i.e. the lines (𝑋𝐿𝑌) and (𝑍𝐿𝑊) are parallel. Theorem 12.17 of [Schwabhauser] p. 125. (Contributed by Thierry Arnoux, 13-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ ∥ = (parlnG‘𝐺) & ⊢ 𝑀 = (midG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺 ∈ TarskiG_E) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ (𝑃 ∖ (𝑋𝐿𝑌))) & ⊢ (𝜑 → 𝑊 ∈ 𝑃) & ⊢ (𝜑 → (𝑋𝑀𝑍) = (𝑌𝑀𝑊)) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) ⇒ ⊢ (𝜑 → (𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊))

13-Jul-2026	prlnginn0 29191	A line 𝐶 intersecting another line 𝐴 also intersects any line 𝐵 parallel to 𝐴. Theorem 12.16 of [Schwabhauser] p. 125. (Contributed by Thierry Arnoux, 13-Jul-2026.)
⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ ∥ = (parlnG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐺 ∈ TarskiG_E) & ⊢ (𝜑 → 𝐻 ∈ ran 𝐸) & ⊢ (𝜑 → 𝐶 ∈ ran 𝐿) & ⊢ (𝜑 → (𝐴 ∩ 𝐶) ≠ ∅) & ⊢ (𝜑 → 𝐴 ≠ 𝐶) & ⊢ (𝜑 → 𝐴 ∥ 𝐵) & ⊢ (𝜑 → 𝐴 ⊆ 𝐻) & ⊢ (𝜑 → 𝐵 ⊆ 𝐻) & ⊢ (𝜑 → 𝐶 ⊆ 𝐻) ⇒ ⊢ (𝜑 → (𝐵 ∩ 𝐶) ≠ ∅)

13-Jul-2026	prlngplngtr 29190	Transitivity of parallelism, for lines in the same plane 𝐻. This is case 1 of Theorem 12.15 of [Schwabhauser] p. 124. (Contributed by Thierry Arnoux, 13-Jul-2026.)
⊢ 𝐸 = (hlG‘𝐺) & ⊢ ∥ = (parlnG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐻 ∈ ran 𝐸) & ⊢ (𝜑 → 𝐴 ⊆ 𝐻) & ⊢ (𝜑 → 𝐴 ∥ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ TarskiG_E) & ⊢ (𝜑 → 𝐶 ⊆ 𝐻) & ⊢ (𝜑 → 𝐵 ∥ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ∥ 𝐶)

13-Jul-2026	prlngpln4 29189	Building a parallel line conserves planes, i.e. given a line 𝐴 and a point 𝑋 not on 𝐴, the (unique) parallel 𝐵 to 𝐴 through 𝑋 lies completely within the plane defined by 𝐴 and 𝑋. Theorem 12.14 of [Schwabhauser] p. 124. (Contributed by Thierry Arnoux, 13-Jul-2026.)
⊢ 𝐸 = (hlG‘𝐺) & ⊢ ∥ = (parlnG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐻 ∈ ran 𝐸) & ⊢ (𝜑 → 𝐴 ⊆ 𝐻) & ⊢ (𝜑 → 𝐴 ∥ 𝐵) & ⊢ (𝜑 → 𝑋 ∈ 𝐻) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝐵 ⊆ 𝐻)

13-Jul-2026	prlngmo2 29187	Playfair's axiom, without the restriction that the point 𝑋 is outside of the line 𝐴. Theorem 12.11 of [Schwabhauser] p. 123. (Contributed by Thierry Arnoux, 13-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ ∥ = (parlnG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝐺 ∈ TarskiG_E) ⇒ ⊢ (𝜑 → ∃*𝑏 ∈ ran 𝐿(𝐴 ∥ 𝑏 ∧ 𝑋 ∈ 𝑏))

13-Jul-2026	dfprlng3 29179	Alternate definition of (strict) parallelism. Theorem 12.7 of [Schwabhauser] p. 122. (Contributed by Thierry Arnoux, 13-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ ∥ = (parlnG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ (𝑃 ∖ {𝑋})) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐴 ≠ (𝑋𝐿𝑌)) ⇒ ⊢ (𝜑 → (𝐴 ∥ (𝑋𝐿𝑌) ↔ (𝑋((hpG‘𝐺)‘𝐴)𝑌 ∧ (𝐴 ∩ (𝑋𝐿𝑌)) = ∅)))

13-Jul-2026	dfprlng2 29178	Alternate definition of (strict) parallelism. Theorem 12.7 of [Schwabhauser] p. 122. (Contributed by Thierry Arnoux, 13-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ ∥ = (parlnG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ (𝑃 ∖ {𝑋})) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝑊 ∈ (𝑃 ∖ {𝑍})) & ⊢ (𝜑 → (𝑋𝐿𝑌) ≠ (𝑍𝐿𝑊)) ⇒ ⊢ (𝜑 → ((𝑋𝐿𝑌) ∥ (𝑍𝐿𝑊) ↔ (𝑍((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑊 ∧ ((𝑋𝐿𝑌) ∩ (𝑍𝐿𝑊)) = ∅)))

13-Jul-2026	ragsupplcgra 29129	An angle ⟨“𝑋𝑌𝑍”⟩ is a right angle exactly when it is congruent to its supplementary angle ⟨“𝑋𝑌𝑊”⟩. Theorem 11.18 of [Schwabhauser] p. 98. (Contributed by Thierry Arnoux, 13-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ (𝑃 ∖ {𝑌})) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ (𝑃 ∖ {𝑌})) & ⊢ (𝜑 → 𝑊 ∈ (𝑃 ∖ {𝑌})) & ⊢ (𝜑 → 𝑌 ∈ (𝑍𝐼𝑊)) ⇒ ⊢ (𝜑 → (⟨“𝑋𝑌𝑍”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝑋𝑌𝑍”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑊”⟩))

13-Jul-2026	cgrarag 29128	Any angle ⟨“𝐴𝐵𝐶”⟩ congruent with a right angle ⟨“𝑋𝑌𝑍”⟩ is a right angle. Theorem 11.17 of [Schwabhauser] p. 98. (Contributed by Thierry Arnoux, 13-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → ⟨“𝑋𝑌𝑍”⟩ ∈ (∟G‘𝐺)) & ⊢ (𝜑 → ⟨“𝑋𝑌𝑍”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩) ⇒ ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))

13-Jul-2026	mirlni 28953	The mirror of a point 𝑋 on a line (𝑌𝐿𝑍) is on the mirrored line ((𝑀‘𝑌)𝐿(𝑀‘𝑍)). (Contributed by Thierry Arnoux, 13-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ 𝑀 = (𝑆‘𝐴) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ≠ 𝑌) & ⊢ (𝜑 → 𝑋 ∈ (𝑌𝐿𝑍)) ⇒ ⊢ (𝜑 → (𝑀‘𝑋) ∈ ((𝑀‘𝑌)𝐿(𝑀‘𝑍)))

13-Jul-2026	mirleqb 28952	Equality theorem for point mirroring. (Contributed by Thierry Arnoux, 13-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ 𝑀 = (𝑆‘𝐴) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝑋 = 𝑌 ↔ (𝑀‘𝑋) = (𝑀‘𝑌)))

13-Jul-2026	hlgrcl2 28854	Reverse closure for rays. (Contributed by Thierry Arnoux, 13-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐴(𝐾‘𝐶)𝐵) ⇒ ⊢ (𝜑 → 𝐵 ∈ 𝑃)

13-Jul-2026	hlgrcl1 28853	Reverse closure for rays. (Contributed by Thierry Arnoux, 13-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐴(𝐾‘𝐶)𝐵) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝑃)

13-Jul-2026	ishlg2 28852	Alternate version of ishlg 28855, including closure. (Contributed by Thierry Arnoux, 13-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐾 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝐴(𝐾‘𝐶)𝐵 ↔ ((𝐴 ∈ 𝑃 ∧ 𝐵 ∈ 𝑃) ∧ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))))))

13-Jul-2026	eldifsnbd 4755	Membership in a set with an element removed implies non-equality with that element. (Contributed by Thierry Arnoux, 13-Jul-2026.)
⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ {𝐶})) ⇒ ⊢ (𝜑 → 𝐴 ≠ 𝐶)

13-Jul-2026	nrmod 3846	Deduce the negation of a restricted "at most one" quantifier. (Contributed by Thierry Arnoux, 13-Jul-2026.)
⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜒)) & ⊢ (𝑥 = 𝑌 → (𝜓 ↔ 𝜃)) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ (𝜑 → 𝜒) & ⊢ (𝜑 → 𝜃) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) ⇒ ⊢ (𝜑 → ¬ ∃*𝑥 ∈ 𝐴 𝜓)

13-Jul-2026	3orim123da 1473	Disjoin antecedents and consequents of three premises. (Contributed by Thierry Arnoux, 13-Jul-2026.)
⊢ (𝜑 → (𝜓 ∨ 𝜃 ∨ 𝜂)) & ⊢ ((𝜑 ∧ 𝜓) → 𝜒) & ⊢ ((𝜑 ∧ 𝜃) → 𝜏) & ⊢ ((𝜑 ∧ 𝜂) → 𝜁) ⇒ ⊢ (𝜑 → (𝜒 ∨ 𝜏 ∨ 𝜁))

12-Jul-2026	rals-no-surprise 50568	Demonstrate that there is never a "surprise" when using the allsome quantifier restricted to a class, that is, it is never possible for the consequent to be both always true and always false of the members of 𝐴 that satisfy the antecedent. This is the restricted counterpart of als-no-surprise 50567, and follows from it by dfrals2 50551. Note that this needs no assumption that 𝐴 is nonempty, because allsome requires a member of 𝐴 satisfying 𝜑, and that member would have to satisfy both 𝜓 and ¬ 𝜓. The ordinary restricted "for all" requires no such member and can be vacuously true, as shown in empty-surprise2 50544; that is the point of allsome. (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ ¬ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ∧ ∀∃𝑥 ∈ 𝐴(𝜑 → ¬ 𝜓))

12-Jul-2026	cbvals 50566	Rule used to change bound variables, using implicit substitution. (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃)) ⇒ ⊢ (∀∃𝑥(𝜑 → 𝜓) ↔ ∀∃𝑦(𝜒 → 𝜃))

12-Jul-2026	nfrals 50565	Bound-variable hypothesis builder for "all some" restricted to a class. (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ Ⅎ𝑥∀∃𝑦 ∈ 𝐴(𝜑 → 𝜓)

12-Jul-2026	nfals 50564	Bound-variable hypothesis builder for "all some". (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ Ⅎ𝑥∀∃𝑦(𝜑 → 𝜓)

12-Jul-2026	alsbid 50563	Deduction form of alsbii 50561. (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 ↔ 𝜃)) & ⊢ (𝜑 → (𝜒 ↔ 𝜏)) ⇒ ⊢ (𝜑 → (∀∃𝑥(𝜓 → 𝜒) ↔ ∀∃𝑥(𝜃 → 𝜏)))

12-Jul-2026	ralsbii 50562	Congruence for "all some" restricted to a class. (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ (𝜑 ↔ 𝜒) & ⊢ (𝜓 ↔ 𝜃) ⇒ ⊢ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀∃𝑥 ∈ 𝐴(𝜒 → 𝜃))

12-Jul-2026	alsbii 50561	Congruence: equivalents may be substituted inside an "all some". (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ (𝜑 ↔ 𝜒) & ⊢ (𝜓 ↔ 𝜃) ⇒ ⊢ (∀∃𝑥(𝜑 → 𝜓) ↔ ∀∃𝑥(𝜒 → 𝜃))

12-Jul-2026	ralsex 50560	The consequent of an "all some" restricted to a class is witnessed: some member of 𝐴 satisfying 𝜑 also satisfies 𝜓. Restricted counterpart of alsex 50559. (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) → ∃𝑥 ∈ 𝐴 𝜓)

12-Jul-2026	alsex 50559	The consequent of an "all some" is witnessed: if 𝜓 holds of every 𝑥 satisfying 𝜑, and some 𝑥 satisfies 𝜑, then some 𝑥 satisfies 𝜓. This is the positive counterpart of als-no-surprise 50567, and it is the property that ordinary "for all" with implication lacks: from ∀𝑥(𝜑 → 𝜓) alone nothing whatever follows about 𝜓, as alimp-surprise 50541 shows. It is the reason the allsome quantifier says what a speaker of "all Martians are green" usually means. (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ (∀∃𝑥(𝜑 → 𝜓) → ∃𝑥𝜓)

12-Jul-2026	ralsn0d 50558	Deduction rule: Given "all some" applied to a class, the class is not the empty set. (Contributed by David A. Wheeler, 23-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
⊢ (𝜑 → ∀∃𝑥 ∈ 𝐴(𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → 𝐴 ≠ ∅)

12-Jul-2026	rals2d 50557	Deduction rule: Given "all some" applied to a class, you can extract the "there exists" part. Note that the witness must satisfy the antecedent 𝜓, not merely be a member of 𝐴. (Contributed by David A. Wheeler, 20-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
⊢ (𝜑 → ∀∃𝑥 ∈ 𝐴(𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)

12-Jul-2026	rals1d 50556	Deduction rule: Given "all some" applied to a class, you can extract the "for all" part. (Contributed by David A. Wheeler, 20-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
⊢ (𝜑 → ∀∃𝑥 ∈ 𝐴(𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))

12-Jul-2026	ralsd 50553	Introduction rule for "all some" restricted to a class. This is the converse of rals1d 50556 and rals2d 50557 taken together. (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒)) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓) ⇒ ⊢ (𝜑 → ∀∃𝑥 ∈ 𝐴(𝜓 → 𝜒))

12-Jul-2026	alsd 50552	Introduction rule: "all some" holds if the "for all" part holds and the antecedent has a witness. This is the converse of als1d 50554 and als2d 50555 taken together, and is what lets an "all some" statement be proved rather than merely taken apart. (Contributed by David A. Wheeler, 12-Jul-2026.)
⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒)) & ⊢ (𝜑 → ∃𝑥𝜓) ⇒ ⊢ (𝜑 → ∀∃𝑥(𝜓 → 𝜒))

12-Jul-2026	dfrals2 50551	The bounded "all some" form is the general form with the class membership folded into the antecedent. (Contributed by David A. Wheeler, 22-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
⊢ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓))

12-Jul-2026	df-rals 50550	Define "all some" applied to a class, which means 𝜓 is true whenever 𝜑 is true for 𝑥 in 𝐴, and there is at least one 𝑥 in 𝐴 where 𝜑 is true. An older definition of the "all some" quantifier when scoped to a class, named df-alsc and now removed, instead applied a bare formula 𝜑 to the members of a class, asserting only (∀𝑥 ∈ 𝐴𝜑 ∧ ∃𝑥𝑥 ∈ 𝐴), that is, that the formula held throughout 𝐴 and that 𝐴 had at least one member. I've now decided that that was a mistake. Its older existence conjunct ∃𝑥𝑥 ∈ 𝐴 did not require any member of 𝐴 to satisfy the antecedent, so if the formula was itself an implication, that inner implication could still be vacuously true, which is precisely what the allsome quantifier exists to prevent. For example, the older definition meant that "among Martians, all tall ones are green" could be considered true if there are Martians, but no tall Martians. This version of the definition instead ensures that claims of the form "among Martians, all tall ones are green" can only be true if all tall Martians are green and that there is at least one tall Martian. (Contributed by David A. Wheeler, 20-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
⊢ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃𝑥 ∈ 𝐴 𝜑))

12-Jul-2026	wrals 50548	Extend wff definition to include "all some" applied to a class, which means 𝜓 is true whenever 𝜑 is true for 𝑥 in 𝐴, and there is at least one 𝑥 in 𝐴 where 𝜑 is true. (Contributed by David A. Wheeler, 20-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
wff ∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓)

12-Jul-2026	wals 50547	Extend wff definition to include "all some" applied to a top-level implication, which means 𝜓 is true whenever 𝜑 is true, and there is at least one 𝑥 where 𝜑 is true. (Contributed by David A. Wheeler, 20-Oct-2018.) (Revised by David A. Wheeler, 12-Jul-2026.)
wff ∀∃𝑥(𝜑 → 𝜓)

12-Jul-2026	idomnzd 49094	A domain has no zero-divisors (besides zero). (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by AV, 12-Jul-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (._r‘𝑅) & ⊢ 0 = (0_g‘𝑅) ⇒ ⊢ ((𝑅 ∈ IDomn ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 · 𝑌) = 0 )) → (𝑋 = 0 ∨ 𝑌 = 0 ))

12-Jul-2026	isidom3 49093	The predicate "is a domain", alternate expression. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by AV, 12-Jul-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (._r‘𝑅) & ⊢ 0 = (0_g‘𝑅) & ⊢ 1 = (1_r‘𝑅) ⇒ ⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 0 ≠ 1 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑎 · 𝑏) = 0 → (𝑎 = 0 ∨ 𝑏 = 0 ))))

12-Jul-2026	bj-vn0ALT 37689	Alternate proof of vn0 4299 which does not use eqabbw 2836 (and is shorter than vn0 4299 when eqabbw 2836 is inlined). (Contributed by BJ, 12-Jul-2026.) Using the same dummy variable for 𝑦 and 𝑧 slightly reduces the proof size. (Proof modification is discouraged.) (New usage is discouraged.)
⊢ V ≠ ∅

12-Jul-2026	cmprmidlmcl 21456	The complement of a prime ideal is multiplicatively closed. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by AV, 12-Jul-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (._r‘𝑅) ⇒ ⊢ (((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (𝐵 ∖ 𝑃) ∧ 𝐽 ∈ (𝐵 ∖ 𝑃))) → (𝐼 · 𝐽) ∈ (𝐵 ∖ 𝑃))

12-Jul-2026	prmidlc2 21455	Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by AV, 12-Jul-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (._r‘𝑅) ⇒ ⊢ (((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (𝐵 ∖ 𝑃) ∧ 𝐽 ∈ 𝐵 ∧ (𝐼 · 𝐽) ∈ 𝑃)) → 𝐽 ∈ 𝑃)

12-Jul-2026	rspsn0 21353	A principal ideal (an ideal generated by one element) in a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by AV, 12-Jul-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (._r‘𝑅) & ⊢ 𝐾 = (RSpan‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝐾‘{𝑋}) = {𝑖 ∈ 𝐵 ∣ ∃𝑥 ∈ 𝐵 𝑖 = (𝑥 · 𝑋)})

12-Jul-2026	nvpss 4371	No class strictly includes the universal class. Dual of npss0 4369. (Contributed by BJ, 12-Jul-2026.)
⊢ ¬ V ⊊ 𝐴

12-Jul-2026	vvin 4367	Two classes are both the universal class if and only if their intersection is the universal class. Dual of un00 4365. (Contributed by BJ, 12-Jul-2026.)
⊢ ((𝐴 = V ∧ 𝐵 = V) ↔ (𝐴 ∩ 𝐵) = V)

12-Jul-2026	vn0 4299	The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.) Avoid ax-8 2145, df-clel 2838. (Revised by GG, 6-Sep-2024.) (Proof shortened by BJ, 12-Jul-2026.)
⊢ V ≠ ∅

10-Jul-2026	25or6to4 42954	Question 67 of 68 from a lecture Prof. Loof Lirpa held last Saturday in Lincoln Park. When asked why the smaller root wasn't reduced to 3/2, Lirpa responded "It really doesn't matter anyhow." (Contributed by Luke Murphy, 10-Jul-2026.)
⊢ (𝜑 → 𝐴 = 1) & ⊢ (𝜑 → 𝐵 = -(;53 / 2)) & ⊢ (𝜑 → 𝐶 = (;75 / 2)) & ⊢ (𝜑 → 𝑋 ∈ ℂ) ⇒ ⊢ (𝜑 → (((𝐴 · (𝑋↑2)) + ((𝐵 · 𝑋) + 𝐶)) = 0 ↔ (𝑋 = ;25 ∨ 𝑋 = (6 / 4))))

10-Jul-2026	quadfac 42953	The solution of a quadratic equation via factoring. (Contributed by Luke Murphy, 10-Jul-2026.)
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → 𝑀 ∈ ℂ) & ⊢ (𝜑 → 𝑁 ∈ ℂ) & ⊢ (𝜑 → (𝑀 + 𝑁) = -(𝐵 / 𝐴)) & ⊢ (𝜑 → (𝑀 · 𝑁) = (𝐶 / 𝐴)) ⇒ ⊢ (𝜑 → (((𝐴 · (𝑋↑2)) + ((𝐵 · 𝑋) + 𝐶)) = 0 ↔ (𝑋 = 𝑀 ∨ 𝑋 = 𝑁)))

10-Jul-2026	nelscottrankgt 35499	If a member of the input set is not a member of the Scott's trick set, then its rank is greater than the rank of a member of the Scott's trick set. (Contributed by BTernaryTau, 10-Jul-2026.)
⊢ ((𝐴 ∈ Scott 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ ¬ 𝐶 ∈ Scott 𝐵) → (rank‘𝐴) ∈ (rank‘𝐶))

10-Jul-2026	scottrankeqel 35498	If a member of the input set has the same rank as a member of the Scott's trick set, then it is also a member of the Scott's trick set. (Contributed by BTernaryTau, 10-Jul-2026.)
⊢ ((𝐴 ∈ Scott 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ (rank‘𝐶) = (rank‘𝐴)) → 𝐶 ∈ Scott 𝐵)

10-Jul-2026	elscott2 35494	Membership in a Scott's trick set. (Contributed by BTernaryTau, 10-Jul-2026.)
⊢ (𝐴 ∈ Scott 𝐵 ↔ (𝐴 ∈ 𝐵 ∧ (rank‘𝐴) = ∩ (rank “ 𝐵)))

10-Jul-2026	dfscott3 35493	Alternate definition of a Scott's trick set. (Contributed by BTernaryTau, 10-Jul-2026.)
⊢ Scott 𝐴 = (𝐴 ∩ (𝑅₁‘suc ∩ (rank “ 𝐴)))

10-Jul-2026	imadifssran 6204	Condition for the range of a class to be the range of one of its restrictions. (Contributed by AV, 4-Oct-2025.) Remove antecedent. (Revised by Eric Schmidt, 10-Jul-2026.)
⊢ ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → ran 𝐹 = ran (𝐹 ↾ 𝐴))

9-Jul-2026	elscottrankeq 35496	Elements in a Scott's trick set have the same rank. (Contributed by BTernaryTau, 9-Jul-2026.)
⊢ ((𝐴 ∈ Scott 𝐶 ∧ 𝐵 ∈ Scott 𝐶) → (rank‘𝐴) = (rank‘𝐵))

8-Jul-2026	rankscott 35503	The rank of a nonempty Scott's trick set. (Contributed by BTernaryTau, 8-Jul-2026.)
⊢ (𝐴 ≠ ∅ → (rank‘Scott 𝐴) = suc ∩ (rank “ 𝐴))

8-Jul-2026	elscottrank 35495	The rank of an element in a Scott's trick set. (Contributed by BTernaryTau, 8-Jul-2026.)
⊢ (𝐴 ∈ Scott 𝐵 → (rank‘𝐴) = ∩ (rank “ 𝐵))

8-Jul-2026	dfscott2 35492	Alternate definition of a Scott's trick set. (Contributed by BTernaryTau, 8-Jul-2026.)
⊢ Scott 𝐴 = {𝑥 ∈ 𝐴 ∣ (rank‘𝑥) = ∩ (rank “ 𝐴)}

7-Jul-2026	hashomiso 45717	The ♯ function yields an order isomorphism between ω and ℕ₀. (Contributed by Eric Schmidt, 7-Jul-2026.)
⊢ (♯ ↾ ω) Isom E , < (ω, ℕ₀)

7-Jul-2026	hashomf1o 45716	The ♯ function yields a bijection from ω to ℕ₀. (Contributed by Eric Schmidt, 7-Jul-2026.)
⊢ (♯ ↾ ω):ω–1-1-onto→ℕ₀

7-Jul-2026	hashnnltb 45715	The ♯ function on ω preserves the ordering. (Contributed by Eric Schmidt, 7-Jul-2026.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ (♯‘𝐴) < (♯‘𝐵)))

7-Jul-2026	hashnnlt 45714	The ♯ function on ω preserves the ordering. (Contributed by Eric Schmidt, 7-Jul-2026.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → (♯‘𝐵) < (♯‘𝐴))

7-Jul-2026	hashnnm 45713	The ♯ function on ω preserves multiplication. (Contributed by Eric Schmidt, 7-Jul-2026.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (♯‘(𝐴 ·_o 𝐵)) = ((♯‘𝐴) · (♯‘𝐵)))

7-Jul-2026	hashnnsuc 45712	The ♯ function on ω turns successor into adding 1. (Contributed by Eric Schmidt, 7-Jul-2026.)
⊢ (𝐴 ∈ ω → (♯‘suc 𝐴) = ((♯‘𝐴) + 1))

7-Jul-2026	hashnna 45711	The ♯ function on ω preserves addition. (Contributed by Eric Schmidt, 7-Jul-2026.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (♯‘(𝐴 +_o 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))

7-Jul-2026	kardexen 35557	One set is equinumerous to another iff an element in its kard cardinality is equinumerous to an element in the second set's kard cardinality. See kardeng 35551 for a version with equality of cardinals. (Contributed by BTernaryTau, 7-Jul-2026.)
⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑥 ∈ (kard‘𝐴)∃𝑦 ∈ (kard‘𝐵)𝑥 ≈ 𝑦)

6-Jul-2026	kardsdom 35556	One set strictly dominates another iff an element in its kard cardinality strictly dominates an element in the second set's kard cardinality. (Contributed by BTernaryTau, 6-Jul-2026.)
⊢ (𝐴 ≺ 𝐵 ↔ ∃𝑥 ∈ (kard‘𝐴)∃𝑦 ∈ (kard‘𝐵)𝑥 ≺ 𝑦)

5-Jul-2026	prlngeu 29186	Given a line 𝐴 and a point 𝑋 not on 𝐴, a unique line parallel to 𝐴 can be drawn through 𝑋. Theorem 12.13 of [Schwabhauser] p. 124. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ ∥ = (parlnG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝑋 ∈ (𝑃 ∖ 𝐴)) & ⊢ (𝜑 → 𝐺 ∈ TarskiG_E) ⇒ ⊢ (𝜑 → ∃!𝑏 ∈ ran 𝐿(𝐴 ∥ 𝑏 ∧ 𝑋 ∈ 𝑏))

5-Jul-2026	prlngmo 29185	Playfair's axiom. Given a line 𝐴 and a point 𝑋 not on 𝐴, at most one line parallel to 𝐴 can be drawn through 𝑋. Theorem 12.11 of [Schwabhauser] p. 123. Note that this is the first instance of a theorem where the geometry is required to be Euclidean, as expressed by 𝐺 ∈ TarskiG_E. Theorem A10 of [Schwabhauser] p. 24 is used, in the form of axtgeucl 28722, in the proof of prlngmolem1 29183. See prlngex 29182 for the corresponding existence theorem. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ ∥ = (parlnG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝑋 ∈ (𝑃 ∖ 𝐴)) & ⊢ (𝜑 → 𝐺 ∈ TarskiG_E) ⇒ ⊢ (𝜑 → ∃*𝑏 ∈ ran 𝐿(𝐴 ∥ 𝑏 ∧ 𝑋 ∈ 𝑏))

5-Jul-2026	prlngmolem2 29184	Lemma for prlngmo 29185. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ ∥ = (parlnG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝑋 ∈ (𝑃 ∖ 𝐴)) & ⊢ (𝜑 → 𝐺 ∈ TarskiG_E) & ⊢ 𝑂 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑃 ∖ 𝑏) ∧ 𝑦 ∈ (𝑃 ∖ 𝑏)) ∧ ∃𝑟 ∈ 𝑏 𝑟 ∈ (𝑥(Itv‘𝐺)𝑦))} & ⊢ 𝑄 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑃 ∖ 𝐴) ∧ 𝑦 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑠 ∈ 𝐴 𝑠 ∈ (𝑥(Itv‘𝐺)𝑦))} ⇒ ⊢ (𝜑 → ∃*𝑏 ∈ ran 𝐿(𝐴 ∥ 𝑏 ∧ 𝑋 ∈ 𝑏))

5-Jul-2026	prlngmolem1 29183	Lemma for prlngmo 29185: Contradiction: Assuming two different parallels 𝐵 and 𝐶 having a common point 𝑋 exist to a line 𝐴, the geometry cannot be Euclidean (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ ∥ = (parlnG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝑋 ∈ (𝑃 ∖ 𝐴)) & ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐵) ∧ 𝑏 ∈ (𝑃 ∖ 𝐵)) ∧ ∃𝑦 ∈ 𝐵 𝑦 ∈ (𝑎𝐼𝑏))} & ⊢ 𝑄 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐴) ∧ 𝑏 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑤 ∈ 𝐴 𝑤 ∈ (𝑎𝐼𝑏))} & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐵 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐶 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐴 ∥ 𝐵) & ⊢ (𝜑 → 𝐴 ∥ 𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ∈ 𝐶) & ⊢ (𝜑 → 𝑇 ∈ 𝐴) & ⊢ (𝜑 → 𝑊 ∈ (𝐶 ∖ 𝐵)) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) & ⊢ (𝜑 → 𝑊𝑂𝑇) ⇒ ⊢ (𝜑 → ¬ 𝐺 ∈ TarskiG_E)

5-Jul-2026	prlngex 29182	There exists at least one parallel line 𝑏 to a given line 𝐴 through a given point 𝑋. Theorem 12.10 of [Schwabhauser] p. 122. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ ∥ = (parlnG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) ⇒ ⊢ (𝜑 → ∃𝑏 ∈ ran 𝐿(𝐴 ∥ 𝑏 ∧ 𝑋 ∈ 𝑏))

5-Jul-2026	perpprlng 29181	If two lines 𝐴 and 𝐵 have a common perpendicular 𝐶 and lie in the same plane 𝐻, then they are parallel. Theorem 12.9 of [Schwabhauser] p. 122. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ ∥ = (parlnG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐻 ∈ ran 𝐸) & ⊢ (𝜑 → 𝐴 ⊆ 𝐻) & ⊢ (𝜑 → 𝐵 ⊆ 𝐻) & ⊢ (𝜑 → 𝐶 ⊆ 𝐻) & ⊢ (𝜑 → 𝐴(⟂G‘𝐺)𝐶) & ⊢ (𝜑 → 𝐵(⟂G‘𝐺)𝐶) ⇒ ⊢ (𝜑 → 𝐴 ∥ 𝐵)

5-Jul-2026	prlngpln3 29180	Two parallel lines are on a common plane. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ ∥ = (parlnG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∥ 𝐵) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝐵 ⊆ (𝐴𝐸𝑋))

5-Jul-2026	prlnghpg 29177	If two lines 𝐴 and 𝐵 are parallel, then any two points 𝑋 and 𝑌 of 𝐵 lie on the same half-plane limited by 𝐴. Theorem 12.6 of [Schwabhauser] p. 122. . (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ ∥ = (parlnG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∥ 𝐵) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝑋((hpG‘𝐺)‘𝐴)𝑌)

5-Jul-2026	prlngpln 29176	Two parallel lines are on a common plane. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ ∥ = (parlnG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∥ 𝐵) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) ⇒ ⊢ (𝜑 → ∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ))

5-Jul-2026	prlngin0 29175	Two parallel lines do not intersect. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝐿 = (LineG‘𝐺) & ⊢ ∥ = (parlnG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∥ 𝐵) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)

5-Jul-2026	prlngrcl2 29174	Reverse closure for parallelism. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝐿 = (LineG‘𝐺) & ⊢ ∥ = (parlnG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∥ 𝐵) ⇒ ⊢ (𝜑 → 𝐵 ∈ ran 𝐿)

5-Jul-2026	prlngrcl1 29173	Reverse closure for parallelism. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝐿 = (LineG‘𝐺) & ⊢ ∥ = (parlnG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∥ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)

5-Jul-2026	perpeq 29132	Uniqueness of the perpendicular to a line 𝐴 within a plane 𝐻 at a point 𝑋. Theorem 11.20 of [Schwabhauser] p. 99. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐻 ∈ ran 𝐸) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝐴 ⊆ 𝐻) & ⊢ (𝜑 → 𝑌 ∈ 𝐻) & ⊢ (𝜑 → 𝑍 ∈ 𝐻) & ⊢ (𝜑 → (𝑋𝐿𝑌)(⟂G‘𝐺)𝐴) & ⊢ (𝜑 → (𝑋𝐿𝑍)(⟂G‘𝐺)𝐴) ⇒ ⊢ (𝜑 → (𝑋𝐿𝑌) = (𝑋𝐿𝑍))

5-Jul-2026	perpeqlem 29131	Lemma for perpeq 29132. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐻 ∈ ran 𝐸) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝐴 ⊆ 𝐻) & ⊢ (𝜑 → 𝑌 ∈ 𝐻) & ⊢ (𝜑 → 𝑍 ∈ 𝐻) & ⊢ (𝜑 → (𝑋𝐿𝑌)(⟂G‘𝐺)𝐴) & ⊢ (𝜑 → (𝑋𝐿𝑍)(⟂G‘𝐺)𝐴) & ⊢ (𝜑 → 𝑌((hpG‘𝐺)‘𝐴)𝑍) ⇒ ⊢ (𝜑 → (𝑋𝐿𝑌) = (𝑋𝐿𝑍))

5-Jul-2026	ragraghl 29130	Drawing two right angles at a point 𝑋 on the same side of a line (𝑋𝐿𝑌) leads to points 𝑊 and 𝑍 on the same ray from 𝑋. Theorem 11.19 of [Schwabhauser] p. 99. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝑊 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) & ⊢ (𝜑 → ⟨“𝑌𝑋𝑍”⟩ ∈ (∟G‘𝐺)) & ⊢ (𝜑 → ⟨“𝑌𝑋𝑊”⟩ ∈ (∟G‘𝐺)) & ⊢ (𝜑 → 𝑍((hpG‘𝐺)‘(𝑌𝐿𝑋))𝑊) ⇒ ⊢ (𝜑 → 𝑍((hlG‘𝐺)‘𝑋)𝑊)

5-Jul-2026	ragcgra 29127	Right angles are congruent with each other. Theorem 11.16 of [Schwabhauser] p. 98. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → ⟨“𝑋𝑌𝑍”⟩ ∈ (∟G‘𝐺)) & ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺)) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) & ⊢ (𝜑 → 𝑌 ≠ 𝑍) ⇒ ⊢ (𝜑 → ⟨“𝑋𝑌𝑍”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)

5-Jul-2026	nhpmirhp 29061	If a point 𝑍 is on the plane defined by a line 𝐴 and a point 𝑌, but not on the same half-plane as 𝑌, then its mirror point (𝑀‘𝑍) by a point 𝑋 on 𝐴 is on the same half-plane as 𝑌. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ 𝑀 = (𝑆‘𝑋) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ (𝑃 ∖ 𝐴)) & ⊢ (𝜑 → 𝑍 ∈ ((𝐴𝐸𝑌) ∖ 𝐴)) & ⊢ (𝜑 → ¬ 𝑌((hpG‘𝐺)‘𝐴)𝑍) ⇒ ⊢ (𝜑 → 𝑌((hpG‘𝐺)‘𝐴)(𝑀‘𝑍))

5-Jul-2026	hpgssplng 29059	Any point 𝑋 on a half plane defined by a line 𝐴 and another point 𝑌 is on the plane defined by 𝐴 and 𝑌. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ (𝑃 ∖ 𝐴)) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋((hpG‘𝐺)‘𝐴)𝑌) ⇒ ⊢ (𝜑 → 𝑋 ∈ (𝐴𝐸𝑌))

5-Jul-2026	mirplncl 29058	The mirror of a point with regard to another point is in the same plane as the two points. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ 𝑀 = (𝑆‘𝑋) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐻 ∈ ran 𝐸) & ⊢ (𝜑 → 𝑋 ∈ 𝐻) & ⊢ (𝜑 → 𝑌 ∈ 𝐻) ⇒ ⊢ (𝜑 → (𝑀‘𝑌) ∈ 𝐻)

5-Jul-2026	plngmiropp 29057	Given a line 𝐴 and a point 𝑋 not on 𝐴, then a point 𝑌 on the plane defined by 𝐴 and 𝑋 is either opposite to 𝑋, or opposite to the mirror point of 𝑋 by any point 𝑍 of 𝐴. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ 𝑀 = (𝑆‘𝑍) & ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐴) ∧ 𝑏 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑡 ∈ 𝐴 𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝑋 ∈ (𝑃 ∖ 𝐴)) & ⊢ (𝜑 → 𝑌 ∈ ((𝐴𝐸𝑋) ∖ 𝐴)) & ⊢ (𝜑 → 𝑍 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝑋𝑂𝑌 ∨ (𝑀‘𝑋)𝑂𝑌))

5-Jul-2026	oppmir 29017	The mirror point with regard to a point 𝑋 on a line 𝐴 lies on the other side of 𝐴. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝑆 = (pInvG‘𝐺) & ⊢ 𝑀 = (𝑆‘𝑋) & ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐴) ∧ 𝑏 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑡 ∈ 𝐴 𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ (𝑃 ∖ 𝐴)) & ⊢ 𝐿 = (LineG‘𝐺) ⇒ ⊢ (𝜑 → 𝑌𝑂(𝑀‘𝑌))

5-Jul-2026	perpin 28986	If two lines 𝐴 and 𝐵 are perpendicular, then they intersect. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴(⟂G‘𝐺)𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∩ 𝐵) ≠ ∅)

5-Jul-2026	ecase33d 1504	Deduction for elimination by cases. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ (𝜑 → ¬ 𝜓) & ⊢ (𝜑 → ¬ 𝜒) & ⊢ (𝜑 → (𝜓 ∨ 𝜒 ∨ 𝜃)) ⇒ ⊢ (𝜑 → 𝜃)

5-Jul-2026	3anasss 1380	Associative law for conjunction applied to antecedent (eliminates syllogism). Converse of 3anassrs 1381. (Contributed by Thierry Arnoux, 5-Jul-2026.)
⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏) ⇒ ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) → 𝜏)

4-Jul-2026	1enumkard 35566	The Fundamental Theorem of Enumeration (see https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/enu.pdf), extended to all sets. The expression ∪ 𝑥 ∈ 𝐴({𝑥} × 𝐵) can be thought of as expressing an indexed disjoint union ⊔ 𝑥 ∈ 𝐴𝐵 where each 𝐵 has its elements tagged with the set 𝑥 that generated it. See the comment directly before undjudom 10152 for context on disjoint union as a representation of cardinal addition. This theorem is not limited to numerable sets, but it also does not depend on AC. See 1enumen 35466 for a version that uses equinumerosity , 1enumcard 35467 for a version that uses the card function, and 1enum 35590 for a version that uses an explicit sum of complex number 1s. (Contributed by BTernaryTau, 4-Jul-2026.)
⊢ (𝐴 ∈ V → (kard‘𝐴) = (kard‘∪ 𝑥 ∈ 𝐴 ({𝑥} × 1_o)))

4-Jul-2026	rankkardu 35565	An upper bound on the rank of a kard cardinal. (Contributed by BTernaryTau, 4-Jul-2026.)
⊢ (rank‘(kard‘𝐴)) ⊆ suc (rank‘𝐴)

4-Jul-2026	karddom 35555	One set dominates another iff an element in its kard cardinality dominates an element in the second set's kard cardinality. (Contributed by BTernaryTau, 4-Jul-2026.)
⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑥 ∈ (kard‘𝐴)∃𝑦 ∈ (kard‘𝐵)𝑥 ≼ 𝑦)

4-Jul-2026	kardsn 35554	A singleton has cardinality one. (Contributed by BTernaryTau, 4-Jul-2026.)
⊢ (𝐴 ∈ 𝑉 → (kard‘{𝐴}) = (kard‘1_o))

4-Jul-2026	kardenir 35552	If two sets are equinumerous, then their kard cardinal numbers are equal. (Contributed by BTernaryTau, 4-Jul-2026.)
⊢ (𝐴 ≈ 𝐵 → (kard‘𝐴) = (kard‘𝐵))

4-Jul-2026	rankscottu 35504	An upper bound on the rank of a Scott's trick set. (Contributed by BTernaryTau, 4-Jul-2026.)
⊢ (𝐴 ∈ 𝐵 → (rank‘Scott 𝐵) ⊆ suc (rank‘𝐴))

3-Jul-2026	kardfi 35564	The kard cardinal number of a finite set is finite. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ (𝐴 ∈ Fin → (kard‘𝐴) ∈ Fin)

3-Jul-2026	kardnnfi 35563	The kard cardinal number of a finite ordinal is finite. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ (𝐴 ∈ ω → (kard‘𝐴) ∈ Fin)

3-Jul-2026	kardcard 35562	Two sets have equal kard cardinalities iff they have equal card cardinalities. This theorem depends on the Axiom of Choice. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((kard‘𝐴) = (kard‘𝐵) ↔ (card‘𝐴) = (card‘𝐵)))

3-Jul-2026	ackardcard 35561	The Axiom of Choice implies that two sets have equal kard cardinalities iff they have equal card cardinalities. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ (CHOICE → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((kard‘𝐴) = (kard‘𝐵) ↔ (card‘𝐴) = (card‘𝐵))))

3-Jul-2026	kardcard2 35560	Two numerable sets have equal kard cardinalities iff they have equal card cardinalities. This theorem does not depend on the Axiom of Choice. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((kard‘𝐴) = (kard‘𝐵) ↔ (card‘𝐴) = (card‘𝐵)))

3-Jul-2026	kardcard2b 35559	If two sets have equal kard cardinalities, then they have equal card cardinalities. This theorem does not depend on the Axiom of Choice. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ ((kard‘𝐴) = (kard‘𝐵) → (card‘𝐴) = (card‘𝐵))

3-Jul-2026	kardcard2a 35558	If two sets have equal nonzero card cardinalities, then they have equal kard cardinalities. This theorem does not depend on the Axiom of Choice. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ (((card‘𝐴) = (card‘𝐵) ∧ (card‘𝐴) ≠ ∅) → (kard‘𝐴) = (kard‘𝐵))

3-Jul-2026	kard0b 35553	The empty set is the only set with cardinality zero. This is the kard version of cardeq0 10537. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ ((kard‘𝐴) = (kard‘∅) ↔ 𝐴 = ∅)

3-Jul-2026	kardeng 35551	Two sets are equinumerous iff their kard cardinal numbers are equal. Unlike carden 10536, this theorem does not depend on the Axiom of Choice, but it does depend on the Axiom of Regularity and the Axiom of Infinity. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ (𝐴 ∈ 𝑉 → ((kard‘𝐴) = (kard‘𝐵) ↔ 𝐴 ≈ 𝐵))

3-Jul-2026	kardeq0 35550	Applying kard to a class yields the empty set iff the class is a proper class. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ ((kard‘𝐴) = ∅ ↔ ¬ 𝐴 ∈ V)

3-Jul-2026	elkarden 35549	Any member of the kard cardinal number of a set is equinumerous to the set. Contrast with cardne 9952 for card cardinals. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ (𝐴 ∈ (kard‘𝐵) → 𝐴 ≈ 𝐵)

3-Jul-2026	kard0 35548	The kard cardinality of the empty set is the singleton of the empty set. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ (kard‘∅) = {∅}

3-Jul-2026	kardval2 35547	The value of the kard function. This theorem depends on the Axiom of Regularity and the Axiom of Infinity, but it does not depend on the Axiom of Choice. See also kardval 35546. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ (kard‘𝐴) = {𝑥 ∣ (𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))}

3-Jul-2026	kardval 35546	The value of the kard function. This theorem depends on the Axiom of Regularity and the Axiom of Infinity, but it does not depend on the Axiom of Choice. See also kardval2 35547. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ (kard‘𝐴) = Scott {𝑥 ∣ 𝑥 ≈ 𝐴}

3-Jul-2026	kardfn 35545	The kard class is a function on the universe. This theorem depends on the Axiom of Regularity and the Axiom of Infinity, but it does not depend on the Axiom of Choice. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ kard Fn V

3-Jul-2026	acnum 35506	The Axiom of Choice implies that any set is numerable. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ (CHOICE → (𝐴 ∈ 𝑉 → 𝐴 ∈ dom card))

3-Jul-2026	scottssr1 35505	Relationship between a Scott's trick set and the cumulative hierarchy. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ (𝐴 ∈ 𝐵 → Scott 𝐵 ⊆ (𝑅₁‘suc (rank‘𝐴)))

3-Jul-2026	scott0b 35502	Applying Scott's trick yields the empty set iff it was applied to the empty set. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ (𝐴 = ∅ ↔ Scott 𝐴 = ∅)

3-Jul-2026	scottsn 35501	Applying Scott's trick to a singleton leaves it unchanged. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ Scott {𝐴} = {𝐴}

3-Jul-2026	scott0i 35500	Applying Scott's trick to the empty set leaves it unchanged. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ Scott ∅ = ∅

3-Jul-2026	elscottrankss 35497	Relationship between the ranks of an element in a Scott's trick set and an element in the input set. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ ((𝐴 ∈ Scott 𝐵 ∧ 𝐶 ∈ 𝐵) → (rank‘𝐴) ⊆ (rank‘𝐶))

3-Jul-2026	elscott 35491	Membership in a Scott's trick set. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ (𝐴 ∈ Scott 𝐵 ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (rank‘𝐴) ⊆ (rank‘𝑥)))

3-Jul-2026	scotteqi 35490	Equality theorem for the Scott operation. Inference form of scotteq 9865. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ 𝐴 = 𝐵 ⇒ ⊢ Scott 𝐴 = Scott 𝐵

3-Jul-2026	onrankid 35475	The rank of an ordinal number is itself. (Contributed by BTernaryTau, 3-Jul-2026.)
⊢ (𝐴 ∈ On ↔ (rank‘𝐴) = 𝐴)

2-Jul-2026	df-kard 35544	Define the alternative cardinal number function. Under this definition, the cardinal number of a set is the set of all sets equinumerous to it and having the least possible rank. Definition of [Enderton] p. 222. See kardval 35546 for its value. The principal theorem relating this type of cardinality to equinumerosity is kardeng 35551. Our notation is from Enderton and differentiates this function from the standard cardinal size function defined in df-card 9926. (Contributed by BTernaryTau, 2-Jul-2026.)
⊢ kard = (𝑥 ∈ V ↦ Scott {𝑦 ∣ 𝑦 ≈ 𝑥})

30-Jun-2026	isfieldidl2 21368	Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 6-Jan-2011.) (Revised by AV, 30-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0_g‘𝑅) & ⊢ 𝐼 = (LIdeal‘𝑅) ⇒ ⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ CRing ∧ 𝐵 ≠ { 0 } ∧ 𝐼 = {{ 0 }, 𝐵}))

30-Jun-2026	isfieldidl 21367	Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by AV, 30-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0_g‘𝑅) & ⊢ 𝐼 = (LIdeal‘𝑅) & ⊢ 1 = (1_r‘𝑅) ⇒ ⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ CRing ∧ 0 ≠ 1 ∧ 𝐼 = {{ 0 }, 𝐵}))

30-Jun-2026	rspprop 21351	Properties of a class to be the ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by AV, 30-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐼 = (LIdeal‘𝑅) & ⊢ 𝐾 = (RSpan‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → ((𝐾‘𝑆) = 𝑋 ↔ (𝑋 ∈ 𝐼 ∧ 𝑆 ⊆ 𝑋 ∧ ∀𝑖 ∈ 𝐼 (𝑆 ⊆ 𝑖 → 𝑋 ⊆ 𝑖))))

30-Jun-2026	rspvalint 21350	The ideal generated by a subset of a ring as intersection of ideals including the subset. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by AV, 30-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐼 = (LIdeal‘𝑅) & ⊢ 𝐾 = (RSpan‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → (𝐾‘𝑆) = ∩ {𝑡 ∈ 𝐼 ∣ 𝑆 ⊆ 𝑡})

28-Jun-2026	inlidl 33710	The intersection of two ideals is an ideal. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by AV, 28-Jun-2026.)
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅)) → (𝐼 ∩ 𝐽) ∈ (LIdeal‘𝑅))

28-Jun-2026	unichnlidl 21343	The union of a nonempty chain of ideals is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by AV, 28-Jun-2026.)
⊢ ((𝑅 ∈ Ring ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅) ∧ ∀𝑖 ∈ 𝐶 ∀𝑗 ∈ 𝐶 (𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖))) → ∪ 𝐶 ∈ (LIdeal‘𝑅))

28-Jun-2026	lidlunin0 21342	The union of a nonempty subset of ideals in a ring is nonempty. (Contributed by AV, 28-Jun-2026.)
⊢ ((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) → ∪ 𝐶 ≠ ∅)

27-Jun-2026	isidom2 49092	The predicate "is an integral domain": An integral domain is a commutative prime ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by AV, 27-Jun-2026.)
⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ PrmRing ∧ 𝑅 ∈ CRing))

27-Jun-2026	dfidom2 49091	Alternate definition of the class of integral domains. An integral domain is a commutative prime ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by AV, 27-Jun-2026.)
⊢ IDomn = (PrmRing ∩ CRing)

27-Jun-2026	crngprmringdom 49090	A commutative ring is a prime ring if and only if it is a domain. (Contributed by AV, 27-Jun-2026.)
⊢ (𝑅 ∈ CRing → (𝑅 ∈ PrmRing ↔ 𝑅 ∈ Domn))

27-Jun-2026	crngprmringidom 49089	A commutative ring is a prime ring if and only if it is an integral domain. (Contributed by AV, 27-Jun-2026.)
⊢ (𝑅 ∈ CRing → (𝑅 ∈ PrmRing ↔ 𝑅 ∈ IDomn))

27-Jun-2026	df2idl2crng 21402	The predicate "is an ideal of the commutative ring 𝑅". (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by AV, 27-Jun-2026.)
⊢ 𝑈 = (2Ideal‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (._r‘𝑅) ⇒ ⊢ (𝑅 ∈ CRing → (𝐼 ∈ 𝑈 ↔ (𝐼 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 (𝑥 · 𝑦) ∈ 𝐼)))

27-Jun-2026	lidlbasel 21318	An element of an ideal is an element of the ring. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by AV, 27-Jun-2026.)
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐼 = (LIdeal‘𝑊) ⇒ ⊢ ((𝑈 ∈ 𝐼 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝐵)

27-Jun-2026	0ring01eqbi2 20617	In a ring, 0 = 1 iff the ring contains only 0. (Contributed by Jeff Madsen, 6-Jan-2011.) (Revised by AV, 27-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0_g‘𝑅) & ⊢ 1 = (1_r‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (𝐵 = { 0 } ↔ 1 = 0 ))

26-Jun-2026	prmrngring 49086	A prime ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by AV, 18-Jun-2026.) (Proof shortened by AV, 26-Jun-2026.)
⊢ (𝑅 ∈ PrmRing → 𝑅 ∈ Ring)

26-Jun-2026	prmringnzring 49085	A prime ring is a nonzero ring. (Contributed by AV, 26-Jun-2026.)
⊢ (𝑅 ∈ PrmRing → 𝑅 ∈ NzRing)

26-Jun-2026	1enum 35590	The Fundamental Theorem of Enumeration. According to Doron Zeilberger (in https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/enu.pdf), this theorem was independently discovered by several anonymous cave-dwellers. Zeilberger also states that "While this formula is still useful after all these years, enumerating specific finite sets is no longer considered mathematics. A genuine mathematical fact has to incorporate infinitely many facts". Fortunately, theorems in Metamath are actually theorem schemes that correspond to an infinite number of object-language theorems, so this concern does not apply to us. See 1enumen 35466 for a version that uses equinumerosity , 1enumcard 35467 for a version that uses the card function, and 1enumkard 35566 for a version that uses the kard function. (Contributed by BTernaryTau, 26-Jun-2026.)
⊢ (𝐴 ∈ Fin → (♯‘𝐴) = Σ𝑎 ∈ 𝐴 1)

26-Jun-2026	1enumcard 35467	The Fundamental Theorem of Enumeration (see https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/enu.pdf), extended to all sets. The expression ∪ 𝑥 ∈ 𝐴({𝑥} × 𝐵) can be thought of as expressing an indexed disjoint union ⊔ 𝑥 ∈ 𝐴𝐵 where each 𝐵 has its elements tagged with the set 𝑥 that generated it. See the comment directly before undjudom 10152 for context on disjoint union as a representation of cardinal addition. This theorem does not depend on AC, but it is only meaningful for numerable sets. See 1enumen 35466 and 1enumkard 35566 for versions that are meaningful for non-numerable sets, and see 1enum 35590 for a version that uses an explicit sum of complex number 1s. (Contributed by BTernaryTau, 26-Jun-2026.)
⊢ (𝐴 ∈ V → (card‘𝐴) = (card‘∪ 𝑥 ∈ 𝐴 ({𝑥} × 1_o)))

26-Jun-2026	1enumen 35466	The Fundamental Theorem of Enumeration (see https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/enu.pdf), extended to all sets. The expression ∪ 𝑥 ∈ 𝐴({𝑥} × 𝐵) can be thought of as expressing an indexed disjoint union ⊔ 𝑥 ∈ 𝐴𝐵 where each 𝐵 has its elements tagged with the set 𝑥 that generated it. See the comment directly before undjudom 10152 for context on disjoint union as a representation of cardinal addition. This theorem is not limited to numerable sets, but it also does not depend on AC. See 1enumcard 35467 for a version that uses the card function, 1enumkard 35566 for a version that uses the kard function , and 1enum 35590 for a version that uses an explicit sum of complex number 1s. (Contributed by BTernaryTau, 26-Jun-2026.)
⊢ (𝐴 ∈ V → 𝐴 ≈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 1_o))

26-Jun-2026	4anpull2 1382	An equivalence of two four-terms conjunctions with the terms regrouped (here, the second sub-conjunct of the first term is pulled separately). (Contributed by Zhi Wang, 4-Sep-2024.) (Proof shortened by Garrett Katz, 26-Jun-2026.)
⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒 ∧ 𝜃) ∧ 𝜓))

25-Jun-2026	ifpdfbi 1086	Define the biconditional as conditional logic operator. (Contributed by RP, 20-Apr-2020.) (Proof shortened by Wolf Lammen, 30-Apr-2024.) (Proof shortened by Garrett Katz, 25-Jun-2026.)
⊢ ((𝜑 ↔ 𝜓) ↔ if-(𝜑, 𝜓, ¬ 𝜓))

24-Jun-2026	inv2 35448	The intersection of the universal class with a class is itself. A commuted form of inv1 4356. (Contributed by BTernaryTau, 24-Jun-2026.)
⊢ (V ∩ 𝐴) = 𝐴

23-Jun-2026	rankfn 35487	The rank function is a function on the universe. (Contributed by BTernaryTau, 23-Jun-2026.)
⊢ rank Fn V

23-Jun-2026	rankfo 35486	The rank function maps the universe onto the ordinals. (Contributed by BTernaryTau, 23-Jun-2026.)
⊢ rank:V–onto→On

23-Jun-2026	fnfvintima 35457	Condition for a function value to equal the intersection of an image that contains it. (Contributed by BTernaryTau, 23-Jun-2026.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ∈ 𝐵) → ((𝐹‘𝐶) = ∩ (𝐹 “ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝐶) ⊆ (𝐹‘𝑥)))

21-Jun-2026	sgnrn 15137	The range of the signum function. (Contributed by AV, 16-Jun-2026.) (Proof shortened by TA, 21-Jun-2026.)
⊢ ran sgn = {-1, 0, 1}

20-Jun-2026	reldmun 6035	Split a relation into two parts based on its domain. (Contributed by Thierry Arnoux, 9-Oct-2023.) Remove requirement that 𝐴 and 𝐵 are disjoint. (Revised by Eric Schmidt, 20-Jun-2026.)
⊢ ((Rel 𝑅 ∧ dom 𝑅 = (𝐴 ∪ 𝐵)) → 𝑅 = ((𝑅 ↾ 𝐴) ∪ (𝑅 ↾ 𝐵)))

19-Jun-2026	ringen1zr0 20862	The only unital ring with one element is the zero ring (at least if its operations are internal binary operations). This holds already for nonunital rings, see rngen1zr0 20263, and semirings, see srgen1zr0 20299. (Contributed by FL, 15-Feb-2010.) (Revised by AV, 25-Jan-2020.) (Proof shortened by AV, 19-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+_g‘𝑅) & ⊢ ∗ = (._r‘𝑅) & ⊢ 𝑍 = (0_g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) → (𝐵 ≈ 1_o ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))

19-Jun-2026	srg1zr 20298	The only semiring with a base set consisting of one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.) (Proof shortened by AV, 19-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+_g‘𝑅) & ⊢ ∗ = (._r‘𝑅) ⇒ ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))

18-Jun-2026	drngprmrng 49088	A division ring is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.) (Revised by AV, 18-Jun-2026.)
⊢ (𝑅 ∈ DivRing → 𝑅 ∈ PrmRing)

18-Jun-2026	smprngprmrng 49087	A simple ring (a nonzero ring whose only ideals are 0 and 𝑅) is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.) (Revised by AV, 18-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0_g‘𝑅) & ⊢ 𝑈 = (LIdeal‘𝑅) ⇒ ⊢ ((𝑅 ∈ NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) → 𝑅 ∈ PrmRing)

18-Jun-2026	isprmrng 49084	The predicate "is a prime ring". (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by AV, 18-Jun-2026.)
⊢ 0 = (0_g‘𝑅) & ⊢ 𝑃 = (PrmIdeal‘𝑅) ⇒ ⊢ (𝑅 ∈ PrmRing ↔ (𝑅 ∈ Ring ∧ { 0 } ∈ 𝑃))

18-Jun-2026	df-prmring 49083	Define the class of prime rings. A ring is prime if the zero ideal is a prime ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by AV, 18-Jun-2026.)
⊢ PrmRing = {𝑟 ∈ Ring ∣ {(0_g‘𝑟)} ∈ (PrmIdeal‘𝑟)}

18-Jun-2026	prlngsym 29172	Parallelism is symmetric. Theorem 12.5 of [Schwabhauser] p. 122. (Contributed by Thierry Arnoux, 18-Jun-2026.)
⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ ∥ = (parlnG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∥ 𝐵) ⇒ ⊢ (𝜑 → 𝐵 ∥ 𝐴)

18-Jun-2026	prlngref 29171	Parallelism is reflexive. Theorem 12.4 of [Schwabhauser] p. 122. (Contributed by Thierry Arnoux, 18-Jun-2026.)
⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ ∥ = (parlnG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) ⇒ ⊢ (𝜑 → 𝐴 ∥ 𝐴)

18-Jun-2026	prlngd 29170	Deduce parallelism between two lines 𝐴 and 𝐵. (Contributed by Thierry Arnoux, 18-Jun-2026.)
⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ ∥ = (parlnG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐵 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐻 ∈ ran 𝐸) & ⊢ (𝜑 → 𝐴 ⊆ 𝐻) & ⊢ (𝜑 → 𝐵 ⊆ 𝐻) & ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) ⇒ ⊢ (𝜑 → 𝐴 ∥ 𝐵)

18-Jun-2026	brprlng 29169	Property of two lines 𝐴 and 𝐵 to be parallel. (Contributed by Thierry Arnoux, 18-Jun-2026.)
⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ ∥ = (parlnG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐴 ∥ 𝐵 ↔ ((𝐴 ∈ ran 𝐿 ∧ 𝐵 ∈ ran 𝐿) ∧ (𝐴 = 𝐵 ∨ (∃ℎ ∈ ran 𝐸(𝐴 ⊆ ℎ ∧ 𝐵 ⊆ ℎ) ∧ (𝐴 ∩ 𝐵) = ∅)))))

18-Jun-2026	rngen1zr0 20263	The only ring with one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 15-Feb-2010.) (Revised by AV, 18-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+_g‘𝑅) & ⊢ ∗ = (._r‘𝑅) & ⊢ 0 = (0_g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Rng ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) → (𝐵 ≈ 1_o ↔ ( + = {⟨⟨ 0 , 0 ⟩, 0 ⟩} ∧ ∗ = {⟨⟨ 0 , 0 ⟩, 0 ⟩})))

18-Jun-2026	rngen1zr 20262	The only ring with one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 14-Feb-2010.) (Revised by AV, 18-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+_g‘𝑅) & ⊢ ∗ = (._r‘𝑅) ⇒ ⊢ (((𝑅 ∈ Rng ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 ≈ 1_o ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))

18-Jun-2026	rng1zr 20261	The only ring with a base set consisting of one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 13-Feb-2010.) (Revised by AV, 18-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+_g‘𝑅) & ⊢ ∗ = (._r‘𝑅) ⇒ ⊢ (((𝑅 ∈ Rng ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))

18-Jun-2026	rng1zrlem 20260	Lemma for rng1zr 20261 and srg1zr 20298. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 18-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+_g‘𝑅) & ⊢ ∗ = (._r‘𝑅) ⇒ ⊢ (((𝑅 ∈ Mgm ∧ (mulGrp‘𝑅) ∈ Mgm) ∧ ( + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))

17-Jun-2026	df-prlng 29168	Define the parallel relation for lines. Definition 12.2 of [Schwabhauser] p. 121. Note that the textbook first defines a "strict" parallelism where equal lines are not considered parallel in the strict sense: here we jump directly to the more common definition which allows equality. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ parlnG = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ (𝑎 = 𝑏 ∨ (∃ℎ ∈ ran (hlG‘𝑔)(𝑎 ⊆ ℎ ∧ 𝑏 ⊆ ℎ) ∧ (𝑎 ∩ 𝑏) = ∅)))})

17-Jun-2026	plng3p 29060	If 𝐻 is a plane containing a line 𝐴 and a point 𝑅 not on 𝐴, then 𝐻 is the plane defined by 𝐴 and 𝑅. Theorem 9.26 of [Schwabhauser] p. 76. See tglinethru 28890 for the 2-point line equivalent. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐻 ∈ ran 𝐸) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝑅 ∈ (𝐻 ∖ 𝐴)) & ⊢ (𝜑 → 𝐴 ⊆ 𝐻) ⇒ ⊢ (𝜑 → 𝐻 = (𝐴𝐸𝑅))

17-Jun-2026	lnssplng1 29056	A line defined by two points 𝑋 and 𝑌, both on a plane 𝐻, is entirely contained in 𝐻. First part of Theorem 9.25 of [Schwabhauser] p. 75. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐻 ∈ ran 𝐸) & ⊢ (𝜑 → 𝑋 ∈ 𝐻) & ⊢ (𝜑 → 𝑌 ∈ 𝐻) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) ⇒ ⊢ (𝜑 → (𝑋𝐿𝑌) ⊆ 𝐻)

17-Jun-2026	lnssplng 29055	A line defined by two points 𝑋 and 𝑌, both on a plane 𝐻, is entirely contained in 𝐻. Theorem 9.25 of [Schwabhauser] p. 75. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐻 ∈ ran 𝐸) & ⊢ (𝜑 → 𝑋 ∈ 𝐻) & ⊢ (𝜑 → 𝑌 ∈ 𝐻) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) ⇒ ⊢ (𝜑 → ((𝑋𝐿𝑌) ⊆ 𝐻 ∧ ∃𝑠 ∈ (𝑃 ∖ (𝑋𝐿𝑌))𝐻 = ((𝑋𝐿𝑌)𝐸𝑠)))

17-Jun-2026	lnssplnglem 29054	Lemma for lnssplng 29055. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ (𝐴𝐸𝑅)) & ⊢ (𝜑 → 𝑌 ∈ (𝐴𝐸𝑅)) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝑅 ∈ (𝑃 ∖ 𝐴)) & ⊢ (𝜑 → 𝐴 ≠ (𝑋𝐿𝑌)) & ⊢ (𝜑 → ¬ 𝑌 ∈ 𝐴) ⇒ ⊢ (𝜑 → ((𝑋𝐿𝑌) ⊆ (𝐴𝐸𝑅) ∧ ∃𝑠 ∈ (𝑃 ∖ (𝑋𝐿𝑌))(𝐴𝐸𝑅) = ((𝑋𝐿𝑌)𝐸𝑠)))

17-Jun-2026	plngrot 29053	The plane defined by a line (𝑋𝐿𝑌) and a point 𝑍 is also defined by the line (𝑍𝐿𝑌) and the point 𝑋. See first part of Theorem 9.24 of [Schwabhauser] p. 74. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ (𝑃 ∖ (𝑍𝐿𝑌))) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ (𝑃 ∖ (𝑋𝐿𝑌))) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) ⇒ ⊢ (𝜑 → ((𝑋𝐿𝑌)𝐸𝑍) = ((𝑍𝐿𝑌)𝐸𝑋))

17-Jun-2026	plngrotlem3 29052	Lemma for plngrot 29053. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ (𝑃 ∖ (𝑍𝐿𝑌))) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ (𝑃 ∖ (𝑋𝐿𝑌))) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) & ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑌)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑌))) ∧ ∃𝑡 ∈ (𝑋𝐿𝑌)𝑡 ∈ (𝑎𝐼𝑏))} ⇒ ⊢ (𝜑 → ((𝑋𝐿𝑌)𝐸𝑍) ⊆ ((𝑍𝐿𝑌)𝐸𝑋))

17-Jun-2026	plngrotlem2 29051	Lemma for plngrot 29053. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ (𝑃 ∖ (𝑍𝐿𝑌))) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ (𝑃 ∖ (𝑋𝐿𝑌))) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) & ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑌)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑌))) ∧ ∃𝑡 ∈ (𝑋𝐿𝑌)𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ (𝜑 → 𝑊 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ (𝑍𝐼𝑊)) & ⊢ (𝜑 → 𝑌 ≠ 𝑊) ⇒ ⊢ (𝜑 → ((𝑋𝐿𝑌)𝐸𝑍) ⊆ ((𝑍𝐿𝑌)𝐸𝑋))

17-Jun-2026	plngrotlem1 29050	Lemma for plngrot 29053. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ (𝑃 ∖ (𝑍𝐿𝑌))) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ (𝑃 ∖ (𝑋𝐿𝑌))) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) & ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑋𝐿𝑌)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑋𝐿𝑌))) ∧ ∃𝑡 ∈ (𝑋𝐿𝑌)𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ (𝜑 → 𝑊 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ (𝑍𝐼𝑊)) & ⊢ (𝜑 → 𝑌 ≠ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ ((𝑋𝐿𝑌)𝐸𝑍)) & ⊢ (𝜑 → (𝑆 ∈ (𝑋𝐿𝑌) ∨ 𝑆((hpG‘𝐺)‘(𝑋𝐿𝑌))𝑍)) ⇒ ⊢ (𝜑 → 𝑆 ∈ ((𝑍𝐿𝑌)𝐸𝑋))

17-Jun-2026	plngcp 29049	The plane defined by a line 𝐴 and a point 𝑅 can also be defined using a different point 𝑅 on the same plane: changes the point used to define the plane. Theorem 9.21 of [Schwabhauser] p. 74. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝑅 ∈ (𝑃 ∖ 𝐴)) & ⊢ (𝜑 → 𝑆 ∈ ((𝐴𝐸𝑅) ∖ 𝐴)) ⇒ ⊢ (𝜑 → (𝐴𝐸𝑅) = (𝐴𝐸𝑆))

17-Jun-2026	plngcplem 29048	Lemma for plngcp 29049. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝑅 ∈ (𝑃 ∖ 𝐴)) & ⊢ (𝜑 → 𝑆 ∈ ((𝐴𝐸𝑅) ∖ 𝐴)) & ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐴) ∧ 𝑏 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑡 ∈ 𝐴 𝑡 ∈ (𝑎𝐼𝑏))} ⇒ ⊢ (𝜑 → (𝐴𝐸𝑅) = (𝐴𝐸𝑆))

17-Jun-2026	lnincplng 29047	If two lines 𝐴 and 𝐵 intersect, then 𝐵 is in a plane defined by 𝐴 and any point of 𝐵. Lemma 9.22 of [Schwabhauser] p. 74. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐵 ∈ ran 𝐿) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) & ⊢ (𝜑 → (𝐴 ∩ 𝐵) = {𝑌}) ⇒ ⊢ (𝜑 → 𝐵 ⊆ (𝐴𝐸𝑋))

17-Jun-2026	elplnglnid 29046	The line 𝐴 itself is a subset of a plane defined by the line 𝐴 and a point 𝑅. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝑅 ∈ (𝑃 ∖ 𝐴)) ⇒ ⊢ (𝜑 → 𝐴 ⊆ (𝐴𝐸𝑅))

17-Jun-2026	elplngid 29045	The point 𝑅 is itself an element of a plane defined by a line 𝐴 and the point 𝑅. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝑅 ∈ (𝑃 ∖ 𝐴)) ⇒ ⊢ (𝜑 → 𝑅 ∈ (𝐴𝐸𝑅))

17-Jun-2026	plngssp 29044	Planes are sets of points. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝑅 ∈ (𝑃 ∖ 𝐴)) & ⊢ (𝜑 → 𝑋 ∈ (𝐴𝐸𝑅)) ⇒ ⊢ (𝜑 → 𝑋 ∈ 𝑃)

17-Jun-2026	elplng 29043	Elementhood in the plane defined by a line 𝐴 and a point 𝑅. Definition 9.20 of [Schwabhauser] p. 74. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝑅 ∈ (𝑃 ∖ 𝐴)) & ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐴) ∧ 𝑏 ∈ (𝑃 ∖ 𝐴)) ∧ ∃𝑡 ∈ 𝐴 𝑡 ∈ (𝑎𝐼𝑏))} & ⊢ (𝜑 → 𝑋 ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝐴𝐸𝑅) ↔ (𝑋 ∈ 𝐴 ∨ 𝑋((hpG‘𝐺)‘𝐴)𝑅 ∨ 𝑋𝑂𝑅)))

17-Jun-2026	plngrnssp 29042	Planes are sets of points. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐻 ∈ ran 𝐸) & ⊢ (𝜑 → 𝑋 ∈ 𝐻) ⇒ ⊢ (𝜑 → 𝑋 ∈ 𝑃)

17-Jun-2026	isplng 29041	The property of being a plane. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐻 ∈ ran 𝐸) ⇒ ⊢ (𝜑 → ∃𝑎 ∈ ran 𝐿∃𝑟 ∈ (𝑃 ∖ 𝑎)𝐻 = (𝑎𝐸𝑟))

17-Jun-2026	plngval 29040	The plane defined by a line 𝐴 and a point 𝑅 outside of 𝐴. This is defined as the union of 3 parts: the line itself, the open half-plane containing 𝑅, and the points opposite to 𝑅 (see islnopp 29001). (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝑅 ∈ (𝑃 ∖ 𝐴)) ⇒ ⊢ (𝜑 → (𝐴𝐸𝑅) = {𝑥 ∈ 𝑃 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥((hpG‘𝐺)‘𝐴)𝑅 ∨ ∃𝑡 ∈ 𝐴 𝑡 ∈ (𝑥𝐼𝑅))})

17-Jun-2026	tgelrnpln 29039	The property of being a plane, generated by a line and a point. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝑅 ∈ (𝑃 ∖ 𝐴)) ⇒ ⊢ (𝜑 → (𝐴𝐸𝑅) ∈ ran 𝐸)

17-Jun-2026	tgplnfn 29038	The plane generating function as a function. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ 𝐸 = (hlG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐸 Fn ((ran 𝐿 × 𝑃) ∖ ^◡ E ))

17-Jun-2026	df-plng 29037	Define the function building a plane from a line and a point not on that line. Definition 9.20 of [Schwabhauser] p. 74. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ hlG = (𝑔 ∈ V ↦ (𝑎 ∈ ran (LineG‘𝑔), 𝑟 ∈ ((Base‘𝑔) ∖ 𝑎) ↦ {𝑥 ∈ (Base‘𝑔) ∣ (𝑥 ∈ 𝑎 ∨ 𝑥((hpG‘𝑔)‘𝑎)𝑟 ∨ ∃𝑡 ∈ 𝑎 𝑡 ∈ (𝑥(Itv‘𝑔)𝑟))}))

17-Jun-2026	tglnpt4 28909	Find a second point on a line, outside of a second line. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐵 ∈ ran 𝐿) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) ⇒ ⊢ (𝜑 → ∃𝑧 ∈ (𝐴 ∖ 𝐵)𝑧 ≠ 𝑋)

17-Jun-2026	tglnpt3 28908	Find a third point on a line. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) ⇒ ⊢ (𝜑 → ∃𝑧 ∈ 𝐴 (𝑧 ≠ 𝑋 ∧ 𝑧 ≠ 𝑌))

17-Jun-2026	tglineinsn 28898	If two distinct lines intersect, it is at a single point. Theorem 6.21 of [Schwabhauser] p. 46. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐵 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵)) ⇒ ⊢ (𝜑 → (𝐴 ∩ 𝐵) = {𝑋})

17-Jun-2026	tglinesseq 28894	If a line is a subset of another line, they are equal. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐵 ∈ ran 𝐿) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵)

17-Jun-2026	mpoexd 8078	Existence of an operation class abstraction. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V)

17-Jun-2026	xpdifcnvepel 6168	The set of couples in a Cartesian product, where the second is not an element of the first. (Contributed by Thierry Arnoux, 17-Jun-2026.)
⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐵 ∖ 𝑥)) = ((𝐴 × 𝐵) ∖ ^◡ E )

16-Jun-2026	sgnfo 15138	The signum function as onto function. (Contributed by AV, 16-Jun-2026.)
⊢ sgn:ℝ^*–onto→{-1, 0, 1}

16-Jun-2026	sgndm 15135	The domain of the signum function. (Contributed by AV, 16-Jun-2026.)
⊢ dom sgn = ℝ^*

16-Jun-2026	resdmdfsn 6033	Restricting a class to its domain without a set is the same as restricting the class to the universe without this set. (Contributed by AV, 2-Dec-2018.) Remove antecedent. (Revised by Eric Schmidt, 16-Jun-2026.)
⊢ (𝑅 ↾ (V ∖ {𝑋})) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋}))

16-Jun-2026	resindm 6031	When restricting a class, intersecting with the domain of the class has no effect. (Contributed by FL, 6-Oct-2008.) Remove antecedent. (Revised by Eric Schmidt, 16-Jun-2026.)
⊢ (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ 𝐵)

16-Jun-2026	3jaao 1460	Inference conjoining and disjoining the antecedents of three implications. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Garrett Katz, 16-Jun-2026.)
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜃 → (𝜏 → 𝜒)) & ⊢ (𝜂 → (𝜁 → 𝜒)) ⇒ ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜂) → ((𝜓 ∨ 𝜏 ∨ 𝜁) → 𝜒))

16-Jun-2026	3jaoi 1454	Disjunction of three antecedents (inference). (Contributed by NM, 12-Sep-1995.) (Proof shortened by Garrett Katz, 16-Jun-2026.)
⊢ (𝜑 → 𝜓) & ⊢ (𝜒 → 𝜓) & ⊢ (𝜃 → 𝜓) ⇒ ⊢ ((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓)

15-Jun-2026	anbiim 652	Adding biconditional when antecedents are conjuncted. (Contributed by metakunt, 16-Apr-2024.) (Proof shortened by Wolf Lammen, 7-May-2025.) (Proof shortened by Garrett Katz, 15-Jun-2026.)
⊢ (𝜑 → (𝜒 → 𝜃)) & ⊢ (𝜓 → (𝜃 → 𝜒)) ⇒ ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))

15-Jun-2026	imbibi 395	The antecedent of one side of a biconditional can be moved out of the biconditional to become the antecedent of the remaining biconditional. (Contributed by BJ, 1-Jan-2025.) (Proof shortened by Wolf Lammen, 5-Jan-2025.) (Proof shortened by Garrett Katz, 15-Jun-2026.)
⊢ (((𝜑 → 𝜓) ↔ 𝜒) → (𝜑 → (𝜓 ↔ 𝜒)))

13-Jun-2026	ad5ant135 1394	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (Proof shortened by Garrett Katz, 13-Jun-2026.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ (((((𝜑 ∧ 𝜏) ∧ 𝜓) ∧ 𝜂) ∧ 𝜒) → 𝜃)

13-Jun-2026	ad5ant134 1392	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (Proof shortened by Garrett Katz, 13-Jun-2026.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ (((((𝜑 ∧ 𝜏) ∧ 𝜓) ∧ 𝜒) ∧ 𝜂) → 𝜃)

13-Jun-2026	ad5ant125 1390	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (Proof shortened by Garrett Katz, 13-Jun-2026.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝜏) ∧ 𝜂) ∧ 𝜒) → 𝜃)

13-Jun-2026	ad5ant124 1388	Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) (Proof shortened by Garrett Katz, 13-Jun-2026.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) ⇒ ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝜏) ∧ 𝜒) ∧ 𝜂) → 𝜃)

12-Jun-2026	nmulcld 36666	Closure law for natural multiplication. Deduction form. (Contributed by Scott Fenton, 12-Jun-2026.)
⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → 𝐵 ∈ On) ⇒ ⊢ (𝜑 → (𝐴 ·_no 𝐵) ∈ On)

12-Jun-2026	onvfowev 35581	If 𝐹 maps the ordinals onto the universe, then 𝑅 well-orders the universe. This is the ZFC version of (8 → 3) in https://tinyurl.com/hamkins-gblac. Note that in NBG set theory the antecedent would be something like ∀𝑋(𝑋 ≠ ∅ → ∃𝐹𝐹:On–onto→𝑋), but since we cannot quantify over classes, we instead consider only the case 𝑋 = V which is sufficient for this proof. (Contributed by BTernaryTau, 12-Jun-2026.)
⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐻‘𝑥) ∈ (𝐻‘𝑦)} & ⊢ 𝐻 = (𝑧 ∈ V ↦ ∩ (^◡𝐹 “ {𝑧})) ⇒ ⊢ (𝐹:On–onto→V → 𝑅 We V)

11-Jun-2026	vonf1oonfo 35580	If 𝐹 is a bijection from the ordinals to the universe and 𝐴 is non-empty, then 𝐻 maps the ordinals onto 𝐴. This is the ZFC version of (5 → 8) in https://tinyurl.com/hamkins-gblac, though it neglects to specify that 𝐴 must be non-empty. Note that in NBG set theory the antecedent would be something like ∀𝑋(¬ 𝑋 ∈ V → ∃𝐹𝐹:𝑋–1-1-onto→On), but since we cannot quantify over classes, we instead consider only the case 𝑋 = V which is sufficient for this proof. This theorem can also be viewed as (2 → 8). (Contributed by BTernaryTau, 11-Jun-2026.)
⊢ 𝐻 = (𝑥 ∈ On ↦ if((𝐹‘𝑥) ∈ 𝐴, (𝐹‘𝑥), 𝐷)) & ⊢ 𝐷 = (𝐹‘∩ {𝑦 ∈ On ∣ (𝐹‘𝑦) ∈ 𝐴}) ⇒ ⊢ ((𝐹:On–1-1-onto→V ∧ 𝐴 ≠ ∅) → 𝐻:On–onto→𝐴)

11-Jun-2026	vonf1owev 35574	If 𝐹 is a bijection from the universe to the ordinals, then 𝑅 well-orders the universe. This is the ZFC version of (2 → 3) in https://tinyurl.com/hamkins-gblac. (Contributed by BTernaryTau, 6-Dec-2025.) (Proof shortened by BTernaryTau, 11-Jun-2026.)
⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)} ⇒ ⊢ (𝐹:V–1-1-onto→On → 𝑅 We V)

11-Jun-2026	vonf1wev 35573	If 𝐹 maps the universe one-to-one into the ordinals, then 𝑅 well-orders the universe. This is the ZFC version of (6 → 3) which is used in place of (7 → 3) in https://tinyurl.com/hamkins-gblac. Note that in NBG set theory the antecedent would be something like ∀𝑋∃𝐹𝐹:𝑋–1-1→On, but since we cannot quantify over classes, we instead consider only the case 𝑋 = V which is sufficient for this proof. (Contributed by BTernaryTau, 11-Jun-2026.)
⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)} ⇒ ⊢ (𝐹:V–1-1→On → 𝑅 We V)

10-Jun-2026	nmull0 36669	Natural multiplication by zero. (Contributed by Scott Fenton, 10-Jun-2026.)
⊢ (𝐴 ∈ On → (∅ ·_no 𝐴) = ∅)

10-Jun-2026	nmulr0 36668	Natural multiplication by zero. (Contributed by Scott Fenton, 10-Jun-2026.)
⊢ (𝐴 ∈ On → (𝐴 ·_no ∅) = ∅)

10-Jun-2026	nmulcom 36667	Natural multiplication is commutative. (Contributed by Scott Fenton, 10-Jun-2026.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·_no 𝐵) = (𝐵 ·_no 𝐴))

10-Jun-2026	nmulval 36665	Show the value of natural multiplication. (Contributed by Scott Fenton, 10-Jun-2026.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·_no 𝐵) = ∩ {𝑥 ∈ On ∣ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 ((𝑎 ·_no 𝐵) +no (𝐴 ·_no 𝑏)) ∈ (𝑥 +no (𝑎 ·_no 𝑏))})

10-Jun-2026	nmulcl 36664	Closure law for natural multiplication. (Contributed by Scott Fenton, 10-Jun-2026.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·_no 𝐵) ∈ On)

10-Jun-2026	vonf1oonf1 35579	If 𝐹 is a bijection from the universe to the ordinals, then 𝐻 maps 𝐴 one-to-one into the ordinals. This is the ZFC version of (5 → 6) in https://tinyurl.com/hamkins-gblac. Note that in NBG set theory the antecedent would be something like ∀𝑋(¬ 𝑋 ∈ V → ∃𝐹𝐹:𝑋–1-1-onto→On), but since we cannot quantify over classes, we instead consider only the case 𝑋 = V which is sufficient for this proof. This theorem can also be viewed as (2 → 6). (Contributed by BTernaryTau, 10-Jun-2026.)
⊢ 𝐻 = (𝐹 ↾ 𝐴) ⇒ ⊢ (𝐹:V–1-1-onto→On → 𝐻:𝐴–1-1→On)

10-Jun-2026	bdaydm 27923	The birthday function's domain is No . (Contributed by Scott Fenton, 14-Jun-2011.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ dom bday = No

10-Jun-2026	pige0 26605	π is nonnegative. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 0 ≤ π

10-Jun-2026	2picn 26603	(2 · π) is a complex number. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
⊢ (2 · π) ∈ ℂ

10-Jun-2026	2pire 26601	(2 · π) is a real number. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
⊢ (2 · π) ∈ ℝ

10-Jun-2026	10nprm 17174	10 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ ¬ ;10 ∈ ℙ

10-Jun-2026	1lt10 12857	1 is less than 10. (Contributed by NM, 7-Nov-2012.) (Revised by Mario Carneiro, 9-Mar-2015.) (Revised by AV, 8-Sep-2021.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 1 < ;10

10-Jun-2026	2lt10 12856	2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 2 < ;10

10-Jun-2026	3lt10 12855	3 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 3 < ;10

10-Jun-2026	4lt10 12854	4 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 4 < ;10

10-Jun-2026	5lt10 12853	5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 5 < ;10

10-Jun-2026	6lt10 12852	6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 6 < ;10

10-Jun-2026	7lt10 12851	7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 7 < ;10

10-Jun-2026	8lt10 12850	8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 8-Sep-2021.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 8 < ;10

10-Jun-2026	9t11e99 12847	9 times 11 equals 99. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 6-Sep-2021.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ (9 · ;11) = ;99

10-Jun-2026	11nn 12737	11 is a positive integer. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
⊢ ;11 ∈ ℕ

10-Jun-2026	25nn0 12731	25 is a nonnegative integer. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
⊢ ;25 ∈ ℕ₀

10-Jun-2026	16nn0 12730	16 is a nonnegative integer. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
⊢ ;16 ∈ ℕ₀

10-Jun-2026	12nn0 12729	12 is a nonnegative integer. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
⊢ ;12 ∈ ℕ₀

10-Jun-2026	11nn0 12728	11 is a nonnegative integer. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
⊢ ;11 ∈ ℕ₀

10-Jun-2026	2le3 12416	2 is less than or equal to 3. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 2 ≤ 3

10-Jun-2026	2t4e8 12411	2 times 4 equals 8. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
⊢ (2 · 4) = 8

10-Jun-2026	2t3e6 12408	2 times 3 equals 6. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
⊢ (2 · 3) = 6

10-Jun-2026	2m1e1 12366	2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 12396. (Contributed by David A. Wheeler, 4-Jan-2017.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ (2 − 1) = 1

10-Jun-2026	9pos 12358	The number 9 is positive. (Contributed by NM, 27-May-1999.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 0 < 9

10-Jun-2026	8pos 12357	The number 8 is positive. (Contributed by NM, 27-May-1999.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 0 < 8

10-Jun-2026	7pos 12356	The number 7 is positive. (Contributed by NM, 27-May-1999.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 0 < 7

10-Jun-2026	6pos 12355	The number 6 is positive. (Contributed by NM, 27-May-1999.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 0 < 6

10-Jun-2026	5pos 12354	The number 5 is positive. (Contributed by NM, 27-May-1999.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 0 < 5

10-Jun-2026	4pos 12352	The number 4 is positive. (Contributed by NM, 27-May-1999.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 0 < 4

10-Jun-2026	3pos 12350	The number 3 is positive. (Contributed by NM, 27-May-1999.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 0 < 3

10-Jun-2026	2thalfe1 12349	2 times one half equals 1. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
⊢ (2 · (1 / 2)) = 1

10-Jun-2026	2pos 12346	The number 2 is positive. (Contributed by NM, 27-May-1999.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 0 < 2

10-Jun-2026	0le2 12344	The number 0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 0 ≤ 2

10-Jun-2026	1eltp012 12312	1 is an element of {0, 1, 2}. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 1 ∈ {0, 1, 2}

10-Jun-2026	1elpr01 11205	1 is an element of {0, 1}. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 1 ∈ {0, 1}

10-Jun-2026	0elpr01 11202	0 is an element of {0, 1}. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 0 ∈ {0, 1}

10-Jun-2026	cfon 10239	The cofinality of any set is an ordinal (although it only makes sense when 𝐴 is an ordinal). (Contributed by Mario Carneiro, 9-Mar-2013.) Avoid ax-pow 5338 and ax-un 7734. (Revised by Umit Teoman Dogan, 10-Jun-2026.)
⊢ (cf‘𝐴) ∈ On

10-Jun-2026	1n0 8473	Ordinal one is not equal to ordinal zero. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 1_o ≠ ∅

10-Jun-2026	1oelpr 8465	1_o is an element of {∅, 1_o}. (Contributed by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 1_o ∈ {∅, 1_o}

10-Jun-2026	tfrlem6 8369	Lemma for transfinite recursion. The union of all acceptable functions is a relation. (Contributed by NM, 8-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.) Avoid ax-10 2176, ax-nul 5270, ax-pr 5406, ax-sep 5258 and ax-un 7734. (Revised by Umit Teoman Dogan, 10-Jun-2026.)
⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} ⇒ ⊢ Rel recs(𝐹)

10-Jun-2026	pwuninel 8272	The powerclass of the union of a class does not belong to that class. This theorem provides a way of constructing a new set that does not belong to a given set. See also pwuninel2 8271. (Contributed by NM, 27-Jun-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.) Avoid ax-pr 5406 and ax-un 7734. (Revised by Umit Teoman Dogan, 10-Jun-2026.)
⊢ ¬ 𝒫 ∪ 𝐴 ∈ 𝐴

10-Jun-2026	peano3 7888	The successor of any natural number is not zero. One of Peano's five postulates for arithmetic. Proposition 7.30(3) of [TakeutiZaring] p. 42. (Contributed by NM, 3-Sep-2003.) Avoid ax-nul 5270. (Revised by Umit Teoman Dogan, 10-Jun-2026.)
⊢ (𝐴 ∈ ω → suc 𝐴 ≠ ∅)

10-Jun-2026	f1odm 6826	The domain of a one-to-one onto mapping. (Contributed by NM, 8-Mar-2014.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ (𝐹:𝐴–1-1-onto→𝐵 → dom 𝐹 = 𝐴)

10-Jun-2026	f1rel 6780	A one-to-one onto mapping is a relation. (Contributed by NM, 8-Mar-2014.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ (𝐹:𝐴–1-1→𝐵 → Rel 𝐹)

10-Jun-2026	f1fun 6778	A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ (𝐹:𝐴–1-1→𝐵 → Fun 𝐹)

10-Jun-2026	ffun 6710	A mapping is a function. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)

10-Jun-2026	cnvcnvss 6194	The double converse of a class is a subclass. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 23-Jul-2004.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ ^◡^◡𝐴 ⊆ 𝐴

10-Jun-2026	rnin 6145	The range of an intersection belongs the intersection of ranges. Theorem 9 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.) Avoid ax-pr 5406 and ax-sep 5258. (Revised by Umit Teoman Dogan, 10-Jun-2026.)
⊢ ran (𝐴 ∩ 𝐵) ⊆ (ran 𝐴 ∩ ran 𝐵)

10-Jun-2026	nrelv 5788	The universal class is not a relation. (Contributed by Thierry Arnoux, 23-Jan-2022.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ ¬ Rel V

10-Jun-2026	uniin 4897	The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. See uniinqs 8796 for a condition where equality holds. (Contributed by NM, 4-Dec-2003.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ ∪ (𝐴 ∩ 𝐵) ⊆ (∪ 𝐴 ∩ ∪ 𝐵)

10-Jun-2026	pwidg 4583	A set is an element of its power set. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝐴)

10-Jun-2026	rab0 4343	Any restricted class abstraction restricted to the empty set is empty. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof shortened by JJ, 14-Jul-2021.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅

10-Jun-2026	ssinss1 4199	Intersection preserves subclass relationship. (Contributed by NM, 14-Sep-1999.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶)

10-Jun-2026	pssirr 4058	Proper subclass is irreflexive. Theorem 7 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ ¬ 𝐴 ⊊ 𝐴

10-Jun-2026	nbbn 386	Move negation outside of biconditional. Compare Theorem *5.18 of [WhiteheadRussell] p. 124. (Contributed by NM, 27-Jun-2002.) (Proof shortened by Wolf Lammen, 20-Sep-2013.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)
⊢ ((¬ 𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))

9-Jun-2026	wevonprcf1o 35578	If 𝑅 is a set-like well-ordering of the universe and 𝐴 is a proper class, then 𝐹 is a bijection from the ordinals to 𝐴. This is the ZFC version of (4 → 5) in https://tinyurl.com/hamkins-gblac. (Contributed by BTernaryTau, 9-Jun-2026.)
⊢ 𝐹 = OrdIso(𝑅, 𝐴) ⇒ ⊢ ((𝑅 We V ∧ 𝑅 Se V ∧ ¬ 𝐴 ∈ V) → 𝐹:On–1-1-onto→𝐴)

9-Jun-2026	ordtypeon 35462	A proper class with a set-like well-ordering is isomorphic to the proper class of all ordinal numbers. (Contributed by BTernaryTau, 9-Jun-2026.)
⊢ 𝐹 = OrdIso(𝑅, 𝐴) ⇒ ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ¬ 𝐴 ∈ V) → 𝐹 Isom E , 𝑅 (On, 𝐴))

9-Jun-2026	ordprcon 35459	If an ordinal class is not a set, then it must be the proper class of all ordinals. (Contributed by BTernaryTau, 9-Jun-2026.)
⊢ ((Ord 𝐴 ∧ ¬ 𝐴 ∈ V) → 𝐴 = On)

8-Jun-2026	vonf1osev 35577	If 𝐹 is a bijection from the universe to the ordinals, then 𝑅 is a set-like well-ordering of the universe. This is the ZFC version of (2 → 4) which is used in place of (3 → 4) in https://tinyurl.com/hamkins-gblac. This proof takes advantage of the fact that the well-order constructed in (2 → 3) is also set-like. (Contributed by BTernaryTau, 8-Jun-2026.)
⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥) ∈ (𝐹‘𝑦)} ⇒ ⊢ (𝐹:V–1-1-onto→On → (𝑅 We V ∧ 𝑅 Se V))

7-Jun-2026	pm2.65i 196	Inference for proof by contradiction. (Contributed by NM, 18-May-1994.) (Proof shortened by Wolf Lammen, 11-Sep-2013.) (Proof shortened by Garrett Katz, 7-Jun-2026.)
⊢ (𝜑 → 𝜓) & ⊢ (𝜑 → ¬ 𝜓) ⇒ ⊢ ¬ 𝜑

6-Jun-2026	rlocisunit 33577	Characterize the units of the localization 𝐿 of a ring 𝑅 at 𝑆 as the elements with a "numerator" 𝑃 in the saturation 𝑇 of 𝑆. (Contributed by Thierry Arnoux, 6-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (._r‘𝑅) & ⊢ 𝐿 = (𝑅 RLocal 𝑆) & ⊢ 𝑊 = (Unit‘𝐿) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅))) & ⊢ ∼ = (𝑅 ~_RL 𝑆) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → 𝑄 ∈ 𝑆) & ⊢ 𝑇 = {𝑟 ∈ 𝐵 ∣ ∃𝑠 ∈ 𝐵 (𝑟 · 𝑠) ∈ 𝑆} ⇒ ⊢ (𝜑 → ([⟨𝑃, 𝑄⟩] ∼ ∈ 𝑊 ↔ 𝑃 ∈ 𝑇))

6-Jun-2026	rlocinvunit 33576	In the localization of a ring 𝑅 at 𝑆, inverses of elements of 𝑆 are units. (Contributed by Thierry Arnoux, 6-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ 1 = (1_r‘𝑅) & ⊢ ∼ = (𝑅 ~_RL 𝑆) & ⊢ 𝐿 = (𝑅 RLocal 𝑆) & ⊢ 𝑊 = (Unit‘𝐿) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅))) & ⊢ (𝜑 → 𝑄 ∈ 𝑆) ⇒ ⊢ (𝜑 → [⟨ 1 , 𝑄⟩] ∼ ∈ 𝑊)

6-Jun-2026	isunitc 33542	Characterize units in a commutative ring. (Contributed by Thierry Arnoux, 6-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ · = (._r‘𝑅) & ⊢ 1 = (1_r‘𝑅) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑅 ∈ CRing) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ ∃𝑦 ∈ 𝐵 (𝑋 · 𝑦) = 1 ))

6-Jun-2026	prmidlsubm 21468	The complement of a prime ideal is multiplicatively closed. Converse of ssdifidlprm 21467. (Contributed by Thierry Arnoux, 6-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑃 ∈ (PrmIdeal‘𝑅)) ⇒ ⊢ (𝜑 → (𝐵 ∖ 𝑃) ∈ (SubMnd‘(mulGrp‘𝑅)))

6-Jun-2026	prmidlprop 21457	Property of prime ideals. (Contributed by Thierry Arnoux, 6-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (._r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑃 ∈ (PrmIdeal‘𝑅)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝑃 ∨ 𝑌 ∈ 𝑃))

5-Jun-2026	sbrimvw 2125	Substitution in an implication with a variable not free in the antecedent affects only the consequent. Version of sbrim 2339 based on fewer axioms, but with more disjoint variable conditions. (Contributed by Wolf Lammen, 29-Jan-2024.) Remove DV condition. (Revised by Wolf Lammen, 5-Jun-2026.)
⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))

4-Jun-2026	morleylemrneab 35039	Lemma for morley . (Contributed by TA and SS, 4-Jun-2026.)
⊢ 𝑆 = (Base‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ ∼ = (cgrA‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → 𝐶 ∈ 𝑆) & ⊢ (𝜑 → 𝑃 ∈ 𝑆) & ⊢ (𝜑 → 𝑄 ∈ 𝑆) & ⊢ (𝜑 → 𝑅 ∈ 𝑆) & ⊢ (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)) & ⊢ (𝜑 → ⟨“𝐶𝐴𝑄”⟩ ∼ ⟨“𝑄𝐴𝑅”⟩) & ⊢ (𝜑 → ⟨“𝑅𝐴𝐵”⟩ ∼ ⟨“𝑄𝐴𝑅”⟩) & ⊢ (𝜑 → ⟨“𝐴𝐵𝑅”⟩ ∼ ⟨“𝑅𝐵𝑃”⟩) & ⊢ (𝜑 → ⟨“𝑃𝐵𝐶”⟩ ∼ ⟨“𝑅𝐵𝑃”⟩) & ⊢ (𝜑 → ⟨“𝐵𝐶𝑃”⟩ ∼ ⟨“𝑃𝐶𝑄”⟩) & ⊢ (𝜑 → ⟨“𝑄𝐶𝐴”⟩ ∼ ⟨“𝑃𝐶𝑄”⟩) ⇒ ⊢ (𝜑 → ¬ 𝑅 ∈ (𝐴𝐿𝐵))

4-Jun-2026	btwnlng13 35038	If 𝑍 is between 𝑋 and 𝑌, or 𝑌 is between 𝑋 and 𝑍, then 𝑍 lies on the line 𝑋𝑌. (Contributed by SS, 4-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ 𝐿 = (LineG‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) & ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))) ⇒ ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌))

4-Jun-2026	cgranbtwn 35037	Null angle implies betweenness. (Contributed by SS, 4-Jun-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩) & ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐼𝐶)) ⇒ ⊢ (𝜑 → (𝐷 ∈ (𝐸𝐼𝐹) ∨ 𝐹 ∈ (𝐸𝐼𝐷)))

4-Jun-2026	csbcnv 5874	Move class substitution in and out of the converse of a relation. (Contributed by Thierry Arnoux, 8-Feb-2017.) (Revised by NM, 23-Aug-2018.) Remove dependency on ax-sep 5258 and ax-pr 5406. (Revised by Eric Schmidt, 4-Jun-2026.)
⊢ ^◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌^◡𝐹

4-Jun-2026	abab 839	Introduce one conjunct as equivalent to the other. "abab" stands for "and, biconditional, and, biconditional". (Contributed by Wolf Lammen, 4-Jun-2026.)
⊢ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ (𝜑 ↔ 𝜓)))

3-Jun-2026	fldlring 33770	A field is a local ring. (Contributed by Thierry Arnoux, 3-Jun-2026.) (Proof modification is discouraged.)
⊢ (𝜑 → 𝐹 ∈ Field) ⇒ ⊢ (𝜑 → 𝐹 ∈ LRing)

2-Jun-2026	nmulprop 36663	Show closure and value of natural multiplication. (Contributed by Scott Fenton, 2-Jun-2026.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·_no 𝐵) ∈ On ∧ (𝐴 ·_no 𝐵) = ∩ {𝑥 ∈ On ∣ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 ((𝑎 ·_no 𝐵) +no (𝐴 ·_no 𝑏)) ∈ (𝑥 +no (𝑎 ·_no 𝑏))}))

2-Jun-2026	nmulfn 36662	Natural multiplication is a function over pairs of ordinals. (Contributed by Scott Fenton, 2-Jun-2026.)
⊢ ·_no Fn (On × On)

2-Jun-2026	df-nmul 36661	Define natural ordinal multiplication. This is the corresponding operation to df-nadd 8653. (Contributed by Scott Fenton, 2-Jun-2026.)
⊢ ·_no = frecs({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1^st ‘𝑥) E (1^st ‘𝑦) ∨ (1^st ‘𝑥) = (1^st ‘𝑦)) ∧ ((2^nd ‘𝑥) E (2^nd ‘𝑦) ∨ (2^nd ‘𝑥) = (2^nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}, (On × On), (𝑝 ∈ V, 𝑚 ∈ V ↦ ⦋(1^st ‘𝑝) / 𝑎⦌⦋(2^nd ‘𝑝) / 𝑏⦌∩ {𝑧 ∈ On ∣ ∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 ((𝑐𝑚𝑏) +no (𝑎𝑚𝑑)) ∈ (𝑧 +no (𝑐𝑚𝑑))}))

2-Jun-2026	dflring4 33769	Alternate definition of a local ring: the set (𝐵 ∖ 𝑈) of non-units is an ideal. (Contributed by Thierry Arnoux, 2-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) ⇒ ⊢ (𝑅 ∈ CRing → (𝑅 ∈ LRing ↔ (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅)))

2-Jun-2026	dflring3 33768	Alternate definition of a local ring: local rings have a single maximal ideal. (Contributed by Thierry Arnoux, 2-Jun-2026.)
⊢ (𝑅 ∈ CRing → (𝑅 ∈ LRing ↔ (MaxIdeal‘𝑅) ≈ 1_o))

2-Jun-2026	dflringlem3 33767	Lemma for dflring3 33768. In a commutative local ring 𝑅, the set (𝐵 ∖ 𝑈) of non-units is a maximal ideal. (Contributed by Thierry Arnoux, 2-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ LRing) ⇒ ⊢ (𝜑 → (𝐵 ∖ 𝑈) ∈ (MaxIdeal‘𝑅))

2-Jun-2026	dflringlem2 33766	Lemma for dflring3 33768. In a commutative local ring 𝑅, the set (𝐵 ∖ 𝑈) of non-units is an ideal. (Contributed by Thierry Arnoux, 2-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ LRing) ⇒ ⊢ (𝜑 → (𝐵 ∖ 𝑈) ∈ (LIdeal‘𝑅))

2-Jun-2026	dflringlem 33765	Lemma for dflring3 33768. If a ring 𝑅 has a single maximal ideal 𝑀, then any element 𝑋 outside of 𝑀 is a unit. (Contributed by Thierry Arnoux, 2-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑀 ∈ (MaxIdeal‘𝑅)) & ⊢ (𝜑 → (MaxIdeal‘𝑅) = {𝑀}) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ 𝑀)) ⇒ ⊢ (𝜑 → 𝑋 ∈ 𝑈)

2-Jun-2026	dflring2 33764	Alternate definition of a local ring. (Contributed by Thierry Arnoux, 2-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 1 = (1_r‘𝑅) & ⊢ − = (-_g‘𝑅) ⇒ ⊢ (𝑅 ∈ LRing ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝑈 ∨ ( 1 − 𝑥) ∈ 𝑈)))

2-Jun-2026	drnglring 33763	A division ring is a local ring. (Contributed by Thierry Arnoux, 2-Jun-2026.)
⊢ (𝜑 → 𝐹 ∈ DivRing) ⇒ ⊢ (𝜑 → 𝐹 ∈ LRing)

26-May-2026	axpowg3 35542	A generalization of ax-pow 5338 that combines axpowg 35540 and axpowg2 35541 into a single theorem scheme. Unlike ax-pow 5338, this scheme lacks a distinct variable condition for 𝑦 and 𝑤 as well as for 𝑥 and 𝑤. Usage of this theorem is discouraged because it depends on ax-13 2404. (Contributed by BTernaryTau, 26-May-2026.) (New usage is discouraged.)
⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)

26-May-2026	axpowg2 35541	A generalization of ax-pow 5338 in which 𝑥 and 𝑤 need not be distinct. This theorem scheme bundles ax-pow 5338 with the degenerate instance ∃𝑦∀𝑧(∀𝑥(𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑥) → 𝑧 ∈ 𝑦) which is satisfied by the existence of a set that contains all empty sets (see axprlem1 5396). Usage of this theorem is discouraged because it depends on ax-13 2404. (Contributed by BTernaryTau, 26-May-2026.) (New usage is discouraged.)
⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)

26-May-2026	axpowg 35540	A generalization of ax-pow 5338 that combines it and zfpow 5339 into a single theorem scheme. Unlike ax-pow 5338, this scheme lacks a distinct variable condition for 𝑦 and 𝑤. (Contributed by BTernaryTau, 26-May-2026.)
⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)

25-May-2026	expt 178	Exportation theorem pm3.3 453 (closed form of ex 417) expressed with primitive connectives. (Contributed by NM, 28-Dec-1992.) (Proof shortened by Garrett Katz, 25-May-2026.)
⊢ ((¬ (𝜑 → ¬ 𝜓) → 𝜒) → (𝜑 → (𝜓 → 𝜒)))

24-May-2026	axsepg5 35538	A generalization of ax-sep 5258 that combines axsepg 5259, axsepg2 35534, and axsepg3 35535 into a single theorem scheme. Unlike ax-sep 5258, this scheme lacks a distinct variable condition for 𝜑 and 𝑧, for 𝑥 and 𝑧, and for 𝑦 and 𝑧. Usage of this theorem is discouraged because it depends on ax-13 2404. (Contributed by BTernaryTau, 24-May-2026.) (New usage is discouraged.)
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))

24-May-2026	axsepg4 35537	A generalization of ax-sep 5258 that combines axsepg 5259 and axsepg2 35534 into a single theorem scheme. Unlike ax-sep 5258, this scheme lacks a distinct variable condition for 𝜑 and 𝑧 as well as for 𝑥 and 𝑧. Usage of this theorem is discouraged because it depends on ax-13 2404. (Contributed by BTernaryTau, 24-May-2026.) (New usage is discouraged.)
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))

23-May-2026	elirrv 9560	The membership relation is irreflexive: no set is a member of itself. Theorem 105 of [Suppes] p. 54. This is trivial to prove from zfregfr 9574 and efrirr 5643 (see elirrvALT 9575), but this proof is direct from ax-reg 9555. (Contributed by NM, 19-Aug-1993.) Reduce axiom dependencies and make use of ax-reg 9555 directly. (Revised by BTernaryTau, 27-Dec-2025.) Avoid ax-pr 5406. (Revised by BTernaryTau, 21-May-2026.) (Proof shortened by Matthew House, 23-May-2026.)
⊢ ¬ 𝑥 ∈ 𝑥

22-May-2026	bilanri 511	Inference adding a conjunct to the right-hand side of a biconditional. (Contributed by Matthew House, 22-May-2026.)
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ ((𝜒 ∧ 𝜓) → 𝜑)

22-May-2026	biranri 510	Inference adding a conjunct to the right-hand side of a biconditional. (Contributed by Matthew House, 22-May-2026.)
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ ((𝜓 ∧ 𝜒) → 𝜑)

22-May-2026	bilani 509	Inference adding a conjunct to the left-hand side of a biconditional. (Contributed by Matthew House, 22-May-2026.)
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ ((𝜒 ∧ 𝜑) → 𝜓)

22-May-2026	birani 508	Inference adding a conjunct to the left-hand side of a biconditional. (Contributed by Matthew House, 22-May-2026.)
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ ((𝜑 ∧ 𝜒) → 𝜓)

21-May-2026	axsepg2 35534	A generalization of ax-sep 5258 in which 𝑥 and 𝑧 need not be distinct. This theorem scheme bundles ax-sep 5258 with the degenerate instance ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑧 ∧ 𝜑)) which is satisfied by the existence of the empty set. Usage of this theorem is discouraged because it depends on ax-13 2404. (Contributed by BTernaryTau, 21-May-2026.) (New usage is discouraged.)
⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))

9-May-2026	goldratmolem2 47606	Lemma 2 for determining the value of golden ratio. (Contributed by Ender Ting, 9-May-2026.)
⊢ 𝐹 = (2 · (cos‘(π / 5))) ⇒ ⊢ -1 = ((((𝐹↑5) / 2) − (5 · ((𝐹↑3) / 2))) + (5 · (𝐹 / 2)))

9-May-2026	goldracos5teq 47605	Lemma 1 for determining the value of golden ratio. (Contributed by Ender Ting, 9-May-2026.)
⊢ 𝐹 = (2 · (cos‘(π / 5))) ⇒ ⊢ (cos‘π) = (((;16 · ((𝐹 / 2)↑5)) − (;20 · ((𝐹 / 2)↑3))) + (5 · (𝐹 / 2)))

9-May-2026	cos5teq 47600	Five-times-angle formula for cosine, substitution helper. (Contributed by Ender Ting, 9-May-2026.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 = (5 · 𝐴) ∧ 𝐶 = (cos‘𝐴)) → (cos‘𝐵) = (((;16 · (𝐶↑5)) − (;20 · (𝐶↑3))) + (5 · 𝐶)))

9-May-2026	quantgodel 47571	There can be no formula asserting its own non-universality, in parallel to bj-babygodel 37177; proof path is shorter but relying on a property of specialization which provability predicates do not have. For a matching proof, see quantgodelALT 47572. (Contributed by Ender Ting, 9-May-2026.)
⊢ (𝜑 ↔ ¬ ∀𝑥𝜑) ⇒ ⊢ ⊥

9-May-2026	funopsn 7146	If a function is an ordered pair then it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.) (Proof shortened by AV, 15-Jul-2021.) (Proof shortened by Eric Schmidt, 9-May-2026.) A function is a class of ordered pairs, so the fact that an ordered pair may sometimes be itself a function is an "accident" depending on the specific encoding of ordered pairs as classes (in set.mm, the Kuratowski encoding). A more meaningful statement is funsng 6589, as relsnopg 5792 is to relop 5838. (New usage is discouraged.)
⊢ 𝑋 ∈ V & ⊢ 𝑌 ∈ V ⇒ ⊢ ((Fun 𝐹 ∧ 𝐹 = ⟨𝑋, 𝑌⟩) → ∃𝑎(𝑋 = {𝑎} ∧ 𝐹 = {⟨𝑎, 𝑎⟩}))

9-May-2026	iunopeqop 5506	Implication of an ordered pair being equal to an indexed union of singletons of ordered pairs. (Contributed by AV, 20-Sep-2020.) Remove antecedent. (Revised by Eric Schmidt, 9-May-2026.) (Avoid depending on this detail.)
⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ (∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} = ⟨𝐶, 𝐷⟩ → ∃𝑧 𝐴 = {𝑧})

7-May-2026	quantgodelALT 47572	There can be no formula asserting its own non-universality; follows the steps of bj-babygodel 37177. (Contributed by Ender Ting, 7-May-2026.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝜑 ↔ ¬ ∀𝑥𝜑) ⇒ ⊢ ⊥

4-May-2026	mplidom 33899	The multivariate polynomials over an integral domain form an integral domain. See ply1idom 26263. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ IDomn) ⇒ ⊢ (𝜑 → 𝑃 ∈ IDomn)

4-May-2026	mplidomlem 33898	Lemma for mplidom 33899. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ 𝐻 = (𝑓 ∈ 𝐶 ↦ (𝑛 ∈ (ℕ₀ ↑_m 1_o) ↦ (((((𝑗 ∪ {𝑥}) selectVars 𝑅)‘{𝑥})‘𝑓)‘{⟨𝑥, (𝑛‘∅)⟩}))) & ⊢ 𝐶 = (Base‘𝑆) & ⊢ 𝑆 = ((𝑗 ∪ {𝑥}) mPoly 𝑅) & ⊢ 𝑈 = (((𝑗 ∪ {𝑥}) ∖ {𝑥}) mPoly 𝑅) & ⊢ 𝑄 = (Poly₁‘𝑈) ⇒ ⊢ (𝜑 → 𝑃 ∈ IDomn)

4-May-2026	selvply1rhm0 33897	The ring homomorphism 𝐻 built in selvply1rhm 33896 is injective. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝑈 = ((𝐼 ∖ {𝑋}) mPoly 𝑅) & ⊢ 𝑄 = (Poly₁‘𝑈) & ⊢ 𝐻 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ₀ ↑_m 1_o) ↦ ((((𝐼 selectVars 𝑅)‘{𝑋})‘𝑓)‘{⟨𝑋, (𝑛‘∅)⟩}))) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ 0 = (0_g‘𝑄) & ⊢ 𝑍 = (0_g‘𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → (𝐻‘𝐹) = 0 ) ⇒ ⊢ (𝜑 → 𝐹 = 𝑍)

4-May-2026	selvply1rhm 33896	Build a ring homomorphism 𝐻 between the multivariate polynomials 𝑃 with variables in 𝐼 and the univariate polynomials 𝑄 in a single variable 𝑋 element of 𝐼. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝑈 = ((𝐼 ∖ {𝑋}) mPoly 𝑅) & ⊢ 𝑄 = (Poly₁‘𝑈) & ⊢ 𝐻 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ₀ ↑_m 1_o) ↦ ((((𝐼 selectVars 𝑅)‘{𝑋})‘𝑓)‘{⟨𝑋, (𝑛‘∅)⟩}))) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) & ⊢ (𝜑 → 𝑅 ∈ CRing) ⇒ ⊢ (𝜑 → 𝐻 ∈ (𝑃 RingHom 𝑄))

4-May-2026	selvply1rhmlem5 33895	Lemma for selvply1rhm 33896. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝑈 = ((𝐼 ∖ {𝑋}) mPoly 𝑅) & ⊢ 𝑄 = (Poly₁‘𝑈) & ⊢ 𝐻 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ₀ ↑_m 1_o) ↦ ((((𝐼 selectVars 𝑅)‘{𝑋})‘𝑓)‘{⟨𝑋, (𝑛‘∅)⟩}))) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ 𝑀 = (𝑞 ∈ (Base‘({𝑋} mPoly 𝑈)) ↦ (𝑠 ∈ (ℕ₀ ↑_m 1_o) ↦ (𝑞‘{⟨𝑋, (𝑠‘∅)⟩}))) ⇒ ⊢ (𝜑 → (𝐻‘𝐹) = (𝑀‘(((𝐼 selectVars 𝑅)‘{𝑋})‘𝐹)))

4-May-2026	selvply1rhmlem4 33894	Lemma for selvply1rhm 33896: The mapping 𝐻 is linear. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝑈 = ((𝐼 ∖ {𝑋}) mPoly 𝑅) & ⊢ 𝑄 = (Poly₁‘𝑈) & ⊢ 𝐻 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ₀ ↑_m 1_o) ↦ ((((𝐼 selectVars 𝑅)‘{𝑋})‘𝑓)‘{⟨𝑋, (𝑛‘∅)⟩}))) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐻‘(𝐹(+_g‘𝑃)𝐺)) = ((𝐻‘𝐹)(+_g‘𝑄)(𝐻‘𝐺)))

4-May-2026	selvply1rhmlem3 33893	Lemma for selvply1rhm 33896. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝑈 = ((𝐼 ∖ {𝑋}) mPoly 𝑅) & ⊢ 𝑄 = (Poly₁‘𝑈) & ⊢ 𝐻 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ₀ ↑_m 1_o) ↦ ((((𝐼 selectVars 𝑅)‘{𝑋})‘𝑓)‘{⟨𝑋, (𝑛‘∅)⟩}))) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝑁 ∈ (ℕ₀ ↑_m 1_o)) ⇒ ⊢ (𝜑 → ((𝐻‘𝐹)‘𝑁) = ((((𝐼 selectVars 𝑅)‘{𝑋})‘𝐹)‘{⟨𝑋, (𝑁‘∅)⟩}))

4-May-2026	selvply1rhmlem2 33892	Lemma for selvply1rhm 33896: Image of the ring unit by the mapping 𝐻 (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝑈 = ((𝐼 ∖ {𝑋}) mPoly 𝑅) & ⊢ 𝑄 = (Poly₁‘𝑈) & ⊢ 𝐻 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ₀ ↑_m 1_o) ↦ ((((𝐼 selectVars 𝑅)‘{𝑋})‘𝑓)‘{⟨𝑋, (𝑛‘∅)⟩}))) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) & ⊢ (𝜑 → 𝑅 ∈ CRing) ⇒ ⊢ (𝜑 → (𝐻‘(1_r‘𝑃)) = (1_r‘𝑄))

4-May-2026	selvply1rhmlem1 33891	Lemma for selvply1rhm 33896. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝑈 = ((𝐼 ∖ {𝑋}) mPoly 𝑅) & ⊢ 𝑄 = (Poly₁‘𝑈) & ⊢ 𝐻 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ₀ ↑_m 1_o) ↦ ((((𝐼 selectVars 𝑅)‘{𝑋})‘𝑓)‘{⟨𝑋, (𝑛‘∅)⟩}))) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) & ⊢ (𝜑 → 𝑅 ∈ CRing) ⇒ ⊢ (𝜑 → 𝐻:𝐵⟶(Base‘𝑄))

4-May-2026	selvply1rhmlemb 33890	Lemma for selvply1rhm 33896. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑃 = ({𝑋} mPoly 𝑅) & ⊢ · = (._r‘𝑃) & ⊢ × = (._r‘𝑄) & ⊢ 𝑄 = (Poly₁‘𝑅) & ⊢ 𝑀 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ₀ ↑_m 1_o) ↦ (𝑓‘{⟨𝑋, (𝑛‘∅)⟩}))) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑀‘(𝐹 · 𝐺)) = ((𝑀‘𝐹) × (𝑀‘𝐺)))

4-May-2026	selvply1rhmlema 33889	Lemma for selvply1rhm 33896. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑃 = ({𝑋} mPoly 𝑅) & ⊢ · = (._r‘𝑃) & ⊢ × = (._r‘𝑄) & ⊢ 𝑄 = (Poly₁‘𝑅) & ⊢ 𝑀 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ₀ ↑_m 1_o) ↦ (𝑓‘{⟨𝑋, (𝑛‘∅)⟩}))) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑀‘𝐹) ∈ (Base‘𝑄))

4-May-2026	selvascl 33888	The "variable selection" function evaluated at a scalar. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐴 = (algSc‘𝑃) & ⊢ 𝐶 = (algSc‘𝑇) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅) & ⊢ 𝑇 = (𝐽 mPoly 𝑈) & ⊢ 𝐷 = (𝐶 ∘ (algSc‘𝑈)) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐽 ⊆ 𝐼) ⇒ ⊢ (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘(𝐴‘𝑋)) = (𝐷‘𝑋))

4-May-2026	mplasclco 33887	Case where composing an algebra scalar lifting functions with a scalar leads to a scalar. This is useful when working with selectVars. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝑆 = (Base‘𝑅) & ⊢ 𝑂 = (𝐽 mPoly 𝑅) & ⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝑄 = (𝐼 mPoly 𝑂) & ⊢ 𝐴 = (algSc‘𝑂) & ⊢ 𝐵 = (algSc‘𝑃) & ⊢ 𝐶 = (algSc‘𝑄) & ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} & ⊢ 𝐸 = {𝑗 ∈ (ℕ₀ ↑_m 𝐽) ∣ (^◡𝑗 “ ℕ) ∈ Fin} & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐽 ⊆ 𝐼) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝐴 ∘ (𝐵‘𝑋)) = (𝐶‘(𝐴‘𝑋)))

4-May-2026	0mplric 33886	Multivariate polynomials with no variables are isomorphic with the underlying ring. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑃 = (∅ mPoly 𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → 𝑃 ≃_𝑟 𝑅)

4-May-2026	0mplrim 33885	Build a ring isomorphism between multivariate polynomials with no variables and the underlying ring. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑃 = (∅ mPoly 𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐹 = (𝑝 ∈ 𝐵 ↦ (𝑝‘∅)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝑃 RingIso 𝑅))

4-May-2026	mplnzr 33884	The multivariate polynomials over a nonzero ring form a nonzero ring. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ NzRing) ⇒ ⊢ (𝜑 → 𝑃 ∈ NzRing)

4-May-2026	psrnzr 33883	The ring of power series over a nonzero ring form a nonzero ring. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ NzRing) ⇒ ⊢ (𝜑 → 𝑆 ∈ NzRing)

4-May-2026	ricdomn 33591	A ring is a domain if and only if an isomorphic ring is a domain. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ (𝑅 ≃_𝑟 𝑆 → (𝑅 ∈ Domn ↔ 𝑆 ∈ Domn))

4-May-2026	ricdomn1 33590	A ring isomorphism maps a domain to a domain. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ ((𝑅 ≃_𝑟 𝑆 ∧ 𝑅 ∈ Domn) → 𝑆 ∈ Domn)

4-May-2026	ricnzr1 33589	A ring isomorphism maps a nonzero ring to a nonzero ring. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ ((𝑅 ≃_𝑟 𝑆 ∧ 𝑅 ∈ NzRing) → 𝑆 ∈ NzRing)

4-May-2026	grpidcld 33340	The identity element of a group belongs to the group. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) ⇒ ⊢ (𝜑 → 0 ∈ 𝐵)

4-May-2026	ififcom 32877	Commute two nested conditionals. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ if(𝜑, if(𝜓, 𝐴, 𝐵), 𝐵) = if(𝜓, if(𝜑, 𝐴, 𝐵), 𝐵)

2-May-2026	copsexgw 5474	Version of copsexg 5476 with a disjoint variable condition, which does not require ax-13 2404. (Contributed by GG, 26-Jan-2024.) Shorten proof and remove dependency on ax-10 2176. (Revised by Eric Schmidt, 2-May-2026.)
⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))

1-May-2026	vprc 5284	The universal class is not a member of itself (and thus is not a set). Proposition 5.21 of [TakeutiZaring] p. 21; our proof, however, does not depend on the Axiom of Regularity. (Contributed by NM, 23-Aug-1993.) (Proof shortened by BJ, 1-May-2026.)
⊢ ¬ V ∈ V

1-May-2026	nvel 5283	The universal class does not belong to any class. (Contributed by FL, 31-Dec-2006.) Prove it without using vprc 5284, which is then proved as an instance of it. (Revised by BJ, 1-May-2026.)
⊢ ¬ V ∈ 𝐴

27-Apr-2026	qdiffALT 37953	Alternate proof of qdiff 37952. This is a proof from irrdiff 37951 using excluded middle in a variety of places. (Contributed by Jim Kingdon, 27-Apr-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ ℝ → (𝐴 ∈ ℚ ↔ ∃𝑞 ∈ ℚ ∃𝑟 ∈ ℚ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))))

25-Apr-2026	unidif0 5332	The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.) (Proof shortened by Eric Schmidt, 25-Apr-2026.)
⊢ ∪ (𝐴 ∖ {∅}) = ∪ 𝐴

25-Apr-2026	vnex 5281	The universal class does not exist as a set. (Contributed by NM, 4-Jul-2005.) (Proof shortened by BJ, 25-Apr-2026.)
⊢ ¬ ∃𝑥 𝑥 = V

25-Apr-2026	vneqv 5280	The universal class is not equal to any setvar. (Contributed by NM, 4-Jul-2005.) Extract from vnex 5281 and shorten proof. (Revised by BJ, 25-Apr-2026.)
⊢ ¬ 𝑥 = V

24-Apr-2026	qdiff 37952	The rationals are exactly those reals for which there exist two distinct rationals that are the same distance from the original number. Similar to irrdiff 37951 but here proved with a proof which would also work in constructive mathematics. From an online post by Ingo Blechschmidt. For a proof using irrdiff 37951, see qdiffALT 37953. (Contributed by Jim Kingdon, 24-Apr-2026.)
⊢ (𝐴 ∈ ℝ → (𝐴 ∈ ℚ ↔ ∃𝑞 ∈ ℚ ∃𝑟 ∈ ℚ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))))

22-Apr-2026	sucprcreg 9569	A class is equal to its successor iff it is a proper class (assuming the Axiom of Regularity). (Contributed by NM, 9-Jul-2004.) (Proof shortened by BJ, 16-Apr-2019.) (Proof shortened by SN, 22-Apr-2026.)
⊢ (¬ 𝐴 ∈ V ↔ suc 𝐴 = 𝐴)

22-Apr-2026	nelaneq 9565	A class is not an element of and equal to a class at the same time. Variant of elneq 9564 analogously to elnotel 9580 and en2lp 9576. (Proposed by BJ, 18-Jun-2022.) (Contributed by AV, 18-Jun-2022.) (Proof shortened by TM, 31-Dec-2025.) (Proof shortened by SN, 22-Apr-2026.)
⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐴 = 𝐵)

20-Apr-2026	cos5t 47599	Five-times-angle formula for cosine, in pure cosine form. (Contributed by Ender Ting, 20-Apr-2026.)
⊢ (𝐴 ∈ ℂ → (cos‘(5 · 𝐴)) = (((;16 · ((cos‘𝐴)↑5)) − (;20 · ((cos‘𝐴)↑3))) + (5 · (cos‘𝐴))))

19-Apr-2026	trun 5230	The union of transitive classes is transitive. (Contributed by Eric Schmidt, 19-Apr-2026.)
⊢ ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴 ∪ 𝐵))

17-Apr-2026	sin5t 47598	Five-times-angle formula for sine, in pure sine form. (Contributed by Ender Ting, 17-Apr-2026.)
⊢ (𝐴 ∈ ℂ → (sin‘(5 · 𝐴)) = (((;16 · ((sin‘𝐴)↑5)) − (;20 · ((sin‘𝐴)↑3))) + (5 · (sin‘𝐴))))

17-Apr-2026	sin5tlem5 47597	Lemma 5 for quintupled angle sine calculation: sine of triple-angle and double-angle sum, as a polynomial in sine straight. (Contributed by Ender Ting, 17-Apr-2026.)
⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ (𝑁↑2) = (1 − (𝑀↑2))) → ((((3 · 𝑀) − (4 · (𝑀↑3))) · (1 − (2 · (𝑀↑2)))) + (((4 · (𝑁↑3)) − (3 · 𝑁)) · (2 · (𝑀 · 𝑁)))) = (((;16 · (𝑀↑5)) − (;20 · (𝑀↑3))) + (5 · 𝑀)))

17-Apr-2026	sin5tlem4 47596	Lemma 4 for quintupled angle sine calculation: expanding lemma 3 result to difference of polynomials. (Contributed by Ender Ting, 17-Apr-2026.)
⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ (𝑁↑2) = (1 − (𝑀↑2))) → (((4 · (𝑁↑3)) − (3 · 𝑁)) · (2 · (𝑀 · 𝑁))) = ((((8 · (𝑀↑5)) − (;16 · (𝑀↑3))) + (8 · 𝑀)) − ((6 · 𝑀) − (6 · (𝑀↑3)))))

16-Apr-2026	goldrarp 47604	The golden ratio is a positive real. (Contributed by Ender Ting, 16-Apr-2026.)
⊢ 𝐹 = (2 · (cos‘(π / 5))) ⇒ ⊢ 𝐹 ∈ ℝ⁺

16-Apr-2026	goldrapos 47603	Golden ratio is positive. (Contributed by Ender Ting, 16-Apr-2026.)
⊢ 𝐹 = (2 · (cos‘(π / 5))) ⇒ ⊢ 0 < 𝐹

16-Apr-2026	sin5tlem3 47595	Lemma 3 for quintupled angle sine calculation, multiplicating triple angle cosine by double angle sine. (Contributed by Ender Ting, 16-Apr-2026.)
⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ (𝑁↑2) = (1 − (𝑀↑2))) → (((4 · (𝑁↑3)) − (3 · 𝑁)) · (2 · (𝑀 · 𝑁))) = (((4 · ((1 − (2 · (𝑀↑2))) + (𝑀↑4))) − (3 · (1 − (𝑀↑2)))) · (2 · 𝑀)))

16-Apr-2026	sin5tlem2 47594	Lemma 2 for quintupled angle sine calculation, multiplicating triple angle cosine by cosine straight and converting into sine. (Contributed by Ender Ting, 16-Apr-2026.)
⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ (𝑁↑2) = (1 − (𝑀↑2))) → (((4 · (𝑁↑3)) − (3 · 𝑁)) · 𝑁) = ((4 · ((1 − (2 · (𝑀↑2))) + (𝑀↑4))) − (3 · (1 − (𝑀↑2)))))

13-Apr-2026	wl-dfclel 38142	The defining characterization of class membership. Unlike the forms on which it is based, it is unrestricted. Proven in Tarski's FOL, from the axiom of (set) extensionality (ax-ext 2735), the definitions df-clel 2838 and df-cleq . (Contributed by BJ, 27-Jun-2019.) Base on wl-dfclel.just 38140. (Revised by Wolf Lammen, 13-Apr-2026.)
⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))

13-Apr-2026	mh-infprim3bi 37040	An axiom of infinity in primitive symbols not requiring ax-reg 9555. This version of the axiom was designed by Stefan O'Rear for his zf2.nql program, see https://github.com/sorear/metamath-turing-machines 9555. It directly implies ax-inf 9608, but deriving ax-inf2 9611 requires ax-ext 2735 and ax-rep 5239, see mh-inf3sn 37034. (Contributed by Matthew House, 13-Apr-2026.)
⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 {𝑧} ∈ 𝑦) ↔ ¬ ∀𝑦 ¬ ¬ (𝑥 ∈ 𝑦 → ¬ ∀𝑥(𝑥 ∈ 𝑦 → ¬ ∀𝑧 ¬ ¬ (𝑧 ∈ 𝑦 → ¬ ∀𝑦 ¬ ((𝑦 ∈ 𝑧 → 𝑦 = 𝑥) → ¬ (𝑦 = 𝑥 → 𝑦 ∈ 𝑧))))))

13-Apr-2026	mh-infprim2bi 37039	Shortest possible axiom of infinity in primitive symbols not requiring ax-reg 9555. Deriving ax-inf 9608 or ax-inf2 9611 from this axiom requires ax-ext 2735 and ax-rep 5239, see mh-inf3sn 37034 and inf0 9591. (Contributed by Matthew House, 13-Apr-2026.)
⊢ (∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) ↔ ¬ ∀𝑥 ¬ ∀𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑦 → ¬ (𝑤 ∈ 𝑥 → ¬ 𝑤 = 𝑧)) → 𝑦 ∈ 𝑥))

13-Apr-2026	mh-infprim1bi 37038	Shortest possible axiom of infinity in primitive symbols. Deriving ax-inf 9608 or ax-inf2 9611 from this axiom requires ax-ext 2735, ax-rep 5239, and ax-reg 9555, see inf3 9605 and inf0 9591. (Contributed by Matthew House, 13-Apr-2026.)
⊢ (∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) ↔ ¬ ∀𝑥 ¬ ∀𝑦 ¬ ∀𝑧((𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑧) → ¬ 𝑧 ∈ 𝑥))

13-Apr-2026	mh-regprimbi 37037	Shortest possible version of ax-reg 9555 in primitive symbols. The equivalence is nontrivial, but it still follows solely from the axioms of predicate calculus. (Contributed by Matthew House, 13-Apr-2026.)
⊢ ((∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥))) ↔ ¬ ∀𝑦 ¬ ∀𝑧((𝑦 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ¬ 𝑧 ∈ 𝑥))

13-Apr-2026	mh-unprimbi 37036	Shortest possible version of ax-un 7734 in primitive symbols. (Contributed by Matthew House, 13-Apr-2026.)
⊢ (∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) ↔ ¬ ∀𝑦 ¬ ∀𝑧(𝑧 ∈ 𝑥 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)))

13-Apr-2026	mh-prprimbi 37035	Shortest possible version of ax-pr 5406 in primitive symbols. (Contributed by Matthew House, 13-Apr-2026.)
⊢ (∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) ↔ ¬ ∀𝑧(𝑥 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑧))

13-Apr-2026	mh-inf3sn 37034	Version of inf3 9605 for the set of Zermelo ordinals ∅, {∅}, {{∅}}, {{{∅}}}, etc., where the successor of 𝑦 is {𝑦}. Unlike inf3 9605, the proof does not require ax-reg 9555, since the singleton properties snnz 4743 and sneqr 4806 are sufficient to guarantee that all elements of the sequence are distinct. (Contributed by Matthew House, 13-Apr-2026.)
⊢ ∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {𝑦} ∈ 𝑥) ⇒ ⊢ ω ∈ V

13-Apr-2026	mh-inf3f1 37033	A variant of inf3 9605. If 𝐹 is a one-to-one function from 𝐴 into itself, and there exists an element 𝐵 not in its range, then (rec(𝐹, 𝐵) ↾ ω) is an infinite sequence of distinct elements from 𝐴. If 𝐴 is a set, we can use this theorem to prove ω ∈ V via f1dmex 7955. (Contributed by Matthew House, 13-Apr-2026.)
⊢ (𝜑 → 𝐹:𝐴–1-1→𝐴) & ⊢ (𝜑 → 𝐵 ∈ (𝐴 ∖ ran 𝐹)) ⇒ ⊢ (𝜑 → (rec(𝐹, 𝐵) ↾ ω):ω–1-1→𝐴)

12-Apr-2026	nalset 5278	No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.) Extract exnelv 5277. (Revised by Matthew House, 12-Apr-2026.)
⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥

12-Apr-2026	exnelv 5277	For any set 𝑥, there is a set not contained in 𝑥. The proof is based on Russell's paradox. (Contributed by NM, 23-Aug-1993.) Remove use of ax-12 2213 and ax-13 2404. (Revised by BJ, 31-May-2019.) Extract from nalset 5278. (Revised by Matthew House, 12-Apr-2026.)
⊢ ∃𝑦 ¬ 𝑦 ∈ 𝑥

11-Apr-2026	indsum 15882	Finite sum of a product with the indicator function / Cartesian product with the indicator function. Note: this theorem cannot be efficiently shortened using sumss2 15779, unless there are some additional auxiliary theorems like (if(𝑥 ∈ 𝐴, 1, 0) · 𝐵) = if(𝑥 ∈ 𝐴, 𝐵, 0). (Contributed by Thierry Arnoux, 14-Aug-2017.) (Proof shortened by AV, 11-Apr-2026.)
⊢ (𝜑 → 𝑂 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ 𝑂) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑂) → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → Σ𝑥 ∈ 𝑂 ((((𝟭‘𝑂)‘𝐴)‘𝑥) · 𝐵) = Σ𝑥 ∈ 𝐴 𝐵)

11-Apr-2026	indval0 12223	The indicator function generator does not generate a (meaningful) indicator function for a class which is not a subset of the domain. (Contributed by AV, 11-Apr-2026.)
⊢ (¬ 𝐴 ⊆ 𝑂 → ((𝟭‘𝑂)‘𝐴) = ∅)

10-Apr-2026	ppivalnn 48367	Value of the prime-counting function pi for positive integers, according to Ján Mináč, see statement in [Ribenboim], p. 181. (Contributed by AV, 10-Apr-2026.)
⊢ (𝑁 ∈ ℕ → (π‘𝑁) = Σ𝑘 ∈ (2...𝑁)(⌊‘((((!‘(𝑘 − 1)) + 1) / 𝑘) − (⌊‘((!‘(𝑘 − 1)) / 𝑘)))))

10-Apr-2026	ppivalnnprm 48360	Value of a term of the prime-counting function pi for positive integers, according to Ján Mináč, for a prime number. (Contributed by AV, 10-Apr-2026.)
⊢ (𝑃 ∈ ℙ → (⌊‘((((!‘(𝑃 − 1)) + 1) / 𝑃) − (⌊‘((!‘(𝑃 − 1)) / 𝑃)))) = 1)

10-Apr-2026	flmrecm1 48063	The floor of an integer minus the reciprocal of a positive integer is the integer minus 1. (Contributed by AV, 10-Apr-2026.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (⌊‘(𝑀 − (1 / 𝑁))) = (𝑀 − 1))

10-Apr-2026	nnge2recfl0 48062	The floor of the reciprocal of an integer greater than 1 is 0. (Contributed by AV, 10-Apr-2026.)
⊢ (𝑁 ∈ (ℤ_≥‘2) → (⌊‘(1 / 𝑁)) = 0)

10-Apr-2026	prmssuz2 16756	The primes are integers greater than 1. (Contributed by AV, 10-Apr-2026.)
⊢ ℙ ⊆ (ℤ_≥‘2)

10-Apr-2026	indsumhash 15883	The finite sum of the indicator function is the number of elements of the corresponding subset. (Contributed by AV, 10-Apr-2026.)
⊢ 1 = ((𝟭‘𝑂)‘𝐴) ⇒ ⊢ ((𝑂 ∈ Fin ∧ 𝐴 ⊆ 𝑂) → Σ𝑘 ∈ 𝑂 ( 1 ‘𝑘) = (♯‘𝐴))

10-Apr-2026	fsumconst1 15844	The sum of 1 over a finite set equals the size of the set. (Contributed by AV, 10-Apr-2026.)
⊢ (𝐴 ∈ Fin → Σ𝑘 ∈ 𝐴 1 = (♯‘𝐴))

10-Apr-2026	nnge2recico01 13535	The reciprocal of an integer greater than 1 is in the right open interval between 0 and 1. (Contributed by AV, 10-Apr-2026.)
⊢ (𝑁 ∈ (ℤ_≥‘2) → (1 / 𝑁) ∈ (0[,)1))

10-Apr-2026	fvindre 12227	The range of the indicator function is a subset of ℝ. (Contributed by AV, 10-Apr-2026.)
⊢ (((𝑂 ∈ Fin ∧ 𝐴 ⊆ 𝑂) ∧ 𝑋 ∈ 𝑂) → (((𝟭‘𝑂)‘𝐴)‘𝑋) ∈ ℝ)

9-Apr-2026	rediv11d 43205	One-to-one relationship for division. (Contributed by SN, 9-Apr-2026.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ≠ 0) ⇒ ⊢ (𝜑 → ((𝐴 /_ℝ 𝐶) = (𝐵 /_ℝ 𝐶) ↔ 𝐴 = 𝐵))

9-Apr-2026	redivdird 43204	Distribution of division over addition. (Contributed by SN, 9-Apr-2026.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ≠ 0) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) /_ℝ 𝐶) = ((𝐴 /_ℝ 𝐶) + (𝐵 /_ℝ 𝐶)))

9-Apr-2026	rediv23d 43203	A "commutative"/associative law for division. (Contributed by SN, 9-Apr-2026.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ≠ 0) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐵) /_ℝ 𝐶) = ((𝐴 /_ℝ 𝐶) · 𝐵))

9-Apr-2026	redivrec2d 43202	Relationship between division and reciprocal. (Contributed by SN, 9-Apr-2026.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ≠ 0) ⇒ ⊢ (𝜑 → (𝐴 /_ℝ 𝐵) = ((1 /_ℝ 𝐵) · 𝐴))

8-Apr-2026	ppivalnnnprm 48363	Value of a term of the prime-counting function pi for positive integers, according to Ján Miná&ccaron, for a non-prime number greater than 1. (Contributed by AV, 8-Apr-2026.)
⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ 𝑁 ∉ ℙ) → (⌊‘((((!‘(𝑁 − 1)) + 1) / 𝑁) − (⌊‘((!‘(𝑁 − 1)) / 𝑁)))) = 0)

8-Apr-2026	ppivalnn4 48362	Value of the term of the prime-counting function pi for positive integers, according to Ján Mináč, for 4. (Contributed by AV, 8-Apr-2026.)
⊢ (⌊‘((((!‘(4 − 1)) + 1) / 4) − (⌊‘((!‘(4 − 1)) / 4)))) = 0

7-Apr-2026	nprmdvdsfacm1 48359	A non-prime integer greater than 5 divides the factorial of the integer decreased by 1 (see remark in [Ribenboim] p. 181). Note: not valid for 𝑁 = 4, but for 𝑁 = 1! (Contributed by AV, 7-Apr-2026.)
⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝑁 ∉ ℙ) → 𝑁 ∥ (!‘(𝑁 − 1)))

7-Apr-2026	nprmdvdsfacm1lem4 48358	Lemma 4 for nprmdvdsfacm1 48359. (Contributed by AV, 7-Apr-2026.)
⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 𝑁 ∥ (!‘(𝑁 − 1)))

7-Apr-2026	nprmdvdsfacm1lem3 48357	Lemma 3 for nprmdvdsfacm1 48359. (Contributed by AV, 7-Apr-2026.)
⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → (2 · 𝐴) < (𝑁 − 1))

7-Apr-2026	nprmdvdsfacm1lem2 48356	Lemma 2 for nprmdvdsfacm1 48359. (Contributed by AV, 7-Apr-2026.)
⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 3 ≤ 𝐴)

7-Apr-2026	nprmdvdsfacm1lem1 48355	Lemma 1 for nprmdvdsfacm1 48359. (Contributed by AV, 7-Apr-2026.)
⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝐴 ∈ (2..^𝑁) ∧ 𝑁 = (𝐴↑2)) → 𝑁 ∥ (𝐴 · (2 · 𝐴)))

7-Apr-2026	2timesltsqm1 48099	Two times an integer greater than 2 is less than the square of the integer minus 1. (Contributed by AV, 7-Apr-2026.)
⊢ (𝐴 ∈ (ℤ_≥‘3) → (2 · 𝐴) < ((𝐴↑2) − 1))

7-Apr-2026	wl-dfcleq 38141	The defining characterization of class equality. This version of df-cleq 2755 has no restrictions, unlike the forms on which it is based. It is proved in Tarski's FOL from the axiom of extensionality (ax-ext 2735), the definition of class equality (df-cleq 2755), and the definition of class membership (df-clel 2838). Its forward implication is known as "class extensionality". (Contributed by NM, 15-Sep-1993.) (Revised by BJ, 24-Jun-2019.) Base on wl-dfcleq.just 38137. (Revised by Wolf Lammen, 7-Apr-2026.)
⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

7-Apr-2026	wl-dfclel.just 38140	Add a hypothesis to wl-dfclel.basic 38139, that permits alpha-renaming. (Contributed by Wolf Lammen, 7-Apr-2026.)
⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ 𝐵)) ⇒ ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))

7-Apr-2026	wl-dfcleq.just 38137	The hypotheses added to this version of df-cleq 2755 address the following: 1. Equality of classes is an equivalence relation, as expected of equality. 2. Equality of classes obeys the Law of Indiscernibles (Leibniz's Law), and is compatible with class membership. 3. Alpha-renaming is explicitly permitted. (Contributed by Wolf Lammen, 7-Apr-2026.)
⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ ∀𝑦(𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) & ⊢ 𝐴 = 𝐴 & ⊢ (𝐴 = 𝐵 → (𝐵 = 𝐶 → 𝐶 = 𝐴)) & ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 → 𝐵 ∈ 𝐶)) & ⊢ (𝐴 = 𝐵 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵)) ⇒ ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

7-Apr-2026	elALTtco 36973	Derivation of el 5421 from ax-tco 36964. Use el 5421 instead. (Contributed by Matthew House, 7-Apr-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑦 𝑥 ∈ 𝑦

7-Apr-2026	axnulregtco 36972	Derivation of ax-nul 5270 from ax-reg 9555 and ax-tco 36964. Use ax-nul 5270 instead. (Contributed by Matthew House, 7-Apr-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥

7-Apr-2026	axtco1 36965	Strong form of the Axiom of Transitive Containment. See ax-tco 36964 for more information. In particular, this theorem generalizes the statement of ax-tco 36964, allowing it to be written with only three variables, since 𝑥 need not be distinct from both 𝑧 and 𝑤. (Contributed by Matthew House, 7-Apr-2026.)
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)))

6-Apr-2026	nprmmul2 48260	Special factorization of a non-prime integer greater than 3. (Contributed by AV, 6-Apr-2026.)
⊢ (𝑁 ∈ (ℤ_≥‘4) → (𝑁 ∉ ℙ ↔ ∃𝑎 ∈ (2..^𝑁)∃𝑏 ∈ (2..^𝑁)(𝑎 ≤ 𝑏 ∧ 𝑁 = (𝑎 · 𝑏))))

6-Apr-2026	muldvdsfacm1 48107	The product of two different positive integers less than a third integer divides the factorial of the third integer decreased by 1. By assumption, the third integer must be greater than 3. (Contributed by AV, 6-Apr-2026.)
⊢ ((𝐴 ∈ (1..^𝐵) ∧ 𝐵 ∈ (1..^𝑁)) → (𝐴 · 𝐵) ∥ (!‘(𝑁 − 1)))

6-Apr-2026	muldvdsfacgt 48106	The product of two different positive integers divides the factorial of the bigger integer. (Contributed by AV, 6-Apr-2026.)
⊢ (𝐴 ∈ (1..^𝐵) → (𝐴 · 𝐵) ∥ (!‘𝐵))

6-Apr-2026	facnn0dvdsfac 48105	The factorial of a nonnegative integer divides the factorial of an integer which is greater than or equal to the first integer. (Contributed by AV, 6-Apr-2026.)
⊢ (𝑀 ∈ (0...𝑁) → (!‘𝑀) ∥ (!‘𝑁))

6-Apr-2026	2timesltsq 48098	Two times an integer greater than 2 is less than the square of the integer. (Contributed by AV, 6-Apr-2026.)
⊢ (𝐴 ∈ (ℤ_≥‘3) → (2 · 𝐴) < (𝐴↑2))

6-Apr-2026	ttc0el 37027	A transitive closure contains ∅ as an element iff it is nonempty, assuming Regularity and Transitive Containment. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ≠ ∅ ↔ ∅ ∈ TC+ 𝐴)

6-Apr-2026	dfttc3g 37026	The transitive closure of a set 𝐴 is (TC‘𝐴), assuming Transitive Containment. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ 𝑉 → TC+ 𝐴 = (TC‘𝐴))

6-Apr-2026	ttcexbi 37025	A class is a set iff its transitive closure is a set, assuming Transitive Containment. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ V ↔ TC+ 𝐴 ∈ V)

6-Apr-2026	ttcexg 37024	The transitive closure of a set is a set, assuming Transitive Containment. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ 𝑉 → TC+ 𝐴 ∈ V)

6-Apr-2026	elttcirr 37023	Irreflexivity of 𝐴 ∈ TC+ 𝐵 relationship. This is a consequence of Regularity, but it does not require Transitive Containment. We use the alternative expression dfttc4 37022 to construct a set in which 𝐴 is both ∈-minimal and not ∈-minimal. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ¬ 𝐴 ∈ TC+ 𝐴

6-Apr-2026	dfttc4 37022	An alternative expression for the transitive closure of a class, assuming Regularity. A set 𝑥 is contained in the transitive closure of 𝐴 iff we can construct an ∈-chain from 𝑥 to an element of 𝐴. This weak definition is primarily useful for proving elttcirr 37023. (Contributed by Matthew House, 6-Apr-2026.)
⊢ TC+ 𝐴 = {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))}

6-Apr-2026	dfttc4lem2 37021	Lemma for dfttc4 37022. (Contributed by Matthew House, 6-Apr-2026.)
⊢ 𝐵 = {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))} ⇒ ⊢ (𝐴 ⊆ 𝐵 ∧ Tr 𝐵)

6-Apr-2026	dfttc4lem1 37020	Lemma for dfttc4 37022. (Contributed by Matthew House, 6-Apr-2026.)
⊢ 𝐵 = {𝑥 ∣ ∃𝑦((𝐴 ∩ 𝑦) ≠ ∅ ∧ ∀𝑧 ∈ 𝑦 ((𝑧 ∩ 𝑦) = ∅ → 𝑧 = 𝑥))} & ⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ (((𝐴 ∩ 𝐶) ≠ ∅ ∧ ∀𝑧 ∈ 𝐶 ((𝑧 ∩ 𝐶) = ∅ → 𝑧 = 𝐷)) → 𝐷 ∈ 𝐵)

6-Apr-2026	ttc0elw 37019	If a transitive closure is a set, then it contains ∅ as an element iff it is nonempty, assuming Regularity. If we also assume Transitive Containment, then we can remove the TC+ 𝐴 ∈ 𝑉 hypothesis, see ttc0el 37027. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (TC+ 𝐴 ∈ 𝑉 → (𝐴 ≠ ∅ ↔ ∅ ∈ TC+ 𝐴))

6-Apr-2026	ttcwf3 37018	The sets whose transitive closures are sets are precisely the well-founded sets, assuming Regularity. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (TC+ 𝐴 ∈ V ↔ 𝐴 ∈ ∪ (𝑅₁ “ On))

6-Apr-2026	ttcwf2 37017	If a transitive closure class is a set, then it is well-founded, assuming Regularity. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (TC+ 𝐴 ∈ V ↔ TC+ 𝐴 ∈ ∪ (𝑅₁ “ On))

6-Apr-2026	ttcwf 37016	A set is well-founded iff its transitive closure is well-founded. As a corollary, the transitive closure of any well-founded set is a set. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ ∪ (𝑅₁ “ On) ↔ TC+ 𝐴 ∈ ∪ (𝑅₁ “ On))

6-Apr-2026	dfttc3gw 37015	If the transitive closure of 𝐴 is a set, then its value is (TC‘𝐴). If we assume Transitive Containment, then we can weaken the hypothesis to 𝐴 ∈ 𝑉, see dfttc3g 37026. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (TC+ 𝐴 ∈ 𝑉 → TC+ 𝐴 = (TC‘𝐴))

6-Apr-2026	ttcsntrsucg 37014	The singleton transitive closure of a transitive set is its successor. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ((𝐴 ∈ 𝑉 ∧ Tr 𝐴) → TC+ {𝐴} = suc 𝐴)

6-Apr-2026	ttcsnexbig 37013	The transitive closure of a set is a set iff its singleton transitive closure is a set. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ 𝑉 → (TC+ 𝐴 ∈ V ↔ TC+ {𝐴} ∈ V))

6-Apr-2026	ttcsnexg 37012	If the transitive closure of a class is a set, then its singleton transitive closure is a set. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (TC+ 𝐴 ∈ 𝑉 → TC+ {𝐴} ∈ V)

6-Apr-2026	ttcsng 37011	Relationship between TC+ {𝐴} and TC+ 𝐴: the former contains the additional element 𝐴. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ 𝑉 → TC+ {𝐴} = (TC+ 𝐴 ∪ {𝐴}))

6-Apr-2026	ttcsnmin 37010	The singleton transitive closure is the minimal transitive class containing 𝐴 as an element. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ((𝐴 ∈ 𝐵 ∧ Tr 𝐵) → TC+ {𝐴} ⊆ 𝐵)

6-Apr-2026	ttcsnidg 37009	The singleton transitive closure contains its argument 𝐴 as an element. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ TC+ {𝐴})

6-Apr-2026	ttcsnssg 37008	The transitive closure is contained in the singleton transitive closure. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ 𝑉 → TC+ 𝐴 ⊆ TC+ {𝐴})

6-Apr-2026	ttcpwss 37007	The transitive closure of a power class is contained in the power class of the transitive closure. (Contributed by Matthew House, 6-Apr-2026.)
⊢ TC+ 𝒫 𝐴 ⊆ 𝒫 TC+ 𝐴

6-Apr-2026	ttciun 37006	Distribute indexed union through a transitive closure. (Contributed by Matthew House, 6-Apr-2026.)
⊢ TC+ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 TC+ 𝐵

6-Apr-2026	ttcuni 37005	Distribute union of a class through a transitive closure. (Contributed by Matthew House, 6-Apr-2026.)
⊢ TC+ ∪ 𝐴 = ∪ TC+ 𝐴

6-Apr-2026	ttcun 37004	Distribute union of two classes through a transitive closure. (Contributed by Matthew House, 6-Apr-2026.)
⊢ TC+ (𝐴 ∪ 𝐵) = (TC+ 𝐴 ∪ TC+ 𝐵)

6-Apr-2026	ttciunun 37003	Relationship between TC+ 𝐴 and ∪ 𝑥 ∈ 𝐴TC+ 𝑥: we can decompose TC+ 𝐴 into the elements of ∪ 𝑥 ∈ 𝐴TC+ 𝑥 plus the elements of 𝐴 itself. (Contributed by Matthew House, 6-Apr-2026.)
⊢ TC+ 𝐴 = (∪ 𝑥 ∈ 𝐴 TC+ 𝑥 ∪ 𝐴)

6-Apr-2026	ttcuniun 37002	Relationship between TC+ 𝐴 and TC+ ∪ 𝐴: we can decompose TC+ 𝐴 into the elements of TC+ ∪ 𝐴 plus the elements of 𝐴 itself. (Contributed by Matthew House, 6-Apr-2026.)
⊢ TC+ 𝐴 = (TC+ ∪ 𝐴 ∪ 𝐴)

6-Apr-2026	csbttc 37001	Distribute proper substitution through a transitive closure. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ⦋𝐴 / 𝑥⦌TC+ 𝐵 = TC+ ⦋𝐴 / 𝑥⦌𝐵

6-Apr-2026	ttc00 37000	A class has an empty transitive closure iff it is the empty set. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 = ∅ ↔ TC+ 𝐴 = ∅)

6-Apr-2026	ttc0 36999	The transitive closure of the empty set is the empty set. (Contributed by Matthew House, 6-Apr-2026.)
⊢ TC+ ∅ = ∅

6-Apr-2026	dfttc2g 36998	A shorter expression for the transitive closure of a set. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ 𝑉 → TC+ 𝐴 = ∪ (rec((𝑥 ∈ V ↦ ∪ 𝑥), 𝐴) “ ω))

6-Apr-2026	elttctr 36997	Transitivity of 𝐴 ∈ TC+ 𝐵 relationship. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ((𝐴 ∈ TC+ 𝐵 ∧ 𝐵 ∈ TC+ 𝐶) → 𝐴 ∈ TC+ 𝐶)

6-Apr-2026	ssttctr 36996	Transitivity of 𝐴 ⊆ TC+ 𝐵 relationship. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ((𝐴 ⊆ TC+ 𝐵 ∧ 𝐵 ⊆ TC+ 𝐶) → 𝐴 ⊆ TC+ 𝐶)

6-Apr-2026	ttcidm 36995	The transitive closure operation is idempotent. (Contributed by Matthew House, 6-Apr-2026.)
⊢ TC+ TC+ 𝐴 = TC+ 𝐴

6-Apr-2026	ttctrid 36994	The transitive closure of a transitive class is the class itself. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (Tr 𝐴 → TC+ 𝐴 = 𝐴)

6-Apr-2026	ttcel2 36993	Elements turn into subclasses upon taking transitive closures. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ 𝐵 → TC+ 𝐴 ⊆ TC+ 𝐵)

6-Apr-2026	ttcel 36992	A transitive closure contains the transitive closures of all its elements. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ TC+ 𝐵 → TC+ 𝐴 ⊆ TC+ 𝐵)

6-Apr-2026	ttcss2 36991	The subclass relationship is inherited by transitive closures. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ⊆ 𝐵 → TC+ 𝐴 ⊆ TC+ 𝐵)

6-Apr-2026	ttcss 36990	A transitive closure contains the transitive closures of all its subclasses. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ⊆ TC+ 𝐵 → TC+ 𝐴 ⊆ TC+ 𝐵)

6-Apr-2026	ttcexrg 36989	If the transitive closure of a class is a set, then the class is a set. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (TC+ 𝐴 ∈ 𝑉 → 𝐴 ∈ V)

6-Apr-2026	ttcmin 36988	The transitive closure of 𝐴 is a subclass of every transitive class containing 𝐴. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → TC+ 𝐴 ⊆ 𝐵)

6-Apr-2026	ttctr3 36987	The transitive closure of a class is transitive. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ∪ TC+ 𝐴 ⊆ TC+ 𝐴

6-Apr-2026	ttctr2 36986	The transitive closure of a class is transitive. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ TC+ 𝐵 → 𝐴 ⊆ TC+ 𝐵)

6-Apr-2026	ttctr 36985	The transitive closure of a class is transitive. (Contributed by Matthew House, 6-Apr-2026.)
⊢ Tr TC+ 𝐴

6-Apr-2026	ttcid 36984	The transitive closure contains its argument as a subclass. (Contributed by Matthew House, 6-Apr-2026.)
⊢ 𝐴 ⊆ TC+ 𝐴

6-Apr-2026	nfttc 36983	Bound-variable hypothesis builder for transitive closure. (Contributed by Matthew House, 6-Apr-2026.)
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥TC+ 𝐴

6-Apr-2026	ttceqd 36982	Equality deduction for transitive closure. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → TC+ 𝐴 = TC+ 𝐵)

6-Apr-2026	ttceqi 36981	Equality inference for transitive closure. (Contributed by Matthew House, 6-Apr-2026.)
⊢ 𝐴 = 𝐵 ⇒ ⊢ TC+ 𝐴 = TC+ 𝐵

6-Apr-2026	ttceq 36980	Equality theorem for transitive closure. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 = 𝐵 → TC+ 𝐴 = TC+ 𝐵)

6-Apr-2026	df-ttc 36979	Transitive closure of a class. Unlike (TC‘𝐴) (see df-tc 9705), this definition works even if 𝐴 or its transitive closure is a proper class. Note that unless we assume Transitive Containment, the transitive closure of a set may be a proper class. If we only assume Regularity, then the class of sets whose transitive closure is a set is precisely the class of well-founded sets, see ttcwf3 37018. (Contributed by Matthew House, 6-Apr-2026.)
⊢ TC+ 𝐴 = ∪ 𝑥 ∈ 𝐴 ∪ (rec((𝑦 ∈ V ↦ ∪ 𝑦), {𝑥}) “ ω)

6-Apr-2026	cttc 36978	Extend class notation with the transitive closure of a class. (Contributed by Matthew House, 6-Apr-2026.)
class TC+ 𝐴

6-Apr-2026	tr0el 36977	Every nonempty transitive class contains the empty set ∅ as an element, a consequence of Regularity and Transitive Containment. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ((𝐴 ≠ ∅ ∧ Tr 𝐴) → ∅ ∈ 𝐴)

6-Apr-2026	tr0elw 36976	Every nonempty transitive set contains the empty set ∅ as an element, a consequence of Regularity. If we assume Transitive Containment, then we can omit the 𝐴 ∈ 𝑉 hypothesis, see tr0el 36977. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅ ∧ Tr 𝐴) → ∅ ∈ 𝐴)

6-Apr-2026	tz9.1tco 36975	Version of tz9.1 9699 derived from ax-tco 36964. (Contributed by Matthew House, 6-Apr-2026.)
⊢ 𝐴 ∈ V ⇒ ⊢ ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦))

6-Apr-2026	tz9.1ctco 36974	Version of tz9.1c 9700 derived from ax-tco 36964. (Contributed by Matthew House, 6-Apr-2026.)
⊢ 𝐴 ∈ V ⇒ ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V

6-Apr-2026	axuntco 36971	Derivation of ax-un 7734 from ax-tco 36964. Use ax-un 7734 instead. (Contributed by Matthew House, 6-Apr-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)

6-Apr-2026	axtcond 36970	A version of the Axiom of Transitive Containment with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2404. (Contributed by Matthew House, 6-Apr-2026.) (New usage is discouraged.)
⊢ ∃𝑦∀𝑧((𝑧 = 𝑥 ∨ 𝑧 ∈ 𝑦) → ∀𝑥(𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦))

6-Apr-2026	axtco2g 36969	Weak form of the Axiom of Transitive Containment using class variables and abbreviations. See ax-tco 36964 for more information. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥))

6-Apr-2026	axtco1g 36968	Strong form of the Axiom of Transitive Containment using class variables and abbreviations. See ax-tco 36964 for more information. (Contributed by Matthew House, 6-Apr-2026.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥(𝐴 ∈ 𝑥 ∧ Tr 𝑥))

6-Apr-2026	axtco1from2 36967	Strong form axtco1 36965 of the Axiom of Transitive Containment, derived from the weak form axtco2 36966. See ax-tco 36964 for more information. As written, the proof uses ax-pr 5406 via el 5421, but we could alternatively use ax-pow 5338 via elALT2 5342. Use axtco1 36965 instead. (Contributed by Matthew House, 6-Apr-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)))

6-Apr-2026	axtco2 36966	Weak form of the Axiom of Transitive Containment. See ax-tco 36964 for more information. In particular, this theorem shows the derivation of the weak form from the strong form. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ∃𝑦∀𝑧((𝑧 = 𝑥 ∨ 𝑧 ∈ 𝑦) → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦))

6-Apr-2026	ax-tco 36964	The Axiom of Transitive Containment of ZF set theory. It was derived as axtco 36963 above and is therefore redundant if we assume ax-ext 2735, ax-rep 5239 and ax-inf2 9611, but we state it as a separate axiom here so that its uses can be identified more easily. It states that a transitive set 𝑦 exists that contains a given set 𝑥. In particular, the transitive closure of 𝑥 is a set, since it is a subset of 𝑦, see df-tc 9705. Traditionally, this statement is not counted as an axiom at all, but as a theorem from Replacement and Infinity. In fact, from the transitive closure of 𝑥 we can construct the set of iterated unions of 𝑥 (and vice versa), and Skolem took the existence of the latter set as a motivation for introducing the Axiom of Replacement. But Transitive Containment is strictly weaker than either of those axioms, so many authors identify it as its own axiom when investigating subsystems of ZF, such as Zermelo set theory or finitist set theory. We follow this separation in order to avoid nonessential usage of the stronger axioms. There are two main versions of this axiom that appear in the literature: the strong form ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ Tr 𝑦), see axtco1 36965 and axtco1g 36968, and the weak form ⊢ ∃𝑦(𝑥 ⊆ 𝑦 ∧ Tr 𝑦), see axtco2 36966 and axtco2g 36969. The weak form follows directly from the strong form, see axtco2 36966. But the strong form only follows from the weak form if we allow el 5421 or one of its variants, see axtco1from2 36967. We take the strong form here as the axiom, since it is slightly shorter when expanded to primitive symbols. Yet the weak form turns out to be more suitable for axtcond 36970 for reasons of syntax. (Contributed by Matthew House, 6-Apr-2026.)
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)))

6-Apr-2026	axtco 36963	Axiom of Transitive Containment, derived as a theorem from ax-ext 2735, ax-rep 5239, and ax-inf2 9611. Use ax-tco 36964 instead. (Contributed by Matthew House, 6-Apr-2026.) (New usage is discouraged.)
⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)))

6-Apr-2026	el 5421	Any set is an element of some other set. See elALT 5425 for a shorter proof using more axioms, and see elALT2 5342 for a proof that uses ax-9 2153 and ax-pow 5338 instead of ax-pr 5406. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) Use ax-pr 5406 instead of ax-9 2153 and ax-pow 5338. (Revised by BTernaryTau, 2-Dec-2024.) (Proof shortened by Matthew House, 6-Apr-2026.)
⊢ ∃𝑦 𝑥 ∈ 𝑦

6-Apr-2026	axprlem1 5396	Lemma for axpr 5400. There exists a set to which all empty sets belong. (Contributed by Rohan Ridenour, 10-Aug-2023.) (Revised by BJ, 13-Aug-2023.) (Proof shortened by Matthew House, 6-Apr-2026.)
⊢ ∃𝑥∀𝑦(∀𝑧 ¬ 𝑧 ∈ 𝑦 → 𝑦 ∈ 𝑥)

5-Apr-2026	nprmmul1 48259	Special factorization of a non-prime integer greater than 3. (Contributed by AV, 5-Apr-2026.)
⊢ (𝑁 ∈ (ℤ_≥‘4) → (𝑁 ∉ ℙ ↔ ∃𝑎 ∈ (2..^𝑁)∃𝑏 ∈ (2..^𝑁)𝑁 = (𝑎 · 𝑏)))

5-Apr-2026	nndivides2 48104	Definition of the divides relation for divisors greater than 1. (Contributed by AV, 5-Apr-2026.)
⊢ ((𝑀 ∈ (2..^𝑁) ∧ 𝑁 ∈ ℕ) → (𝑀 ∥ 𝑁 ↔ ∃𝑛 ∈ (2..^𝑁)(𝑛 · 𝑀) = 𝑁))

5-Apr-2026	nnmul2b 48051	A factor of a product of integers is at least 2 and less then the product iff the second factor is at least 2 and less then the product. (Contributed by AV, 5-Apr-2026.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 · 𝐵) = 𝑁) → (𝐴 ∈ (2..^𝑁) ↔ 𝐵 ∈ (2..^𝑁)))

5-Apr-2026	nnmul2 48050	If one factor of a product of integers is at least 2 and less then the product, so is the second factor. (Contributed by AV, 5-Apr-2026.)
⊢ ((𝐴 ∈ (2..^𝑁) ∧ 𝐵 ∈ ℕ ∧ (𝐴 · 𝐵) = 𝑁) → 𝐵 ∈ (2..^𝑁))

5-Apr-2026	elfzo2nn 48049	A member of a half-open range of integers starting at 2 is a positive integer. (Contributed by AV, 5-Apr-2026.)
⊢ (𝐾 ∈ (2..^𝑁) → 𝐾 ∈ ℕ)

5-Apr-2026	elfz2nn 48042	A member of a finite set of sequential integers starting at 2 is a positive integer. (Contributed by AV, 5-Apr-2026.)
⊢ (𝐾 ∈ (2...𝑁) → 𝐾 ∈ ℕ)

5-Apr-2026	bj-alrimdh 37198	Deduction form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2243 and 19.21h 2322. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 13-May-2011.) State the most general derivable instance. (Revised by BJ, 5-Apr-2026.)
⊢ (𝜑 → ∀𝑥𝜓) & ⊢ (𝜒 → ∀𝑥𝜃) & ⊢ (𝜓 → (𝜃 → 𝜏)) ⇒ ⊢ (𝜑 → (𝜒 → ∀𝑥𝜏))

5-Apr-2026	bj-alimdh 37197	General instance of alimdh 1847. (Contributed by NM, 4-Jan-2002.) State the most general derivable instance. (Revised by BJ, 5-Apr-2026.)
⊢ (𝜑 → ∀𝑥𝜓) & ⊢ (𝜓 → (𝜒 → 𝜃)) ⇒ ⊢ (𝜑 → (∀𝑥𝜒 → ∀𝑥𝜃))

5-Apr-2026	zfrep6 5251	A version of the Axiom of Replacement. Normally 𝜑 would have free variables 𝑥 and 𝑦. Axiom 6 of [Kunen] p. 12. The Separation Scheme ax-sep 5258 cannot be derived from this version and must be stated as a separate axiom in an axiom system (such as Kunen's) that uses this version in place of our ax-rep 5239. (Contributed by NM, 10-Oct-2003.) Shorten proof and reduce axiom dependencies. (Revised by BJ, 5-Apr-2026.)
⊢ (∀𝑥 ∈ 𝑧 ∃!𝑦𝜑 → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑)

5-Apr-2026	replem 5250	A lemma for variants of the axiom of replacement: if we can form the set of images of the functional relation, then we can also form a set containing all its images. The converse requires the axiom of separation. (Contributed by BJ, 5-Apr-2026.)
⊢ ((∀𝑥 ∈ 𝑧 ∃𝑦𝜑 ∧ ∃𝑤∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜑)) → ∃𝑤∀𝑥 ∈ 𝑧 ∃𝑦 ∈ 𝑤 𝜑)

4-Apr-2026	ppi1sum 48366	Value of the prime-counting function pi for 1, according to Ján Mináč. (Contributed by AV, 4-Apr-2026.)
⊢ (π‘1) = Σ𝑘 ∈ ∅ (⌊‘((((!‘(𝑘 − 1)) + 1) / 𝑘) − (⌊‘((!‘(𝑘 − 1)) / 𝑘))))

4-Apr-2026	indprmfz 48365	An indicator function for prime numbers in a finite interval of integers, according to Ján Mináč. (Contributed by AV, 4-Apr-2026.)
⊢ 𝐼 = (2...𝐴) ⇒ ⊢ ((𝟭‘𝐼)‘(𝐼 ∩ ℙ)) = (𝑘 ∈ 𝐼 ↦ (⌊‘((((!‘(𝑘 − 1)) + 1) / 𝑘) − (⌊‘((!‘(𝑘 − 1)) / 𝑘)))))

4-Apr-2026	indprm 48364	An indicator function for prime numbers, according to Ján Mináč. (Contributed by AV, 4-Apr-2026.)
⊢ ((𝟭‘(ℤ_≥‘2))‘ℙ) = (𝑘 ∈ (ℤ_≥‘2) ↦ (⌊‘((((!‘(𝑘 − 1)) + 1) / 𝑘) − (⌊‘((!‘(𝑘 − 1)) / 𝑘)))))

4-Apr-2026	ppivalnnnprmge6 48361	Value of a term of the prime-counting function pi for positive integers, according to Ján Mináč, for a non-prime number greater than 4. (Contributed by AV, 4-Apr-2026.)
⊢ ((𝑁 ∈ (ℤ_≥‘6) ∧ 𝑁 ∉ ℙ) → (⌊‘((((!‘(𝑁 − 1)) + 1) / 𝑁) − (⌊‘((!‘(𝑁 − 1)) / 𝑁)))) = 0)

4-Apr-2026	nprmmul3 48261	Special factorization of a non-prime integer greater than 3. (Contributed by AV, 4-Apr-2026.)
⊢ (𝑁 ∈ (ℤ_≥‘4) → (𝑁 ∉ ℙ ↔ (∃𝑎 ∈ (2..^𝑁)∃𝑏 ∈ (2..^𝑁)(𝑎 < 𝑏 ∧ 𝑁 = (𝑎 · 𝑏)) ∨ ∃𝑎 ∈ (2..^𝑁)𝑁 = (𝑎↑2))))

4-Apr-2026	rerecne0d 43198	The reciprocal of a nonzero number is nonzero. (Contributed by SN, 4-Apr-2026.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≠ 0) ⇒ ⊢ (𝜑 → (1 /_ℝ 𝐴) ≠ 0)

4-Apr-2026	bj-nnf-cbval 37386	Compared with cbvalv1 2373, this saves ax-12 2213. (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → ∀𝑦𝜑) & ⊢ (𝜑 → Ⅎ'𝑦𝜓) & ⊢ (𝜑 → Ⅎ'𝑥𝜒) & ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

4-Apr-2026	bj-nnf-cbvali 37385	Compared with bj-nnf-cbvaliv 37382, replacing the DV condition on 𝑦, 𝜓 with the nonfreeness condition requires ax-11 2192. (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → ∀𝑦𝜑) & ⊢ (𝜑 → Ⅎ'𝑦𝜓) & ⊢ (𝜑 → Ⅎ'𝑥𝜒) & ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))

4-Apr-2026	bj-nnf-cbvaliv 37382	The only DV conditions are those saying that 𝑦 is a fresh variable used to construct 𝜒. (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → Ⅎ'𝑥𝜒) & ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))

4-Apr-2026	bj-nnf-spime 37381	An existential generalization result in deduction form, from ax-1 6-- ax-6 1997, where the only DV condition is on 𝑥, 𝑦, and where 𝑥 should be nonfree in the new proposition 𝜒 (and in the context 𝜑). (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → Ⅎ'𝑥𝜓) & ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (𝜓 → ∃𝑥𝜒))

4-Apr-2026	bj-nnf-spim 37380	A universal specialization result in deduction form, proved from ax-1 6 -- ax-6 1997, where the only DV condition is on 𝑥, 𝑦 and where 𝑥 should be nonfree in the new proposition 𝜒 (and in the context 𝜑). (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → Ⅎ'𝑥𝜒) & ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))

4-Apr-2026	bj-hbex 37318	A more general instance of hbex 2358. (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑥𝜓) ⇒ ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜓)

4-Apr-2026	bj-hbexd 37316	A more general instance of the deduction form of hbex 2358. (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑦𝜓) & ⊢ (𝜓 → (𝜒 → ∀𝑥𝜃)) ⇒ ⊢ (𝜑 → (∃𝑦𝜒 → ∀𝑥∃𝑦𝜃))

4-Apr-2026	bj-hbal 37287	More general instance of hbal 2202. (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑥𝜓) ⇒ ⊢ (∀𝑦𝜑 → ∀𝑥∀𝑦𝜓)

4-Apr-2026	bj-hbald 37285	General statement that hbald 2203 proves . (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑦𝜓) & ⊢ (𝜓 → (𝜒 → ∀𝑥𝜃)) ⇒ ⊢ (𝜑 → (∀𝑦𝜒 → ∀𝑥∀𝑦𝜃))

4-Apr-2026	bj-spim0 37272	A universal specialization result in deduction form, proved from ax-1 6 -- ax-6 1997, where the only DV condition is on 𝑥, 𝑦 and where 𝑥 should be nonfree in the new proposition 𝜒 (and in the context 𝜑). (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → (∃𝑥𝜒 → 𝜒)) & ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))

4-Apr-2026	bj-cbveximdv 37237	A lemma for alpha-renaming of variables bound by an existential quantifier. (Contributed by BJ, 4-Apr-2026.) (Proof modification is discouraged.)
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → ∀𝑦𝜑) & ⊢ (𝜑 → (𝜒 → ∀𝑦𝜒)) & ⊢ (𝜑 → ∀𝑥∃𝑦𝜓) & ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃)) ⇒ ⊢ (𝜑 → (∃𝑥𝜒 → ∃𝑦𝜃))

4-Apr-2026	bj-cbvalimdv 37236	A lemma for alpha-renaming of variables bound by a universal quantifier. (Contributed by BJ, 4-Apr-2026.) (Proof modification is discouraged.)
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → ∀𝑦𝜑) & ⊢ (𝜑 → (∃𝑥𝜃 → 𝜃)) & ⊢ (𝜑 → ∀𝑦∃𝑥𝜓) & ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃)) ⇒ ⊢ (𝜑 → (∀𝑥𝜒 → ∀𝑦𝜃))

4-Apr-2026	bj-cbveximd 37235	A lemma for alpha-renaming of variables bound by an existential quantifier. (Contributed by BJ, 4-Apr-2026.) (Proof modification is discouraged.)
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → ∀𝑦𝜑) & ⊢ (𝜑 → (𝜒 → ∀𝑦𝜒)) & ⊢ (𝜑 → (∃𝑥𝜃 → 𝜃)) & ⊢ (𝜑 → ∀𝑥∃𝑦𝜓) & ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃)) ⇒ ⊢ (𝜑 → (∃𝑥𝜒 → ∃𝑦𝜃))

4-Apr-2026	bj-cbvalimd 37234	A lemma for alpha-renaming of variables bound by a universal quantifier. (Contributed by BJ, 4-Apr-2026.) (Proof modification is discouraged.)
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → ∀𝑦𝜑) & ⊢ (𝜑 → (𝜒 → ∀𝑦𝜒)) & ⊢ (𝜑 → (∃𝑥𝜃 → 𝜃)) & ⊢ (𝜑 → ∀𝑦∃𝑥𝜓) & ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃)) ⇒ ⊢ (𝜑 → (∀𝑥𝜒 → ∀𝑦𝜃))

4-Apr-2026	bj-cbvalimd0 37231	A lemma for alpha-renaming of variables bound by a universal quantifier. In applications, 𝑥 = 𝑦 will be substituted for 𝜓 and ax6ev 1999 will prove Hypothesis bj-cbvalimd0.denote. When ax6ev 1999 is not available but only its universal closure is, then bj-cbvalimd 37234 or bj-cbvalimdv 37236 should be used (see bj-cbvalimdlem 37232, bj-cbval 37249). (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → ∀𝑦𝜑) & ⊢ (𝜑 → (𝜒 → ∀𝑦𝜒)) & ⊢ (𝜑 → (∃𝑥𝜃 → 𝜃)) & ⊢ (𝜑 → ∃𝑥𝜓) & ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃)) ⇒ ⊢ (𝜑 → (∀𝑥𝜒 → ∀𝑦𝜃))

4-Apr-2026	bj-spime 37230	A lemma for existential generalization. In applications, 𝑥 = 𝑦 will be substituted for 𝜓 and ax6ev 1999 will prove Hypothesis bj-spime.denote. (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) & ⊢ (𝜑 → ∃𝑥𝜓) & ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃)) ⇒ ⊢ (𝜑 → (𝜒 → ∃𝑥𝜃))

4-Apr-2026	bj-spim 37229	A lemma for universal specification. In applications, 𝑥 = 𝑦 will be substituted for 𝜓 and ax6ev 1999 will prove Hypothesis bj-spim.denote. (Contributed by BJ, 4-Apr-2026.)
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → (∃𝑥𝜃 → 𝜃)) & ⊢ (𝜑 → ∃𝑥𝜓) & ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃)) ⇒ ⊢ (𝜑 → (∀𝑥𝜒 → 𝜃))

3-Apr-2026	bj-spimenfa 37228	An existential generalization result: if 𝜑 holds and implies 𝜓 for at least one value of 𝑥, and if furthermore 𝑥 is ∀ -weakly nonfree in 𝜑, then 𝜓 holds for at least one value of 𝑥. (Contributed by BJ, 3-Apr-2026.) Proof should not use 19.35 1907. (Proof modification is discouraged.)
⊢ ((𝜑 → ∀𝑥𝜑) → (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓)))

3-Apr-2026	bj-spimnfe 37227	A universal specification result: if 𝜑 is true for all values of 𝑥 and implies 𝜓 for at least one value, and if furthermore 𝑥 is ∃-weakly nonfree in 𝜓, then 𝜓 follows. An intermediate result on the way to prove 19.36i 2267, bj-19.36im 37369, 19.36imv 1975, spimfw 1995... (Contributed by BJ, 3-Apr-2026.) Proof should not use 19.35 1907. (Proof modification is discouraged.)
⊢ ((∃𝑥𝜓 → 𝜓) → (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)))

3-Apr-2026	bj-imim11i 37123	The propositional function ((. → 𝜑) → 𝜓) is increasing. Its associated inference is wl-syls2 38145. (Contributed by BJ, 3-Apr-2026.)
⊢ (𝜑 → 𝜓) ⇒ ⊢ (((𝜑 → 𝜒) → 𝜃) → ((𝜓 → 𝜒) → 𝜃))

3-Apr-2026	bj-imim11 37122	The propositional function ((. → 𝜑) → 𝜓) is increasing. (Contributed by BJ, 3-Apr-2026.)
⊢ ((𝜑 → 𝜓) → (((𝜑 → 𝜒) → 𝜃) → ((𝜓 → 𝜒) → 𝜃)))

2-Apr-2026	hoicvr 47245	𝐼 is a countable set of half-open intervals that covers the whole multidimensional reals. See Definition 1135 (b) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 11-Oct-2020.) Avoid ax-rep 5239 and shorten proof. (Revised by GG, 2-Apr-2026.)
⊢ 𝐼 = (𝑗 ∈ ℕ ↦ (𝑥 ∈ 𝑋 ↦ ⟨-𝑗, 𝑗⟩)) & ⊢ (𝜑 → 𝑋 ∈ Fin) ⇒ ⊢ (𝜑 → (ℝ ↑_m 𝑋) ⊆ ∪ 𝑗 ∈ ℕ X𝑖 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑖))

2-Apr-2026	rerecrecd 43201	A number is equal to the reciprocal of its reciprocal. (Contributed by SN, 2-Apr-2026.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≠ 0) ⇒ ⊢ (𝜑 → (1 /_ℝ (1 /_ℝ 𝐴)) = 𝐴)

2-Apr-2026	sn-redividd 43196	A number divided by itself is 1. (Contributed by SN, 2-Apr-2026.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≠ 0) ⇒ ⊢ (𝜑 → (𝐴 /_ℝ 𝐴) = 1)

2-Apr-2026	sn-rediv0d 43195	Division into zero is zero. (Contributed by SN, 2-Apr-2026.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≠ 0) ⇒ ⊢ (𝜑 → (0 /_ℝ 𝐴) = 0)

2-Apr-2026	sn-rediv1d 43194	A number divided by 1 is itself. (Contributed by SN, 2-Apr-2026.)
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 /_ℝ 1) = 𝐴)

2-Apr-2026	rediveq1d 43193	Equality in terms of unit ratio. (Contributed by SN, 2-Apr-2026.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ≠ 0) ⇒ ⊢ (𝜑 → ((𝐴 /_ℝ 𝐵) = 1 ↔ 𝐴 = 𝐵))

2-Apr-2026	redivne0bd 43192	The ratio of nonzero numbers is nonzero. (Contributed by SN, 2-Apr-2026.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ≠ 0) ⇒ ⊢ (𝜑 → (𝐴 ≠ 0 ↔ (𝐴 /_ℝ 𝐵) ≠ 0))

2-Apr-2026	redivmul2d 43188	Relationship between division and multiplication. (Contributed by SN, 2-Apr-2026.)
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ≠ 0) ⇒ ⊢ (𝜑 → ((𝐴 /_ℝ 𝐶) = 𝐵 ↔ 𝐴 = (𝐶 · 𝐵)))

2-Apr-2026	padct 33044	Index a countable set with integers and pad with 𝑍. (Contributed by Thierry Arnoux, 1-Jun-2020.) Avoid ax-rep 5239. (Revised by GG, 2-Apr-2026.)
⊢ ((𝐴 ≼ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (^◡𝑓 ↾ 𝐴)))

2-Apr-2026	istrkg2ld 28710	Property of fulfilling the lower dimension 2 axiom. (Contributed by Thierry Arnoux, 20-Nov-2019.) Avoid ax-rep 5239. (Revised by GG, 2-Apr-2026.)
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑉 → (𝐺Dim_TarskiG≥2 ↔ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))

2-Apr-2026	smndex1igid 18966	The composition of the modulo function 𝐼 and a constant function (𝐺‘𝐾) results in (𝐺‘𝐾) itself. (Contributed by AV, 14-Feb-2024.) Avoid ax-rep 5239. (Revised by GG, 2-Apr-2026.)
⊢ 𝑀 = (EndoFMnd‘ℕ₀) & ⊢ 𝑁 ∈ ℕ & ⊢ 𝐼 = (𝑥 ∈ ℕ₀ ↦ (𝑥 mod 𝑁)) & ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ₀ ↦ 𝑛)) ⇒ ⊢ (𝐾 ∈ (0..^𝑁) → (𝐼 ∘ (𝐺‘𝐾)) = (𝐺‘𝐾))

2-Apr-2026	smndex1gid 18964	The composition of a constant function (𝐺‘𝐾) with another endofunction on ℕ₀ results in (𝐺‘𝐾) itself. (Contributed by AV, 14-Feb-2024.) Avoid ax-rep 5239. (Revised by GG, 2-Apr-2026.)
⊢ 𝑀 = (EndoFMnd‘ℕ₀) & ⊢ 𝑁 ∈ ℕ & ⊢ 𝐼 = (𝑥 ∈ ℕ₀ ↦ (𝑥 mod 𝑁)) & ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ₀ ↦ 𝑛)) ⇒ ⊢ ((𝐹 ∈ (Base‘𝑀) ∧ 𝐾 ∈ (0..^𝑁)) → ((𝐺‘𝐾) ∘ 𝐹) = (𝐺‘𝐾))

2-Apr-2026	smndex1gbas 18962	The constant functions (𝐺‘𝐾) are endofunctions on ℕ₀. (Contributed by AV, 12-Feb-2024.) Avoid ax-rep 5239 and shorten proof. (Revised by GG, 2-Apr-2026.)
⊢ 𝑀 = (EndoFMnd‘ℕ₀) & ⊢ 𝑁 ∈ ℕ & ⊢ 𝐼 = (𝑥 ∈ ℕ₀ ↦ (𝑥 mod 𝑁)) & ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ₀ ↦ 𝑛)) ⇒ ⊢ (𝐾 ∈ (0..^𝑁) → (𝐺‘𝐾) ∈ (Base‘𝑀))

1-Apr-2026	nowisdomv 30806	One's wisdom on matters of the universe can be refuted on April Fool's day. (Contributed by Prof. Loof Lirpa, 1-Apr-2026.) (New usage is discouraged.)
⊢ ¬ 𝑊⟨“ I 5”⟩dom V

28-Mar-2026	copsex2gd 37763	Implicit substitution inference for ordered pairs. (Contributed by NM, 28-May-1995.) Use a similar proof to copsex4g 5480 to reduce axiom usage. (Revised by SN, 1-Sep-2024.) Adapt copsex2g 5478 $p to deduction form. (Revised by BJ, 28-Mar-2026.) Do not use copsex2g 5478. (Proof modification is discouraged.)
⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒)) ⇒ ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒))

28-Mar-2026	cgsex2gd 37762	Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995.) Adapt cgsex2g 3500 to deduction form. (Revised by BJ, 28-Mar-2026.) Do not use cgsex2g 3500. (Proof modification is discouraged.)
⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝜓) & ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) ⇒ ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (∃𝑥∃𝑦(𝜓 ∧ 𝜒) ↔ 𝜃))

28-Mar-2026	bj-alnnf2 37344	If a proposition holds, then it holds for all values of a given variable if and only if it does not depend on that variable. (Contributed by BJ, 28-Mar-2026.)
⊢ (𝜑 → (∀𝑥𝜑 ↔ Ⅎ'𝑥𝜑))

28-Mar-2026	bj-alnnf 37343	In deduction-style proofs, it is equivalent to assert that the context holds for all values of a variable, or that is does not depend on that variable. (Contributed by BJ, 28-Mar-2026.)
⊢ ((𝜑 → ∀𝑥𝜑) ↔ (𝜑 → Ⅎ'𝑥𝜑))

28-Mar-2026	bj-alsyl 37195	Syllogism under the universal quantifier, in the curried form appearing as Theorem *10.3 of [WhiteheadRussell] p. 145. See alsyl 1923 for the uncurried form. (Contributed by BJ, 28-Mar-2026.)
⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥(𝜓 → 𝜒) → ∀𝑥(𝜑 → 𝜒)))

27-Mar-2026	axnulALT2 35452	Alternate proof of axnul 5269, proved from propositional calculus, ax-gen 1825, ax-4 1839, ax-6 1997, and ax-rep 5239. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by BTernaryTau, 27-Mar-2026.)
⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥

26-Mar-2026	axprALT2 35484	Alternate proof of axpr 5400, proved from predicate calculus, ax-rep 5239, and ax-inf2 9611. (Contributed by BTernaryTau, 26-Mar-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)

21-Mar-2026	bj-nnfbd0 37354	If two formulas are equivalent, then nonfreeness of a variable in one of them is equivalent to nonfreeness in the other, deduction form. The antecedent of the conclusion is in the "strong necessity" modality of modal logic (see also bj-nnftht 37349) in order not to require sp 2219 (modal T). See bj-nnfbi 37353. (Contributed by BJ, 21-Mar-2026.)
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ ((𝜑 ∧ ∀𝑥𝜑) → (Ⅎ'𝑥𝜓 ↔ Ⅎ'𝑥𝜒))

20-Mar-2026	bj-bisimpr 37127	Implication from equivalence with a conjunct. Its associated inference is simprbi 502. (Contributed by BJ, 20-Mar-2026.)
⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) → (𝜑 → 𝜒))

20-Mar-2026	bj-bisimpl 37126	Implication from equivalence with a conjunct. Its associated inference is simplbi 501. (Contributed by BJ, 20-Mar-2026.)
⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) → (𝜑 → 𝜓))

19-Mar-2026	bj-almpig 37194	A partially quantified form of mpi 21 similar to bj-almpi 37193. (Contributed by BJ, 19-Mar-2026.)
⊢ (𝜑 → (𝜒 → 𝜓)) & ⊢ ∀𝑥𝜒 ⇒ ⊢ ∀𝑥(𝜑 → 𝜓)

19-Mar-2026	bj-almpi 37193	A quantified form of mpi 21. See also barbara 2690, bj-ala1i 37192, bj-almp 37185. (Contributed by BJ, 19-Mar-2026.)
⊢ ∀𝑥(𝜑 → (𝜒 → 𝜓)) & ⊢ ∀𝑥𝜒 ⇒ ⊢ ∀𝑥(𝜑 → 𝜓)

19-Mar-2026	bj-alimii 37191	Inference associated with alimi 1841. Double inference associated with alim 1840. The usual proof of an associated inference (here from alimi 1841 and ax-mp 5) has the same size and same number of steps. (Contributed by BJ, 19-Mar-2026.)
⊢ (𝜓 → 𝜑) & ⊢ ∀𝑥𝜓 ⇒ ⊢ ∀𝑥𝜑

19-Mar-2026	bj-almp 37185	A quantified form of ax-mp 5. See also barbara 2690, bj-ala1i 37192, bj-almpi 37193. (Contributed by BJ, 19-Mar-2026.)
⊢ ∀𝑥(𝜓 → 𝜑) & ⊢ ∀𝑥𝜓 ⇒ ⊢ ∀𝑥𝜑

17-Mar-2026	bj-evalf 37697	The evaluation at a class is a function from the universal class into the universal class. (Contributed by BJ, 17-Mar-2026.)
⊢ Slot 𝐴:V⟶V

16-Mar-2026	sin5tlem1 47593	Lemma 1 for quintupled angle sine calculation, expanding triple-angle sine times double-angle cosine. (Contributed by Ender Ting, 16-Mar-2026.)
⊢ (𝑁 ∈ ℂ → (((3 · 𝑁) − (4 · (𝑁↑3))) · (1 − (2 · (𝑁↑2)))) = (((8 · (𝑁↑5)) − (;10 · (𝑁↑3))) + (3 · 𝑁)))

16-Mar-2026	cos3t 47592	Triple-angle formula for cosine, in pure cosine form. (Contributed by Ender Ting, 16-Mar-2026.)
⊢ (𝐴 ∈ ℂ → (cos‘(3 · 𝐴)) = ((4 · ((cos‘𝐴)↑3)) − (3 · (cos‘𝐴))))

16-Mar-2026	sin3t 47591	Triple-angle formula for sine, in pure sine form. (Contributed by Ender Ting, 16-Mar-2026.)
⊢ (𝐴 ∈ ℂ → (sin‘(3 · 𝐴)) = ((3 · (sin‘𝐴)) − (4 · ((sin‘𝐴)↑3))))

16-Mar-2026	esplyfvaln 33945	The last elementary symmetric polynomial is the product of all variables. (Contributed by Thierry Arnoux, 16-Mar-2026.)
⊢ 𝑊 = (𝐼 mPoly 𝑅) & ⊢ 𝑉 = (𝐼 mVar 𝑅) & ⊢ 𝐸 = (𝐼eSymPoly𝑅) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ 𝑁 = (♯‘𝐼) & ⊢ 𝑀 = (mulGrp‘𝑊) ⇒ ⊢ (𝜑 → (𝐸‘𝑁) = (𝑀 Σ_g 𝑉))

16-Mar-2026	esplyfval1 33944	The first elementary symmetric polynomial is the sum of all variables. (Contributed by Thierry Arnoux, 16-Mar-2026.)
⊢ 𝑊 = (𝐼 mPoly 𝑅) & ⊢ 𝑉 = (𝐼 mVar 𝑅) & ⊢ 𝐸 = (𝐼eSymPoly𝑅) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → (𝐸‘1) = (𝑊 Σ_g 𝑉))

16-Mar-2026	mplmonprod 33925	Finite product of monomials. Here the function 𝐺 maps a bag of variables to the corresponding monomial. (Contributed by Thierry Arnoux, 16-Mar-2026.)
⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐷) & ⊢ 1 = (1_r‘𝑅) & ⊢ 0 = (0_g‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑃) & ⊢ 𝐺 = (𝑦 ∈ 𝐷 ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 = 𝑦, 1 , 0 ))) ⇒ ⊢ (𝜑 → (𝑀 Σ_g (𝐺 ∘ 𝐹)) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂ_fld Σ_g (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)‘𝑖))))))

16-Mar-2026	mplgsum 33924	Finite commutative sums of polynomials are taken componentwise. (Contributed by Thierry Arnoux, 16-Mar-2026.)
⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → (𝑃 Σ_g 𝐹) = (𝑦 ∈ 𝐷 ↦ (𝑅 Σ_g (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘)‘𝑦)))))

16-Mar-2026	psrmonprod 33923	Finite product of bags of variables in a power series. Here the function 𝐺 maps a bag of variables to the corresponding monomial. (Contributed by Thierry Arnoux, 16-Mar-2026.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐷) & ⊢ 1 = (1_r‘𝑅) & ⊢ 0 = (0_g‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑆) & ⊢ 𝐺 = (𝑦 ∈ 𝐷 ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 = 𝑦, 1 , 0 ))) ⇒ ⊢ (𝜑 → (𝑀 Σ_g (𝐺 ∘ 𝐹)) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂ_fld Σ_g (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)‘𝑖))))))

16-Mar-2026	psrmonmul2 33922	The product of two power series monomials adds the exponent vectors together. Here, the function 𝐺 is a monomial builder, which maps a bag of variables with the monic monomial with only those variables. (Contributed by Thierry Arnoux, 16-Mar-2026.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 0 = (0_g‘𝑅) & ⊢ 1 = (1_r‘𝑅) & ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ · = (._r‘𝑆) & ⊢ (𝜑 → 𝑌 ∈ 𝐷) & ⊢ 𝐺 = (𝑦 ∈ 𝐷 ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 = 𝑦, 1 , 0 ))) ⇒ ⊢ (𝜑 → ((𝐺‘𝑋) · (𝐺‘𝑌)) = (𝐺‘(𝑋 ∘_f + 𝑌)))

16-Mar-2026	psrmonmul 33921	The product of two power series monomials adds the exponent vectors together. For example, the product of (𝑥↑2)(𝑦↑2) with (𝑦↑1)(𝑧↑3) is (𝑥↑2)(𝑦↑3)(𝑧↑3), where the exponent vectors ⟨2, 2, 0⟩ and ⟨0, 1, 3⟩ are added to give ⟨2, 3, 3⟩. (Contributed by Thierry Arnoux, 16-Mar-2026.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 0 = (0_g‘𝑅) & ⊢ 1 = (1_r‘𝑅) & ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ · = (._r‘𝑆) & ⊢ (𝜑 → 𝑌 ∈ 𝐷) ⇒ ⊢ (𝜑 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘_f + 𝑌), 1 , 0 )))

16-Mar-2026	psrmon 33920	A monomial is a power series. (Contributed by Thierry Arnoux, 16-Mar-2026.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 0 = (0_g‘𝑅) & ⊢ 1 = (1_r‘𝑅) & ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) ⇒ ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ 𝐵)

16-Mar-2026	psrgsum 33919	Finite commutative sums of power series are taken componentwise. (Contributed by Thierry Arnoux, 16-Mar-2026.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → (𝑆 Σ_g 𝐹) = (𝑦 ∈ 𝐷 ↦ (𝑅 Σ_g (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘)‘𝑦)))))

16-Mar-2026	suppgsumssiun 33373	The support of a function defined as a group sum is a subset of the indexed union of the supports. (Contributed by Thierry Arnoux, 16-Mar-2026.)
⊢ 𝑍 = (0_g‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ Mnd) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝑋) ⇒ ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (𝑀 Σ_g (𝑦 ∈ 𝐵 ↦ 𝐶))) supp 𝑍) ⊆ ∪ 𝑦 ∈ 𝐵 ((𝑥 ∈ 𝐴 ↦ 𝐶) supp 𝑍))

15-Mar-2026	goldrasin 47602	Alternative trigonometric formula for the golden ratio. (Contributed by Ender Ting, 15-Mar-2026.)
⊢ 𝐹 = (2 · (cos‘(π / 5))) ⇒ ⊢ 𝐹 = (2 · (sin‘(π · (3 / ;10))))

15-Mar-2026	goldrarr 47601	The golden ratio is a real value. (Contributed by Ender Ting, 15-Mar-2026.)
⊢ 𝐹 = (2 · (cos‘(π / 5))) ⇒ ⊢ 𝐹 ∈ ℝ

14-Mar-2026	bj-axreprepsep 37693	Strong axiom of replacement (universal closure of ax-rep 5239) from the axioms of separation and replacement as written in the theorem's hypotheses. The statement does not require a nonempty universe; most of the proof does not either, except for the use of 19.8a 2217, which could be removed by reworking the proof, since it is applied in a subexpression bound by the variable it introduces. Proof modifications should not introduce steps relying on a nonempty universe, like alrimiv 1957. (Contributed by BJ, 14-Mar-2026.) (Proof modification is discouraged.)
⊢ ∀𝑥∃𝑠∀𝑦(𝑦 ∈ 𝑠 ↔ (𝑦 ∈ 𝑥 ∧ ∃𝑧𝜑)) & ⊢ ∀𝑠(∀𝑦 ∈ 𝑠 ∃!𝑧𝜑 → ∃𝑡∀𝑧(𝑧 ∈ 𝑡 ↔ ∃𝑦 ∈ 𝑠 𝜑)) ⇒ ⊢ ∀𝑥(∀𝑦 ∈ 𝑥 ∃*𝑧𝜑 → ∃𝑡∀𝑧(𝑧 ∈ 𝑡 ↔ ∃𝑦 ∈ 𝑥 𝜑))

14-Mar-2026	bj-axseprep 37692	Axiom of separation (universal closure of ax-sep 5258) from a weak form of the axiom of replacement requiring that the functional relation in it be a (total) function and the weak emptyset axiom (existence of an empty set provided existence of a set), as written in the theorem's hypotheses. This result shows that the weak emptyset axiom is not only the result of a cheap way to avoid an axiom redundancy (in this case, the existence axiom extru 2005) by adding it as an antecedent, but also permits to prove nontrivial results that hold in nonnecessarily nonempty universes. This proof is by cases so is not intuitionistic. The statement does not require a nonempty universe; most of the proof does not either, and the parts that do (e.g., near sb8ef 2387 and sbequ12r 2288 and eueq2 3674) could be reworked to avoid it. Proof modifications should not introduce steps relying on a nonempty universe, like alrimiv 1957. (Contributed by BJ, 14-Mar-2026.) (Proof modification is discouraged.)
⊢ (∃𝑥⊤ → ∃𝑦∀𝑧 ∈ 𝑦 ⊥) & ⊢ ∀𝑥(∀𝑧 ∈ 𝑥 ∃!𝑡𝜓 → ∃𝑦∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 𝜓)) & ⊢ (𝜓 ↔ ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))) ⇒ ⊢ ∀𝑥∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))

14-Mar-2026	bj-rep 37691	Version of the axiom of replacement requiring the functional relation in the axiom to be a (total) function from ax-rep 5239 (in the form of axrep6 5248). (Contributed by BJ, 14-Mar-2026.) The proof proves the statement without the DV condition on 𝑥, 𝜑, but the DV condition is added to this statement to show that this weaker version is sufficient. (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∀𝑥(∀𝑦 ∈ 𝑥 ∃!𝑧𝜑 → ∃𝑡∀𝑧(𝑧 ∈ 𝑡 ↔ ∃𝑦 ∈ 𝑥 𝜑))

14-Mar-2026	bj-cbvaew 37247	Exixtentially quantifying over a non-occurring variable is independent from the variable, under a weaker condition than in bj-cbvexvv 37243. (Contributed by BJ, 14-Mar-2026.) (Proof modification is discouraged.)
⊢ ((∀𝑥𝜑 → ∀𝑦⊥) → (∃𝑦𝜓 → ∃𝑥𝜓))

14-Mar-2026	bj-cbveaw 37246	Universally quantifying over a non-occurring variable is independent from the variable, under a weaker condition than in bj-cbvalvv 37242. (Contributed by BJ, 14-Mar-2026.) (Proof modification is discouraged.)
⊢ ((∃𝑥⊤ → ∃𝑦𝜑) → (∀𝑦𝜓 → ∀𝑥𝜓))

14-Mar-2026	bj-cbvew 37245	Existentially quantifying over a non-occurring variable is independent from the variable, under a weaker condition than in bj-cbvexvv 37243. If ⊤ is substituted for 𝜑, then the statement reads: "existentially quantifying over a non-occurring variable is independent from the variable as soon as that result is true for the True truth constant. The label "cbvew" means "'change bound variable' theorem, 'exists' quantifier, weak version". (Contributed by BJ, 14-Mar-2026.) This proof is intuitionistic. (Proof modification is discouraged.)
⊢ ((∃𝑥⊤ → ∃𝑦𝜑) → (∃𝑥𝜓 → ∃𝑦𝜓))

14-Mar-2026	bj-cbvaw 37244	Universally quantifying over a non-occurring variable is independent from the variable, under a weaker condition than in bj-cbvalvv 37242. If ⊥ is substituted for 𝜑, then the statement reads: "universally quantifying over a non-occurring variable is independent from the variable as soon as that result is true for the False truth constant". The label "cbvaw" means "'change bound variable' theorem, 'all' quantifier, weak version". (Contributed by BJ, 14-Mar-2026.) This proof is not intuitionistic (it uses ja 188); an intuitionistically valid statement is obtained by expressing the antecedent as a disjunction (classically equivalent through imor 866). (Proof modification is discouraged.)
⊢ ((∀𝑥𝜑 → ∀𝑦⊥) → (∀𝑥𝜓 → ∀𝑦𝜓))

14-Mar-2026	bj-exextruan 37241	An equivalent expression for existential quantification over a non-occurring variable proved over ax-1 6-- ax-5 1940. The forward implication can be seen as a strengthening of ax-5 1940 (a conjunct is added to the consequent of the implication). The reverse implication can be strengthened when ax-6 1997 is posited (which implies that models are non-empty), see 19.8v 2013. See bj-alextruim 37240 for a dual statement. An approximate meaning is: the existential quantification of a proposition over a non-occurring variable holds if and only if the proposition holds and the universe is nonempty. (Contributed by BJ, 14-Mar-2026.) (Proof modification is discouraged.)
⊢ (∃𝑥𝜑 ↔ (∃𝑥⊤ ∧ 𝜑))

14-Mar-2026	bj-alextruim 37240	An equivalent expression for universal quantification over a non-occurring variable proved over ax-1 6-- ax-5 1940. The forward implication can be strengthened when ax-6 1997 is posited (which implies that models are non-empty), see spvw 2011. The reverse implication can be seen as a strengthening of ax-5 1940 (since the antecedent of the implication is weakened). See bj-exextruan 37241 for a dual statement. An approximate meaning is: the universal quantification of a proposition over a non-occurring variable holds if and only if the proposition holds in nonempty universes. (Contributed by BJ, 14-Mar-2026.) (Proof modification is discouraged.)
⊢ (∀𝑥𝜑 ↔ (∃𝑥⊤ → 𝜑))

14-Mar-2026	bj-exexalal 37180	A lemma for changing bound variables. Only the forward implication is intuitionistic. (Contributed by BJ, 14-Mar-2026.)
⊢ ((∃𝑥𝜑 → ∃𝑦𝜓) ↔ (∀𝑦 ¬ 𝜓 → ∀𝑥 ¬ 𝜑))

11-Mar-2026	axprglem 5409	Lemma for axprg 5410. (Contributed by GG, 11-Mar-2026.)
⊢ (𝑥 = 𝐴 → ∃𝑧∀𝑤((𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧))

8-Mar-2026	bj-axnul 37690	Over the base theory ax-1 6-- ax-5 1940, the axiom of separation implies the weak emptyset axiom. By "weak emptyset axiom", we mean the axiom asserting existence of an empty set (which can be called "the" empty set when the axiom of extensionality ax-ext 2735 is posited) provided existence of a set (the True truth constant existentially quantified over a fresh variable, extru 2005). This is the conclusion of bj-axnul 37690. Note that the weak emptyset axiom implies ⊢ (∃𝑥⊤ → ∃𝑦⊤) without DV conditions hence also the same statement as the weak emptyset axiom without DV conditions on 𝑥, but only on 𝑦, 𝑧. By "axiom of separation", we mean the universal closure of ax-sep 5258, simulated here by its instance with ⊥ substituted for 𝜑 (and with the variable used to assert existence in the weak emptyset axiom substituted for the containing set) as the hypothesis of bj-axnul 37690. In particular, the axiom of existence extru 2005 and the axiom of separation together imply the emptyset axiom (and conversely, the emptyset axiom implies the axiom of existence). Note: this theorem does not require a disjointness condition on 𝑦, 𝑧, although both axioms should be stated with all variables disjoint. This proof only uses an instance of the axiom of separation with a bounded formula, so is valid in a constructive setting (see the CZF section in the "Intuitionistic Logic Explorer" iset.mm). (Contributed by BJ, 8-Mar-2026.) (Proof modification is discouraged.)
⊢ ∀𝑥∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ ⊥)) ⇒ ⊢ (∃𝑥⊤ → ∃𝑦∀𝑧 ∈ 𝑦 ⊥)

8-Mar-2026	bj-cbvexvv 37243	Existentially quantifying over a non-occurring variable is independent of that variable, over ax-1 6-- ax-5 1940 and the existence axiom extru 2005. See bj-cbvew 37245 for a strengthening. (Contributed by BJ, 8-Mar-2026.) (Proof modification is discouraged.)
⊢ (∃𝑥𝜑 → (∃𝑦𝜓 → ∃𝑥𝜓))

8-Mar-2026	bj-cbvalvv 37242	Universally quantifying over a non-occurring variable is independent of that variable, over ax-1 6-- ax-5 1940 and the existence axiom extru 2005. See bj-cbvaw 37244 for a strengthening. (Contributed by BJ, 8-Mar-2026.) (Proof modification is discouraged.)
⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → ∀𝑦𝜓))

8-Mar-2026	bj-spvew 37239	Version of 19.8v 2013 and 19.9v 2014 proved from ax-1 6-- ax-5 1940. The antecedent can for instance be proved with the existence axiom extru 2005. (Contributed by BJ, 8-Mar-2026.) This could also be proved from bj-spvw 37238 using duality, but that proof would not be intuitionistic, contrary to the present one. (Proof modification is discouraged.)
⊢ (∃𝑥𝜑 → (𝜓 ↔ ∃𝑥𝜓))

8-Mar-2026	bj-spvw 37238	Version of spvw 2011 and 19.3v 2012 proved from ax-1 6-- ax-5 1940. The antecedent can for instance be proved with the existence axiom extru 2005. (Contributed by BJ, 8-Mar-2026.) (Proof modification is discouraged.)
⊢ (∃𝑥𝜑 → (𝜓 ↔ ∀𝑥𝜓))

8-Mar-2026	bj-axdd2ALT 37223	Alternate proof of bj-axdd2 37166 (this should replace bj-axdd2 37166 when bj-exalimi 37219 is moved to the main section). (Contributed by BJ, 8-Mar-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜓))

6-Mar-2026	opex 5447	An ordered pair of classes is a set. Exercise 7 of [TakeutiZaring] p. 16. (Contributed by NM, 18-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.) Avoid ax-nul 5270. (Revised by GG, 6-Mar-2026.)
⊢ ⟨𝐴, 𝐵⟩ ∈ V

6-Mar-2026	snexg 5413	A singleton built on a set is a set. Special case of snex 5412 which is intuitionistically valid. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 19-May-2013.) Extract from snex 5412 and shorten proof. (Revised by BJ, 15-Jan-2025.) (Proof shortened by GG, 6-Mar-2026.)
⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)

6-Mar-2026	snex 5412	A singleton is a set. Theorem 7.12 of [Quine] p. 51, proved using Extensionality, Separation and Pairing. See also snexALT 5356. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 19-May-2013.) Avoid ax-nul 5270 and shorten proof. (Revised by GG, 6-Mar-2026.)
⊢ {𝐴} ∈ V

6-Mar-2026	prex 5411	The Axiom of Pairing using class variables. Theorem 7.13 of [Quine] p. 51. By virtue of its definition, an unordered pair remains a set (even though no longer a pair) even when its components are proper classes (see prprc 4734), so we can dispense with hypotheses requiring them to be sets. (Contributed by NM, 15-Jul-1993.) Avoid ax-nul 5270 and shorten proof. (Revised by GG, 6-Mar-2026.)
⊢ {𝐴, 𝐵} ∈ V

6-Mar-2026	axprg 5410	Derive The Axiom of Pairing with class variables. (Contributed by GG, 6-Mar-2026.)
⊢ ∃𝑧∀𝑤((𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧)

5-Mar-2026	mh-setindnd 37029	A version of mh-setind 37028 with no distinct variable conditions. (Contributed by Matthew House, 5-Mar-2026.) (New usage is discouraged.)
⊢ (∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∀𝑦𝜑)) → 𝜑)

4-Mar-2026	regsfromunir1 37032	Derivation of ax-regs 35520 from unir1 9786. (Contributed by Matthew House, 4-Mar-2026.)
⊢ ∪ (𝑅₁ “ On) = V ⇒ ⊢ (∃𝑥𝜑 → ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))

4-Mar-2026	regsfromsetind 37031	Derivation of ax-regs 35520 from mh-setind 37028. (Contributed by Matthew House, 4-Mar-2026.)
⊢ (∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)) → ¬ 𝜑) ⇒ ⊢ (∃𝑥𝜑 → ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))

4-Mar-2026	regsfromregtco 37030	Derivation of ax-regs 35520 from ax-reg 9555 + ax-tco 36964. (Contributed by Matthew House, 4-Mar-2026.)
⊢ (∃𝑦 𝑦 ∈ 𝑤 → ∃𝑦(𝑦 ∈ 𝑤 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑤))) & ⊢ ∃𝑢(𝑣 ∈ 𝑢 ∧ ∀𝑡(𝑡 ∈ 𝑢 → ∀𝑠(𝑠 ∈ 𝑡 → 𝑠 ∈ 𝑢))) ⇒ ⊢ (∃𝑥𝜑 → ∃𝑦(∀𝑥(𝑥 = 𝑦 → 𝜑) ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ ∀𝑥(𝑥 = 𝑧 → 𝜑))))

4-Mar-2026	mh-setind 37028	Principle of set induction setind 9717, written with primitive symbols. (Contributed by Matthew House, 4-Mar-2026.)
⊢ (∀𝑦(∀𝑥(𝑥 ∈ 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → 𝜑)

27-Feb-2026	eln0s2 28531	A non-negative surreal integer is a surreal ordinal with a finite birthday. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝐴 ∈ ℕ_0s ↔ (𝐴 ∈ On_s ∧ ( bday ‘𝐴) ∈ ω))

27-Feb-2026	peano2n0sd 28505	Peano postulate: the successor of a non-negative surreal integer is a non-negative surreal integer. Deduction form. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ ℕ_0s) ⇒ ⊢ (𝜑 → (𝐴 +_s 1_s ) ∈ ℕ_0s)

27-Feb-2026	divs1d 28379	A surreal divided by one is itself. Deduction version. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ No ) ⇒ ⊢ (𝜑 → (𝐴 /_su 1_s ) = 𝐴)

27-Feb-2026	rightnod 28056	An element of a right set is a surreal. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ ( R ‘𝐵)) ⇒ ⊢ (𝜑 → 𝐴 ∈ No )

27-Feb-2026	rightoldd 28055	An element of a right set is an element of the old set. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ ( R ‘𝐵)) ⇒ ⊢ (𝜑 → 𝐴 ∈ ( O ‘( bday ‘𝐵)))

27-Feb-2026	leftnod 28054	An element of a left set is a surreal. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ ( L ‘𝐵)) ⇒ ⊢ (𝜑 → 𝐴 ∈ No )

27-Feb-2026	leftoldd 28053	An element of a left set is an element of the old set. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ ( L ‘𝐵)) ⇒ ⊢ (𝜑 → 𝐴 ∈ ( O ‘( bday ‘𝐵)))

27-Feb-2026	rightno 28052	An element of a right set is a surreal. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝐴 ∈ ( R ‘𝐵) → 𝐴 ∈ No )

27-Feb-2026	leftno 28051	An element of a left set is a surreal. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝐴 ∈ ( L ‘𝐵) → 𝐴 ∈ No )

27-Feb-2026	rightold 28050	An element of a right set is an element of the old set. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝐴 ∈ ( R ‘𝐵) → 𝐴 ∈ ( O ‘( bday ‘𝐵)))

27-Feb-2026	leftold 28049	An element of a left set is an element of the old set. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝐴 ∈ ( L ‘𝐵) → 𝐴 ∈ ( O ‘( bday ‘𝐵)))

27-Feb-2026	oldmaded 28043	An element of an old set is an element of a made set. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ ( O ‘𝐵)) ⇒ ⊢ (𝜑 → 𝐴 ∈ ( M ‘𝐵))

27-Feb-2026	oldmade 28042	An element of an old set is an element of a made set. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝐴 ∈ ( O ‘𝐵) → 𝐴 ∈ ( M ‘𝐵))

27-Feb-2026	newnod 28022	An element of a new set is a surreal. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ ( N ‘𝐵)) ⇒ ⊢ (𝜑 → 𝐴 ∈ No )

27-Feb-2026	oldnod 28021	An element of an old set is a surreal. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ ( O ‘𝐵)) ⇒ ⊢ (𝜑 → 𝐴 ∈ No )

27-Feb-2026	madenod 28020	An element of a made set is a surreal. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ ( M ‘𝐵)) ⇒ ⊢ (𝜑 → 𝐴 ∈ No )

27-Feb-2026	newno 28019	An element of a new set is a surreal. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝐴 ∈ ( N ‘𝐵) → 𝐴 ∈ No )

27-Feb-2026	oldno 28018	An element of an old set is a surreal. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝐴 ∈ ( O ‘𝐵) → 𝐴 ∈ No )

27-Feb-2026	madeno 28017	An element of a made set is a surreal. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝐴 ∈ ( M ‘𝐵) → 𝐴 ∈ No )

27-Feb-2026	nulsgtsd 27952	The empty set is greater than any set of surreals. Deduction version. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ⊆ No ) ⇒ ⊢ (𝜑 → 𝐴 <<s ∅)

27-Feb-2026	nulsltsd 27951	The empty set is less-than any set of surreals. Deduction version. (Contributed by Scott Fenton, 27-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ⊆ No ) ⇒ ⊢ (𝜑 → ∅ <<s 𝐴)

26-Feb-2026	dfz12s2 28662	The set of dyadic fractions is the same as the old set of ω. (Contributed by Scott Fenton, 26-Feb-2026.)
⊢ ℤ_s[1/2] = ( O ‘ω)

26-Feb-2026	bdayfin 28661	A surreal has a finite birthday iff it is a dyadic fraction. (Contributed by Scott Fenton, 26-Feb-2026.)
⊢ (𝐴 ∈ No → (𝐴 ∈ ℤ_s[1/2] ↔ ( bday ‘𝐴) ∈ ω))

26-Feb-2026	bdayfinlem 28660	Lemma for bdayfin 28661. Handle the non-negative case. (Contributed by Scott Fenton, 26-Feb-2026.)
⊢ ((𝐴 ∈ No ∧ 0_s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → 𝐴 ∈ ℤ_s[1/2])

26-Feb-2026	bdayfinbnd 28643	Given a non-negative integer and a non-negative surreal of lesser or equal birthday, show that the surreal can be expressed as a dyadic fraction with an upper bound on the integer and exponent. This proof follows the proof from Mizar at https://mizar.uwb.edu.pl/version/current/html/surrealn.html. (Contributed by Scott Fenton, 26-Feb-2026.)
⊢ (𝜑 → 𝑁 ∈ ℕ_0s) & ⊢ (𝜑 → 𝑍 ∈ No ) & ⊢ (𝜑 → ( bday ‘𝑍) ⊆ ( bday ‘𝑁)) & ⊢ (𝜑 → 0_s ≤s 𝑍) ⇒ ⊢ (𝜑 → (𝑍 = 𝑁 ∨ ∃𝑥 ∈ ℕ_0s ∃𝑦 ∈ ℕ_0s ∃𝑝 ∈ ℕ_0s (𝑍 = (𝑥 +_s (𝑦 /_su (2_s↑_s𝑝))) ∧ 𝑦 <s (2_s↑_s𝑝) ∧ (𝑥 +_s 𝑝) <s 𝑁)))

26-Feb-2026	bdayfinbndlem2 28642	Lemma for bdayfinbnd 28643. Conduct the induction. (Contributed by Scott Fenton, 26-Feb-2026.)
⊢ (𝑁 ∈ ℕ_0s → ∀𝑧 ∈ No ((( bday ‘𝑧) ⊆ ( bday ‘𝑁) ∧ 0_s ≤s 𝑧) → (𝑧 = 𝑁 ∨ ∃𝑥 ∈ ℕ_0s ∃𝑦 ∈ ℕ_0s ∃𝑝 ∈ ℕ_0s (𝑧 = (𝑥 +_s (𝑦 /_su (2_s↑_s𝑝))) ∧ 𝑦 <s (2_s↑_s𝑝) ∧ (𝑥 +_s 𝑝) <s 𝑁))))

26-Feb-2026	bdayfinbndlem1 28641	Lemma for bdayfinbnd 28643. Show the first half of the inductive step. (Contributed by Scott Fenton, 26-Feb-2026.)
⊢ (𝜑 → 𝑁 ∈ ℕ_0s) & ⊢ (𝜑 → ∀𝑧 ∈ No ((( bday ‘𝑧) ⊆ ( bday ‘𝑁) ∧ 0_s ≤s 𝑧) → (𝑧 = 𝑁 ∨ ∃𝑥 ∈ ℕ_0s ∃𝑦 ∈ ℕ_0s ∃𝑝 ∈ ℕ_0s (𝑧 = (𝑥 +_s (𝑦 /_su (2_s↑_s𝑝))) ∧ 𝑦 <s (2_s↑_s𝑝) ∧ (𝑥 +_s 𝑝) <s 𝑁)))) ⇒ ⊢ (𝜑 → ∀𝑤 ∈ No ((( bday ‘𝑤) ⊆ ( bday ‘(𝑁 +_s 1_s )) ∧ 0_s ≤s 𝑤) → (𝑤 = (𝑁 +_s 1_s ) ∨ ∃𝑎 ∈ ℕ_0s ∃𝑏 ∈ ℕ_0s ∃𝑞 ∈ ℕ_0s (𝑤 = (𝑎 +_s (𝑏 /_su (2_s↑_s𝑞))) ∧ 𝑏 <s (2_s↑_s𝑞) ∧ (𝑎 +_s 𝑞) <s (𝑁 +_s 1_s )))))

25-Feb-2026	dfpeters2 39604	Alternate definition of PetErs in fully modular form. This expands the Ers 𝑛 predicate into: (i) a typedness module ( Rels × CoMembErs ), (ii) an equivalence module for the coset relation ≀ (𝑟 ⋉ (^◡ E ↾ 𝑛)) ∈ EqvRels, (iii) the corresponding quotient-carrier (domain quotient) equation dom ≀ (...) / ≀ (...) = 𝑛. This is the equivalence-side counterpart of the modular decomposition dfpetparts2 39602 on the partition side. (Contributed by Peter Mazsa, 25-Feb-2026.)
⊢ PetErs = ((( Rels × CoMembErs ) ∩ {⟨𝑟, 𝑛⟩ ∣ ≀ (𝑟 ⋉ (^◡ E ↾ 𝑛)) ∈ EqvRels }) ∩ {⟨𝑟, 𝑛⟩ ∣ (dom ≀ (𝑟 ⋉ (^◡ E ↾ 𝑛)) / ≀ (𝑟 ⋉ (^◡ E ↾ 𝑛))) = 𝑛})

25-Feb-2026	dfpet2parts2 39603	Grade stability applied to the decomposed PetParts modules. Pet2Parts is obtained by applying the grade-stability operator SucMap ShiftStable (see df-shiftstable 39112) to the modular intersection from dfpetparts2 39602. This makes the two orthogonal stability axes explicit: (E) semantic stability / equilibrium: BlockLiftFix, (G) grade stability: SucMap ShiftStable, assembled on top of typedness and disjoint-span base modules. This is the principled "extra level" that does not arise for Disjs: disjoint relations already bundle their internal map/carrier consistency via QMap and ElDisjs (see dfdisjs6 39572 / dfdisjs7 39573), while the present construction has an additional external grading axis imposed by the canonical successor map SucMap. (Contributed by Peter Mazsa, 20-Feb-2026.) (Revised by Peter Mazsa, 25-Feb-2026.)
⊢ Pet2Parts = ( SucMap ShiftStable ((( Rels × MembParts ) ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (^◡ E ↾ 𝑛)) ∈ Disjs }) ∩ BlockLiftFix ))

25-Feb-2026	dfpetparts2 39602	Alternate definition of PetParts as typedness + disjoint-span + block-lift equilibrium. This theorem is the key modularization step. It decomposes PetParts into the intersection of three orthogonal modules: (T) typedness: ⟨𝑟, 𝑛⟩ ∈ ( Rels × MembParts ), (D) disjoint-span: (𝑟 ⋉ (^◡ E ↾ 𝑛)) ∈ Disjs, (E) semantic equilibrium: ⟨𝑟, 𝑛⟩ ∈ BlockLiftFix, i.e. the carrier 𝑛 is a fixpoint of the induced block-generation operator. Conceptually, (D) provides the disjointness/quotient discipline for the lifted span, while (E) prevents hidden carrier drift (refinement or coarsening of what counts as a block) by enforcing the fixpoint equation. The point of this theorem is that these constraints can be imposed and reused independently by later constructions, while their intersection recovers the intended Parts-based notion. This mirrors the internal packaging of Disjs (see dfdisjs6 39572 / dfdisjs7 39573): for disjoint relations, the "map layer + carrier layer" decomposition is internal via QMap and ElDisjs; for PetParts, the carrier 𝑛 is an external parameter, so the additional carrier stability must be factored explicitly as BlockLiftFix. (Contributed by Peter Mazsa, 20-Feb-2026.) (Revised by Peter Mazsa, 25-Feb-2026.)
⊢ PetParts = ((( Rels × MembParts ) ∩ {⟨𝑟, 𝑛⟩ ∣ (𝑟 ⋉ (^◡ E ↾ 𝑛)) ∈ Disjs }) ∩ BlockLiftFix )

25-Feb-2026	df-peters 39599	Define the class of equivalence-side general partition-equivalence spans. ⟨𝑟, 𝑛⟩ ∈ PetErs means: (1) 𝑟 is a set-relation (𝑟 ∈ Rels), and (2) 𝑛 is a carrier recognized on the equivalence side of membership (𝑛 ∈ CoMembErs), and (3) the coset relation of the lifted span, ≀ (𝑟 ⋉ (^◡ E ↾ 𝑛)), is an equivalence relation on its natural quotient with carrier 𝑛 (i.e. ≀ (𝑟 ⋉ (^◡ E ↾ 𝑛)) Ers 𝑛). This packages the equivalence-view of the same lifted construction that underlies PetParts. It is designed to be parallel to PetParts so later proofs can freely choose the partition side (Parts) or the equivalence side (Ers) without rebuilding the bridge each time; the identification is provided by petseq 39606 (using typesafepets 39605 and mpets 39586). The explicit typing (𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ) is included for the same reason as in df-petparts 39598: to make typedness a reusable module. (Contributed by Peter Mazsa, 19-Feb-2026.) (Revised by Peter Mazsa, 25-Feb-2026.)
⊢ PetErs = {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ CoMembErs ) ∧ ≀ (𝑟 ⋉ (^◡ E ↾ 𝑛)) Ers 𝑛)}

25-Feb-2026	df-petparts 39598	Define the class of partition-side general partition-equivalence spans. ⟨𝑟, 𝑛⟩ ∈ PetParts means: (1) 𝑟 is a set-relation (𝑟 ∈ Rels), and (2) 𝑛 is a membership block-carrier (𝑛 ∈ MembParts), and (3) the block-lift span (𝑟 ⋉ (^◡ E ↾ 𝑛)) is a generalized partition on its natural quotient-carrier 𝑛 (i.e. (𝑟 ⋉ (^◡ E ↾ 𝑛)) Parts 𝑛). This is the horizontal feasibility base object on the partition side, expressed in the type-safe Parts language. The explicit typing (𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) is included at the definition level so later modular refinements can treat typedness as a first-class component (e.g. intersecting a typedness module with disjointness and equilibrium modules) without repeatedly restating it. In particular, it lets decompositions such as dfpetparts2 39602 be written as clean intersections whose first conjunct is exactly the typedness module ( Rels × MembParts ). (Contributed by Peter Mazsa, 19-Feb-2026.) (Revised by Peter Mazsa, 25-Feb-2026.)
⊢ PetParts = {⟨𝑟, 𝑛⟩ ∣ ((𝑟 ∈ Rels ∧ 𝑛 ∈ MembParts ) ∧ (𝑟 ⋉ (^◡ E ↾ 𝑛)) Parts 𝑛)}

25-Feb-2026	bdayfinbndcbv 28640	Lemma for bdayfinbnd 28643. Change some bound variables. (Contributed by Scott Fenton, 25-Feb-2026.)
⊢ (𝜑 → 𝑁 ∈ ℕ_0s) & ⊢ (𝜑 → ∀𝑧 ∈ No ((( bday ‘𝑧) ⊆ ( bday ‘𝑁) ∧ 0_s ≤s 𝑧) → (𝑧 = 𝑁 ∨ ∃𝑥 ∈ ℕ_0s ∃𝑦 ∈ ℕ_0s ∃𝑝 ∈ ℕ_0s (𝑧 = (𝑥 +_s (𝑦 /_su (2_s↑_s𝑝))) ∧ 𝑦 <s (2_s↑_s𝑝) ∧ (𝑥 +_s 𝑝) <s 𝑁)))) ⇒ ⊢ (𝜑 → ∀𝑤 ∈ No ((( bday ‘𝑤) ⊆ ( bday ‘𝑁) ∧ 0_s ≤s 𝑤) → (𝑤 = 𝑁 ∨ ∃𝑎 ∈ ℕ_0s ∃𝑏 ∈ ℕ_0s ∃𝑞 ∈ ℕ_0s (𝑤 = (𝑎 +_s (𝑏 /_su (2_s↑_s𝑞))) ∧ 𝑏 <s (2_s↑_s𝑞) ∧ (𝑎 +_s 𝑞) <s 𝑁))))

25-Feb-2026	bdaypw2bnd 28639	Birthday bounding rule for non-negative dyadic rationals. (Contributed by Scott Fenton, 25-Feb-2026.)
⊢ (𝜑 → 𝑁 ∈ ℕ_0s) & ⊢ (𝜑 → 𝑋 ∈ ℕ_0s) & ⊢ (𝜑 → 𝑌 ∈ ℕ_0s) & ⊢ (𝜑 → 𝑃 ∈ ℕ_0s) & ⊢ (𝜑 → 𝑌 <s (2_s↑_s𝑃)) & ⊢ (𝜑 → (𝑋 +_s 𝑃) <s 𝑁) ⇒ ⊢ (𝜑 → ( bday ‘(𝑋 +_s (𝑌 /_su (2_s↑_s𝑃)))) ⊆ ( bday ‘𝑁))

25-Feb-2026	onlesd 28444	Less-than or equal is the same as non-strict birthday comparison over surreal ordinals. Deduction version. (Contributed by Scott Fenton, 25-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ On_s) & ⊢ (𝜑 → 𝐵 ∈ On_s) ⇒ ⊢ (𝜑 → (𝐴 ≤s 𝐵 ↔ ( bday ‘𝐴) ⊆ ( bday ‘𝐵)))

25-Feb-2026	onltsd 28443	Less-than is the same as birthday comparison over surreal ordinals. Deduction version. (Contributed by Scott Fenton, 25-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ On_s) & ⊢ (𝜑 → 𝐵 ∈ On_s) ⇒ ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ ( bday ‘𝐴) ∈ ( bday ‘𝐵)))

25-Feb-2026	onles 28442	Less-than or equal is the same as non-strict birthday comparison over surreal ordinals. (Contributed by Scott Fenton, 25-Feb-2026.)
⊢ ((𝐴 ∈ On_s ∧ 𝐵 ∈ On_s) → (𝐴 ≤s 𝐵 ↔ ( bday ‘𝐴) ⊆ ( bday ‘𝐵)))

25-Feb-2026	lestri3d 27904	Trichotomy law for surreal less-than or equal. Deduction version. (Contributed by Scott Fenton, 25-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) ⇒ ⊢ (𝜑 → (𝐴 = 𝐵 ↔ (𝐴 ≤s 𝐵 ∧ 𝐵 ≤s 𝐴)))

25-Feb-2026	lesloed 27903	Surreal less-than or equal in terms of less-than. Deduction version. (Contributed by Scott Fenton, 25-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) ⇒ ⊢ (𝜑 → (𝐴 ≤s 𝐵 ↔ (𝐴 <s 𝐵 ∨ 𝐴 = 𝐵)))

25-Feb-2026	ltsnled 27902	Surreal less-than in terms of less-than or equal. Deduction version. (Contributed by Scott Fenton, 25-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) ⇒ ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ ¬ 𝐵 ≤s 𝐴))

25-Feb-2026	lesnltd 27901	Surreal less-than or equal in terms of less-than. Deduction version. (Contributed by Scott Fenton, 25-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) ⇒ ⊢ (𝜑 → (𝐴 ≤s 𝐵 ↔ ¬ 𝐵 <s 𝐴))

24-Feb-2026	addsge01d 28190	A surreal is less-than or equal to itself plus a non-negative surreal. (Contributed by Scott Fenton, 24-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) ⇒ ⊢ (𝜑 → ( 0_s ≤s 𝐵 ↔ 𝐴 ≤s (𝐴 +_s 𝐵)))

24-Feb-2026	funcnvmpt 6993	Condition for a function in maps-to notation to be single-rooted. (Contributed by Thierry Arnoux, 28-Feb-2017.) (Proof shortened by Peter Mazsa, 24-Feb-2026.)
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐹 & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) ⇒ ⊢ (𝜑 → (Fun ^◡𝐹 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 = 𝐵))

24-Feb-2026	bian1d 590	Adding a superfluous conjunct in a biconditional. (Contributed by Thierry Arnoux, 26-Feb-2017.) (Proof shortened by Hongxiu Chen, 29-Jun-2025.) (Proof shortened by Peter Mazsa, 24-Feb-2026.)
⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃))) ⇒ ⊢ (𝜑 → ((𝜒 ∧ 𝜓) ↔ (𝜒 ∧ 𝜃)))

23-Feb-2026	pw2ltdivmuls2d 28631	Surreal less-than relationship between division and multiplication for powers of two. (Contributed by Scott Fenton, 23-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝑁 ∈ ℕ_0s) ⇒ ⊢ (𝜑 → ((𝐴 /_su (2_s↑_s𝑁)) <s 𝐵 ↔ 𝐴 <s (𝐵 ·_s (2_s↑_s𝑁))))

23-Feb-2026	n0lts1e0 28542	A non-negative surreal integer is less than one iff it is zero. (Contributed by Scott Fenton, 23-Feb-2026.)
⊢ (𝐴 ∈ ℕ_0s → (𝐴 <s 1_s ↔ 𝐴 = 0_s ))

23-Feb-2026	cutminmax 28110	If the left set of 𝑋 has a maximum and the right set of 𝑋 has a minimum, then 𝑋 is equal to the cut of the maximum and the minimum. (Contributed by Scott Fenton, 23-Feb-2026.)
⊢ (𝜑 → 𝐿 ∈ ( L ‘𝑋)) & ⊢ (𝜑 → ∀𝑥 ∈ ( L ‘𝑋)𝑥 ≤s 𝐿) & ⊢ (𝜑 → 𝑅 ∈ ( R ‘𝑋)) & ⊢ (𝜑 → ∀𝑦 ∈ ( R ‘𝑋)𝑅 ≤s 𝑦) ⇒ ⊢ (𝜑 → 𝑋 = ({𝐿} \|s {𝑅}))

23-Feb-2026	sltsbday 28091	Birthday comparison rule for surreals. (Contributed by Scott Fenton, 23-Feb-2026.)
⊢ (𝜑 → 𝐴 = (𝐿 \|s 𝑅)) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐿 <<s {𝐵}) & ⊢ (𝜑 → {𝐵} <<s 𝑅) ⇒ ⊢ (𝜑 → ( bday ‘𝐴) ⊆ ( bday ‘𝐵))

22-Feb-2026	dfblockliftmap 39090	Alternate definition of the block lift map. (Contributed by Peter Mazsa, 29-Jan-2026.) (Revised by Peter Mazsa, 22-Feb-2026.)
⊢ (𝑅 BlockLiftMap 𝐴) = (𝑚 ∈ dom (𝑅 ⋉ (^◡ E ↾ 𝐴)) ↦ [𝑚](𝑅 ⋉ (^◡ E ↾ 𝐴)))

22-Feb-2026	df-blockliftmap 39089	Define the block lift map. Given a relation 𝑅 and a carrier/set 𝐴, we form the block relation (𝑅 ⋉ ^◡ E ) (i.e., "follow both 𝑅 and element"), restricted to 𝐴 (or, equivalently, "follow both 𝑅 and elements-of-A", cf. xrnres2 39056). Then map each domain element 𝑚 to its coset [𝑚] under that restricted block relation. For 𝑚 in the domain, which requires (𝑚 ∈ 𝐴 ∧ 𝑚 ≠ ∅ ∧ [𝑚]𝑅 ≠ ∅) (cf. eldmxrncnvepres 39064), the fiber has the product form [𝑚](𝑅 ⋉ ^◡ E ) = ([𝑚]𝑅 × 𝑚), so the block relation lifts a block 𝑚 to the rectangular grid "external labels × internal members", see dfblockliftmap2 39091. Contrast: while the adjoined lift, via (𝑅 ∪ ^◡ E ), attaches neighbors and members in a single relation (see dfadjliftmap2 39087), the block lift labels each internal member by each external neighbor. For the general case and a two-stage construction (first block lift, then adjoin membership), see the comments to df-adjliftmap 39085. For the equilibrium condition, see df-blockliftfix 39111. (Contributed by Peter Mazsa, 24-Jan-2026.) (Revised by Peter Mazsa, 22-Feb-2026.)
⊢ (𝑅 BlockLiftMap 𝐴) = QMap (𝑅 ⋉ (^◡ E ↾ 𝐴))

22-Feb-2026	dfadjliftmap 39086	Alternate (expanded) definition of the adjoined lift map. (Contributed by Peter Mazsa, 28-Jan-2026.) (Revised by Peter Mazsa, 22-Feb-2026.)
⊢ (𝑅 AdjLiftMap 𝐴) = (𝑚 ∈ dom ((𝑅 ∪ ^◡ E ) ↾ 𝐴) ↦ [𝑚]((𝑅 ∪ ^◡ E ) ↾ 𝐴))

22-Feb-2026	df-adjliftmap 39085	Define the adjoined lift map. Given a relation 𝑅 and a carrier/set 𝐴, we form the adjoined relation (𝑅 ∪ ^◡ E ) (i.e., "follow 𝑅 or follow elements"), restricted to 𝐴, and map each domain element 𝑚 to its coset [𝑚] under that restricted adjoined relation, see its expanded version dfadjliftmap 39086. Thus, for 𝑚 in its domain, we have (𝑚 ∪ [𝑚]𝑅), see dfadjliftmap2 39087. Its key special case is successor: for 𝑅 = I and 𝐴 = dom I, or 𝐴 = V, the adjoined relation is ( I ∪ ^◡ E ), and the coset becomes [𝑚]( I ∪ ^◡ E ) = (𝑚 ∪ {𝑚}). So ( I AdjLiftMap dom I ) or ( I AdjLiftMap V) (see dfsucmap2 39094 and dfsucmap3 39093) are exactly the successor map 𝑚 ↦ suc 𝑚 (cf. dfsucmap4 39095), which is a prerequisite for accepting the adjoining lift as the right generalization of successor. A maximally generic form would be "( R F LiftMap A )" defined as (𝑚 ∈ dom ((𝑅𝐹^◡ E ) ↾ 𝐴) ↦ [𝑚]((𝑅𝐹^◡ E ) ↾ 𝐴)) where 𝐹 is an object-level binary operator on relations (used via df-ov 7415). However, ∪ and ⋉ are introduced in set.mm as class constructors (e.g. df-un 3911), not as an object-level binary function symbol 𝐹 that can be passed as a parameter. To make the generic 𝐹-pattern literally usable, we would need to reify union and ⋉ as function-objects, which is additional infrastructure. To avoid introducing operator-as-function objects solely to support 𝐹, we define: AdjLiftMap directly using df-un 3911, and BlockLiftMap directly using the existing ⋉ constructor dfxrn2 39015, so we treat any "generic 𝐹-LiftMap" as optional future generalization, not a dependency. We prefer to avoid defining too many concepts. For this reason, we will not introduce a named "adjoining relation", a named carrier "adjoining lift" "( R AdjLift A )", in place of ran (𝑅 AdjLiftMap 𝐴), which is (dom ((𝑅 ∪ ^◡ E ) ↾ 𝐴) / ((𝑅 ∪ ^◡ E ) ↾ 𝐴)), cf. dfqs2 8702, or the equilibrium condition "AdjLiftFix" , in place of {⟨𝑟, 𝑎⟩ ∣ (dom ((𝑅 ∪ ^◡ E ) ↾ 𝐴) / ((𝑅 ∪ ^◡ E ) ↾ 𝐴)) = 𝑎} (cf. its analog df-blockliftfix 39111). These are definable by simple expansions and/or domain-quotient theorems when needed. A "two-stage" construction is obtained by first forming the block relation (𝑅 ⋉ ^◡ E ) and then adjoining elements as "BlockAdj" . Combined, it uses the relation ((𝑅 ⋉ ^◡ E ) ∪ ^◡ E ), which for 𝑚 in its domain (𝐴 ∖ {∅}) gives (𝑚 ∪ [𝑚](𝑅 ⋉ ^◡ E )), yielding "BlockAdjLiftMap" (cf. blockadjliftmap 39088) and "BlockAdjLiftFix". We only introduce these if a downstream theorem actually requires them. (Contributed by Peter Mazsa, 24-Jan-2026.) (Revised by Peter Mazsa, 22-Feb-2026.)
⊢ (𝑅 AdjLiftMap 𝐴) = QMap ((𝑅 ∪ ^◡ E ) ↾ 𝐴)

22-Feb-2026	z12bday 28659	A dyadic fraction has a finite birthday. (Contributed by Scott Fenton, 20-Aug-2025.) (Proof shortened by Scott Fenton, 22-Feb-2026.)
⊢ (𝐴 ∈ ℤ_s[1/2] → ( bday ‘𝐴) ∈ ω)

22-Feb-2026	z12bdaylem 28658	Lemma for z12bday 28659. Handle the non-negative case. (Contributed by Scott Fenton, 22-Feb-2026.)
⊢ ((𝐴 ∈ ℤ_s[1/2] ∧ 0_s ≤s 𝐴) → ( bday ‘𝐴) ∈ ω)

22-Feb-2026	z12bdaylem2 28645	Lemma for z12bday 28659. Show the first half of the equality. (Contributed by Scott Fenton, 22-Feb-2026.)
⊢ (𝜑 → 𝑁 ∈ ℕ_0s) & ⊢ (𝜑 → 𝑀 ∈ ℕ_0s) & ⊢ (𝜑 → 𝑃 ∈ ℕ_0s) & ⊢ (𝜑 → ((2_s ·_s 𝑀) +_s 1_s ) <s (2_s↑_s𝑃)) ⇒ ⊢ (𝜑 → ( bday ‘(𝑁 +_s (((2_s ·_s 𝑀) +_s 1_s ) /_su (2_s↑_s𝑃)))) ⊆ ( bday ‘((𝑁 +_s 𝑃) +_s 1_s )))

22-Feb-2026	z12bdaylem1 28644	Lemma for z12bday 28659. Prove an inequality for birthday ordering. (Contributed by Scott Fenton, 22-Feb-2026.)
⊢ (𝜑 → 𝑁 ∈ ℕ_0s) & ⊢ (𝜑 → 𝑀 ∈ ℕ_0s) & ⊢ (𝜑 → 𝑃 ∈ ℕ_0s) & ⊢ (𝜑 → ((2_s ·_s 𝑀) +_s 1_s ) <s (2_s↑_s𝑃)) ⇒ ⊢ (𝜑 → (𝑁 +_s (((2_s ·_s 𝑀) +_s 1_s ) /_su (2_s↑_s𝑃))) ≠ (𝑁 +_s 𝑃))

22-Feb-2026	bdaypw2n0bnd 28638	Upper bound for the birthday of a proper fraction of a power of two. This is actually a strict equality when 𝐴 is odd, but we do not need this for the rest of our development. (Contributed by Scott Fenton, 22-Feb-2026.)
⊢ ((𝐴 ∈ ℕ_0s ∧ 𝑁 ∈ ℕ_0s ∧ 𝐴 <s (2_s↑_s𝑁)) → ( bday ‘(𝐴 /_su (2_s↑_s𝑁))) ⊆ suc ( bday ‘𝑁))

22-Feb-2026	onsbnd2 28456	The surreals of a given birthday are bounded below by the negative of that ordinal. (Contributed by Scott Fenton, 22-Feb-2026.)
⊢ ((𝐴 ∈ On_s ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( -u_s ‘𝐴) ≤s 𝐵)

22-Feb-2026	onsbnd 28455	The surreals of a given birthday are bounded above by that ordinal. (Contributed by Scott Fenton, 22-Feb-2026.)
⊢ ((𝐴 ∈ On_s ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → 𝐵 ≤s 𝐴)

22-Feb-2026	addonbday 28453	The birthday of the sum of two ordinals is the natural sum of their birthdays. (Contributed by Scott Fenton, 22-Feb-2026.)
⊢ ((𝐴 ∈ On_s ∧ 𝐵 ∈ On_s) → ( bday ‘(𝐴 +_s 𝐵)) = (( bday ‘𝐴) +no ( bday ‘𝐵)))

22-Feb-2026	ons2ind 28449	Double induction schema for surreal ordinals. (Contributed by Scott Fenton, 22-Feb-2026.)
⊢ (𝑥 = 𝑥_𝑂 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝑦_𝑂 → (𝜓 ↔ 𝜒)) & ⊢ (𝑥 = 𝑥_𝑂 → (𝜃 ↔ 𝜒)) & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) & ⊢ (𝑦 = 𝐵 → (𝜏 ↔ 𝜂)) & ⊢ ((𝑥 ∈ On_s ∧ 𝑦 ∈ On_s) → ((∀𝑥_𝑂 ∈ On_s ∀𝑦_𝑂 ∈ On_s ((𝑥_𝑂 <s 𝑥 ∧ 𝑦_𝑂 <s 𝑦) → 𝜒) ∧ ∀𝑥_𝑂 ∈ On_s (𝑥_𝑂 <s 𝑥 → 𝜓) ∧ ∀𝑦_𝑂 ∈ On_s (𝑦_𝑂 <s 𝑦 → 𝜃)) → 𝜑)) ⇒ ⊢ ((𝐴 ∈ On_s ∧ 𝐵 ∈ On_s) → 𝜂)

21-Feb-2026	elz12si 28647	Inference form of membership in the dyadic fractions. (Contributed by Scott Fenton, 21-Feb-2026.)
⊢ ((𝐴 ∈ ℤ_s ∧ 𝑁 ∈ ℕ_0s) → (𝐴 /_su (2_s↑_s𝑁)) ∈ ℤ_s[1/2])

21-Feb-2026	bdaypw2n0bndlem 28637	Lemma for bdaypw2n0bnd 28638. Prove the case with a successor. (Contributed by Scott Fenton, 21-Feb-2026.)
⊢ ((𝐴 ∈ ℕ_0s ∧ 𝑁 ∈ ℕ_0s ∧ 𝐴 <s (2_s↑_s(𝑁 +_s 1_s ))) → ( bday ‘(𝐴 /_su (2_s↑_s(𝑁 +_s 1_s )))) ⊆ suc ( bday ‘(𝑁 +_s 1_s )))

21-Feb-2026	pw2divsidd 28630	Identity law for division over powers of two. (Contributed by Scott Fenton, 21-Feb-2026.)
⊢ (𝜑 → 𝑁 ∈ ℕ_0s) ⇒ ⊢ (𝜑 → ((2_s↑_s𝑁) /_su (2_s↑_s𝑁)) = 1_s )

21-Feb-2026	pw2divs0d 28629	Division into zero is zero for a power of two. (Contributed by Scott Fenton, 21-Feb-2026.)
⊢ (𝜑 → 𝑁 ∈ ℕ_0s) ⇒ ⊢ (𝜑 → ( 0_s /_su (2_s↑_s𝑁)) = 0_s )

21-Feb-2026	zcuts0 28582	Either the left or right set of a surreal integer is empty. (Contributed by Scott Fenton, 21-Feb-2026.)
⊢ (𝐴 ∈ ℤ_s → (( L ‘𝐴) = ∅ ∨ ( R ‘𝐴) = ∅))

21-Feb-2026	negright 28233	The right set of the negative of a surreal is the set of negatives of its left set. (Contributed by Scott Fenton, 21-Feb-2026.)
⊢ (𝐴 ∈ No → ( R ‘( -u_s ‘𝐴)) = ( -u_s “ ( L ‘𝐴)))

21-Feb-2026	negleft 28232	The left set of the negative of a surreal is the set of negatives of its right set. (Contributed by Scott Fenton, 21-Feb-2026.)
⊢ (𝐴 ∈ No → ( L ‘( -u_s ‘𝐴)) = ( -u_s “ ( R ‘𝐴)))

20-Feb-2026	df-blockliftfix 39111	Define the equilibrium / fixed-point condition for "block carriers". Start with a candidate block-family 𝑎 (a set whose elements you intend to treat as blocks). Combine it with a relation 𝑟 by forming the block-lift span 𝑇 = (𝑟 ⋉ (^◡ E ↾ 𝑎)). For a block 𝑢 ∈ 𝑎, the fiber [𝑢]𝑇 is the set of all outputs produced from "external targets" of 𝑟 together with "internal members" of 𝑢; in other words, 𝑇 is the mechanism that generates new blocks from old ones. Now apply the standard quotient construction (dom 𝑇 / 𝑇). This produces the family of all T-blocks (the cosets [𝑥]𝑇 of witnesses 𝑥 in the domain of 𝑇). In general, this operation can change your carrier: starting from 𝑎, it may generate a different block-family (dom 𝑇 / 𝑇). The equation (dom (𝑟 ⋉ (^◡ E ↾ 𝑎)) / (𝑟 ⋉ (^◡ E ↾ 𝑎))) = 𝑎 says exactly: if you generate blocks from 𝑎 using the lift determined by 𝑟 (cf. df-blockliftmap 39089), you get back the same 𝑎. So 𝑎 is stable under the block-generation operator induced by 𝑟. This is why it is a genuine fixpoint/equilibrium condition: one application of the "make-the-blocks" operator causes no carrier drift, i.e. no hidden refinement/coarsening of what counts as a block. Here, the quotient (dom 𝑇 / 𝑇) is the standard carrier of 𝑇 -blocks; see dfqs2 8702 for the quotient-as-range viewpoint. This is an untyped equilibrium predicate on pairs ⟨𝑟, 𝑎⟩. No hypothesis 𝑟 ∈ Rels is built into the definition, because the fixpoint equation depends only on those ordered pairs ⟨𝑥, 𝑦⟩ that belong to 𝑟 and hence can witness an atomic instance 𝑥𝑟𝑦; extra non-ordered-pair "junk" elements in 𝑟 are ignored automatically by the relational membership predicate. When later work needs 𝑟 to be relation-typed (e.g. to intersect with ( Rels × V)-style typedness modules, or to apply Rels-based infrastructure uniformly), the additional typing constraint 𝑟 ∈ Rels should be imposed locally as a separate conjunct (rather than being baked into this equilibrium module). (Contributed by Peter Mazsa, 25-Jan-2026.) (Revised by Peter Mazsa, 20-Feb-2026.)
⊢ BlockLiftFix = {⟨𝑟, 𝑎⟩ ∣ (dom (𝑟 ⋉ (^◡ E ↾ 𝑎)) / (𝑟 ⋉ (^◡ E ↾ 𝑎))) = 𝑎}

20-Feb-2026	n0ssoldg 28527	The non-negative surreal integers are a subset of the old set of ω. To avoid the axiom of infinity, we include it as an antecedent. (Contributed by Scott Fenton, 20-Feb-2026.)
⊢ (ω ∈ V → ℕ_0s ⊆ ( O ‘ω))

20-Feb-2026	infinf 10552	Equivalence between two infiniteness criteria for sets. (Contributed by David Moews, 1-May-2017.) (Proof shortened by Scott Fenton, 20-Feb-2026.)
⊢ (𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ Fin ↔ ω ≼ 𝐴))

20-Feb-2026	infinfg 10551	Equivalence between two infiniteness criteria for sets. To avoid the axiom of infinity, we include it as a hypothesis. (Contributed by Scott Fenton, 20-Feb-2026.)
⊢ ((ω ∈ V ∧ 𝐴 ∈ 𝐵) → (¬ 𝐴 ∈ Fin ↔ ω ≼ 𝐴))

19-Feb-2026	pets2eq 39607	Grade-stable generalized partition-equivalence identification. After applying the same grade-stability operator (SucMap ShiftStable) to both sides, the grade-stable pet classes still coincide. Confirms that the grade/tower infrastructure is orthogonal to the partition-vs-equivalence viewpoint: stability is preserved under the PetParts = PetErs identification. This is the level at which we can freely work on whichever side is more convenient (Parts for block discipline, Ers for equivalence reasoning), without changing the stable notion of "pet". (Contributed by Peter Mazsa, 19-Feb-2026.)
⊢ Pet2Parts = Pet2Ers

19-Feb-2026	petseq 39606	Generalized partition-equivalence identification. The partition-side scheme PetParts and the equivalence-side scheme PetErs define the same class of spans (pairs ⟨𝑟, 𝑛⟩). This plays the same organizational role for lifted spans that mpets 39586 plays for carriers: mpets 39586 identifies MembParts with CoMembErs at the membership-carrier level, while petseq 39606 identifies the corresponding span-level predicates built from Parts and Ers. Unlike the earlier broad pets 39596, the bridge used here is the type-safe span theorem typesafepets 39605, which restricts to membership block-carriers. Since typedness (𝑟 ∈ Rels and the appropriate carrier condition) is now built directly into PetParts and PetErs, this theorem can be used downstream without repeatedly re-establishing basic typing premises. (Contributed by Peter Mazsa, 19-Feb-2026.)
⊢ PetParts = PetErs

19-Feb-2026	typesafepets 39605	Type-safe pets 39596 scheme. On a membership block-carrier 𝐴 ∈ MembParts, the lifted span (𝑅 ⋉ (^◡ E ↾ 𝐴)) yields a generalized partition of 𝐴 iff its coset relation yields an equivalence relation on the same carrier 𝐴. This is the type-safe replacement for the earlier broad pets 39596: it explicitly restricts to carriers where 𝐴 is already known to be a block-family (by MembParts). That removes the standard type-safety objection ("are you equating a quotient-carrier of blocks with raw witnesses?") by construction. It is the key bridge used to identify the partition-side and equivalence-side pet classes (petseq 39606), in complete parallel with the membership bridge mpets 39586. This theorem is intentionally not the definition of PetParts; it is the bridge used by petseq 39606 after typedness is enforced by the "Pet*" definitions. (Contributed by Peter Mazsa, 19-Feb-2026.)
⊢ ((𝐴 ∈ MembParts ∧ 𝑅 ∈ 𝑉) → ((𝑅 ⋉ (^◡ E ↾ 𝐴)) Parts 𝐴 ↔ ≀ (𝑅 ⋉ (^◡ E ↾ 𝐴)) Ers 𝐴))

19-Feb-2026	df-pet2ers 39601	Define the class of grade- and blocklift-stable equivalence-side general partition-equivalence spans. The equivalence-side analogue of Pet2Parts: stability of PetErs under one-step grade shift along SucMap. Ensures that the equivalence-side formulation supports the same tower/grade infrastructure as the partition-side formulation. SucMap ShiftStable is the grade axis and does not change the equivalence-vs-partition viewpoint (reinforced by pets2eq 39607). (Contributed by Peter Mazsa, 19-Feb-2026.)
⊢ Pet2Ers = ( SucMap ShiftStable PetErs )

19-Feb-2026	df-pet2parts 39600	Define the class of grade- and blocklift-stable partition-side general partition-equivalence spans. It consists of those ⟨𝑟, 𝑛⟩ ∈ PetParts such that ⟨𝑟, 𝑛⟩ remains in PetParts after shifting one grade along SucMap (via ShiftStable). Concretely: ⟨𝑟, 𝑛⟩ ∈ PetParts and there exists a predecessor 𝑚 with suc 𝑚 = 𝑛 such that ⟨𝑟, 𝑚⟩ ∈ PetParts (encoded by SucMap ∘ PetParts inside ShiftStable). I.e., it introduces the external (tower/grade) stability axis. This is the "4th level" for pet 39595 (see dfpet2parts2 39603): beyond (i) carrier membership partition, (ii) disjointness, and (iii) semantic equilibrium, we require (iv) stability under a canonical grade shift. PetParts already enforces disjointness and the quotient-carrier equation for the lifted span (hence semantic equilibrium via dfpetparts2 39602). Pet2Parts adds the external grade (tower) stability axis via df-shiftstable 39112 with SucMap. This (iv) is why we need explicit second-level Pet2Parts, while Disjs typically does not: Disjs already packages its own internal two-step consistency (carrier + map) by dfdisjs6 39572 / dfdisjs7 39573, whereas pet 39595 has an additional grade axis that must be imposed separately. (Contributed by Peter Mazsa, 19-Feb-2026.)
⊢ Pet2Parts = ( SucMap ShiftStable PetParts )

19-Feb-2026	shiftstableeq2 39113	Equality theorem for shift-stability of two classes. (Contributed by Peter Mazsa, 19-Feb-2026.)
⊢ (𝐹 = 𝐺 → (𝑆 ShiftStable 𝐹) = (𝑆 ShiftStable 𝐺))

19-Feb-2026	ecqmap2 39080	Fiber of QMap equals singleton quotient: a conceptual bridge between "map fibers" and quotients. (Contributed by Peter Mazsa, 19-Feb-2026.)
⊢ (𝐴 ∈ dom 𝑅 → [𝐴] QMap 𝑅 = ({𝐴} / 𝑅))

19-Feb-2026	bj-dfsbc 37255	Proof of df-sbc 3746 when taking bj-df-sb 37253 as definition. (Contributed by BJ, 19-Feb-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ [𝐴 / 𝑥]𝜑)

19-Feb-2026	bj-sbcex 37254	Proof of sbcex 3755 when taking bj-df-sb 37253 as definition. (Contributed by BJ, 19-Feb-2026.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)

19-Feb-2026	bj-df-sb 37253	Proposed definition to replace df-sb 2097 and df-sbc 3746. Proof is therefore unimportant. Contrary to df-sb 2097, this definition makes a substituted formula false when one substitutes a non-existent object for a variable: this is better suited to the "Levy-style" treatment of classes as virtual objects adopted by set.mm. That difference is unimportant since as soon as ax6ev 1999 is posited, all variables "exist". (Contributed by BJ, 19-Feb-2026.)
⊢ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝐴 ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑)))

19-Feb-2026	renod 28667	A surreal real is a surreal number. (Contributed by Scott Fenton, 19-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ ℝ_s) ⇒ ⊢ (𝜑 → 𝐴 ∈ No )

19-Feb-2026	reno 28666	A surreal real is a surreal number. (Contributed by Scott Fenton, 19-Feb-2026.)
⊢ (𝐴 ∈ ℝ_s → 𝐴 ∈ No )

19-Feb-2026	oldfib 28551	The old set of an ordinal is finite iff the ordinal is finite. (Contributed by Scott Fenton, 19-Feb-2026.)
⊢ (𝐴 ∈ On → (𝐴 ∈ ω ↔ ( O ‘𝐴) ∈ Fin))

19-Feb-2026	ordfin 9201	A generalization of onfin 9200 to include the class of all ordinals. (Contributed by Scott Fenton, 19-Feb-2026.)
⊢ (Ord 𝐴 → (𝐴 ∈ Fin ↔ 𝐴 ∈ ω))

18-Feb-2026	suceldisj 39448	Disjointness of successor enforces element-carrier separation: If 𝐵 is the successor of 𝐴 and 𝐵 is element-disjoint as a family, then no element of 𝐴 can itself be a member of 𝐴 (equivalently, every 𝑥 ∈ 𝐴 has empty intersection with the carrier 𝐴). Provides a clean bridge between "disjoint family at the next grade" and "no block contains a block of the same family" at the previous grade: MembPart alone does not enforce this, see dfmembpart2 39503 (it gives disjoint blocks and excludes the empty block, but does not prevent 𝑢 ∈ 𝑚 from also being a member of the carrier 𝑚). This lemma is used to justify when grade-stability (via successor-shift) supplies the extra separation axioms needed in roof/root-style carrier reasoning. (Contributed by Peter Mazsa, 18-Feb-2026.)
⊢ ((𝐴 ∈ 𝑉 ∧ ElDisj 𝐵 ∧ suc 𝐴 = 𝐵) → ∀𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)

18-Feb-2026	wl-eujustlem1 38224	Version of cbvexvw 2067 with references to ax-6 1997 listed as antecedents. (Contributed by Wolf Lammen, 18-Feb-2026.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((∀𝑦∃𝑥 𝑥 = 𝑦 ∧ ∀𝑥∃𝑦 𝑥 = 𝑦) → (∃𝑥𝜑 ↔ ∃𝑦𝜓))

18-Feb-2026	noinfepregs 35527	There are no infinite descending ∈-chains, proven using ax-regs 35520. (Contributed by BTernaryTau, 18-Feb-2026.)
⊢ ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥)

18-Feb-2026	noinfepfnregs 35526	There are no infinite descending ∈-chains, proven using ax-regs 35520. (Contributed by BTernaryTau, 18-Feb-2026.)
⊢ (𝐹 Fn ω → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥))

18-Feb-2026	fineqvinfep 35519	A counterexample demonstrating that tz9.1 9699 does not hold when all sets are finite and an infinite descending ∈-chain exists. (Contributed by BTernaryTau, 18-Feb-2026.)
⊢ 𝐴 = {(𝐹‘∅)} ⇒ ⊢ ((Fin = V ∧ 𝐹:ω–1-1→V ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ¬ ∃𝑦(𝐴 ⊆ 𝑦 ∧ Tr 𝑦))

18-Feb-2026	1reno 28671	Surreal one is a surreal real. (Contributed by Scott Fenton, 18-Feb-2026.)
⊢ 1_s ∈ ℝ_s

16-Feb-2026	dfdisjs7 39573	Alternate definition of the class of disjoints (via carrier disjointness + unique representatives). Ideology-free normal form of dfdisjs6 39572: "blocks cover their elements" (∃) and "each block has a unique generator" (∃!), expressed entirely at the quotient-carrier level. Same class as dfdisjs6 39572, but presented in fully expanded ∃ / ∃! form over the quotient-carrier (dom 𝑟 / 𝑟). Makes explicit (a) element-disjointness of the quotient-carrier and (b) unique representative existence for each block. These are exactly the two conditions that rule out type-confusions (blocks vs witnesses) and ensure canonical decomposition. This is the form that best supports analogy arguments with df-petparts 39598 and with successor-style uniqueness patterns. (Contributed by Peter Mazsa, 16-Feb-2026.)
⊢ Disjs = {𝑟 ∈ Rels ∣ (∀𝑥∃*𝑢 ∈ (dom 𝑟 / 𝑟)𝑥 ∈ 𝑢 ∧ ∀𝑢 ∈ (dom 𝑟 / 𝑟)∃!𝑡 ∈ dom 𝑟 𝑢 = [𝑡]𝑟)}

16-Feb-2026	dfdisjs6 39572	Alternate definition of the class of disjoints (via quotient-map stability). Disjs is the class of relations 𝑟 whose quotient-map QMap 𝑟 is again disjoint and whose induced quotient-carrier is element-disjoint. This is the definitional "stability-by-decomposition" packaging of disjointness: it builds Disjs from two internal layers (i) a carrier-layer constraint and (ii) a map-layer closure constraint. This is deliberately different from "u R x" style definitions: it makes the carrier of blocks and the uniqueness-of-representatives discipline first-class and reusable (via QMap) rather than implicit. (Contributed by Peter Mazsa, 16-Feb-2026.)
⊢ Disjs = {𝑟 ∈ Rels ∣ (ran QMap 𝑟 ∈ ElDisjs ∧ QMap 𝑟 ∈ Disjs )}

16-Feb-2026	eldisjs7 39571	Elementhood in the class of disjoints. 𝑅 ∈ Disjs iff: 𝑅 ∈ Rels, and every 𝑥 belongs to at most one block 𝑢 in the quotient-carrier (dom 𝑅 / 𝑅) (element-disjointness at the carrier), and every block 𝑢 in the quotient-carrier has a unique representative 𝑡 ∈ dom 𝑅 such that 𝑢 = [𝑡]𝑅. Provides the "fully expanded" quantifier characterization of the same decomposition as eldisjs6 39570, but without explicitly mentioning QMap. This is the "E*/E!"" view that is closest in spirit to suc11reg 9589-style injectivity and to the "unique generator per block" narrative. It is also the right contrast-point to older one-line criteria like dfdisjs4 39426 (the "u R x" style), because it makes the carrier and representation discipline explicit and type-safe. (Contributed by Peter Mazsa, 16-Feb-2026.)
⊢ (𝑅 ∈ Disjs ↔ (𝑅 ∈ Rels ∧ (∀𝑥∃*𝑢 ∈ (dom 𝑅 / 𝑅)𝑥 ∈ 𝑢 ∧ ∀𝑢 ∈ (dom 𝑅 / 𝑅)∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅)))

16-Feb-2026	eldisjs6 39570	Elementhood in the class of disjoints. A relation 𝑅 is in Disjs iff: it is relation-typed, and its quotient-map QMap 𝑅 is itself disjoint, and its quotient-carrier ran QMap 𝑅 = (dom 𝑅 / 𝑅) lies in ElDisjs (element-disjoint carriers). This is the central "stability-by-decomposition" theorem for Disjs: it explains why Disjs is internally well-behaved without adding an external stability clause. It is the exact template that PetParts imitates: for pet 39595, the analogue of "map layer" is the disjointness of the lifted span, the analogue of "carrier layer" is the block-lift fixpoint (BlockLiftFix), and then adds external grade stability (SucMap ShiftStable) which Disjs does not need. (Contributed by Peter Mazsa, 16-Feb-2026.)
⊢ (𝑅 ∈ Disjs ↔ (𝑅 ∈ Rels ∧ (ran QMap 𝑅 ∈ ElDisjs ∧ QMap 𝑅 ∈ Disjs )))

16-Feb-2026	rnqmapeleldisjsim 39492	Element-disjointness of the quotient carrier forces coset disjointness. Supplies the "cosets don't overlap unless equal" direction, but expressed via ran QMap 𝑅 (the quotient carrier) and ElDisjs. This is the structural reason Disjs needs a "carrier disjointness" level distinct from the "unique representatives" level. (Contributed by Peter Mazsa, 16-Feb-2026.)
⊢ ((𝑅 ∈ 𝑉 ∧ ran QMap 𝑅 ∈ ElDisjs ∧ (𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅)) → (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ → [𝐴]𝑅 = [𝐵]𝑅))

16-Feb-2026	qmapeldisjsbi 39491	Injectivity of coset map from QMap being disjoint (biconditional form). Convenience version of qmapeldisjsim 39490. (Contributed by Peter Mazsa, 16-Feb-2026.)
⊢ ((𝑅 ∈ 𝑉 ∧ QMap 𝑅 ∈ Disjs ∧ (𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅)) → ([𝐴]𝑅 = [𝐵]𝑅 ↔ 𝐴 = 𝐵))

16-Feb-2026	qmapeldisjsim 39490	Injectivity of coset map from QMap being disjoint (implication form): under the Disjs condition on QMap 𝑅, the coset assignment is injective on dom 𝑅. (Contributed by Peter Mazsa, 16-Feb-2026.)
⊢ ((𝑅 ∈ 𝑉 ∧ QMap 𝑅 ∈ Disjs ∧ (𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅)) → ([𝐴]𝑅 = [𝐵]𝑅 → 𝐴 = 𝐵))

16-Feb-2026	disjimeceqbi2 39437	Injectivity of the block constructor under disjointness. suc11reg 9589 analogue: under disjointness, equal blocks force equal generators (on dom 𝑅). (Contributed by Peter Mazsa, 16-Feb-2026.)
⊢ ( Disj 𝑅 → ((𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅) → ([𝐴]𝑅 = [𝐵]𝑅 ↔ 𝐴 = 𝐵)))

16-Feb-2026	disjimeceqim2 39435	Disj implies injectivity (pairwise form). The same content as disjimeceqim 39434 but packaged for direct use with explicit hypotheses (𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅). (Contributed by Peter Mazsa, 16-Feb-2026.)
⊢ ( Disj 𝑅 → ((𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅) → ([𝐴]𝑅 = [𝐵]𝑅 → 𝐴 = 𝐵)))

16-Feb-2026	falseral0 4476	A false statement can only be true for elements of an empty set. (Contributed by AV, 30-Oct-2020.) (Proof shortened by TM, 16-Feb-2026.)
⊢ ((∀𝑥 ¬ 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜑) → 𝐴 = ∅)

16-Feb-2026	r19.3rzv 4465	Restricted quantification of wff not containing quantified variable. (Contributed by NM, 10-Mar-1997.) Avoid ax-12 2213. (Revised by TM, 16-Feb-2026.)
⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))

15-Feb-2026	eldisjsim5 39569	Disjs is closed under QMap. If a relation is "disjoint-structured" (Disjs), then its canonical block map is also "disjoint-structured". This is the second "structure level" in Disjs: it expresses that the property is stable under passing to the canonical block map, a theme that mirrors Pet-grade stability at a different axis. (Contributed by Peter Mazsa, 15-Feb-2026.)
⊢ (𝑅 ∈ Disjs → QMap 𝑅 ∈ Disjs )

15-Feb-2026	eldisjsim4 39568	Disjs implies element-disjoint range of QMap. Same as eldisjsim3 39567 but expressed using the block-map range ran QMap 𝑅 (often the more modular expression). (Contributed by Peter Mazsa, 15-Feb-2026.)
⊢ (𝑅 ∈ Disjs → ran QMap 𝑅 ∈ ElDisjs )

15-Feb-2026	vieta 33951	Vieta's Formulas: Coefficients of a monic polynomial 𝐹 expressed as a product of linear polynomials of the form 𝑋 − 𝑍 can be expressed in terms of elementary symmetric polynomials. The formulas appear in Chapter 6 of [Lang], p. 190. Theorem vieta1 26454 is a special case for the complex numbers, for the case 𝐾 = 1. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝑊 = (Poly₁‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ − = (-_g‘𝑊) & ⊢ 𝑀 = (mulGrp‘𝑊) & ⊢ 𝑄 = (𝐼 eval 𝑅) & ⊢ 𝐸 = (𝐼eSymPoly𝑅) & ⊢ 𝑁 = (inv_g‘𝑅) & ⊢ 1 = (1_r‘𝑅) & ⊢ · = (._r‘𝑅) & ⊢ 𝑋 = (var₁‘𝑅) & ⊢ 𝐴 = (algSc‘𝑊) & ⊢ ↑ = (._g‘(mulGrp‘𝑅)) & ⊢ 𝐻 = (♯‘𝐼) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝑍:𝐼⟶𝐵) & ⊢ 𝐹 = (𝑀 Σ_g (𝑛 ∈ 𝐼 ↦ (𝑋 − (𝐴‘(𝑍‘𝑛))))) & ⊢ (𝜑 → 𝐾 ∈ (0...𝐻)) & ⊢ 𝐶 = (coe₁‘𝐹) ⇒ ⊢ (𝜑 → (𝐶‘(𝐻 − 𝐾)) = ((𝐾 ↑ (𝑁‘ 1 )) · ((𝑄‘(𝐸‘𝐾))‘𝑍)))

15-Feb-2026	vietalem 33950	Lemma for vieta 33951: induction step. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝑊 = (Poly₁‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ − = (-_g‘𝑊) & ⊢ 𝑀 = (mulGrp‘𝑊) & ⊢ 𝑄 = (𝐼 eval 𝑅) & ⊢ 𝐸 = (𝐼eSymPoly𝑅) & ⊢ 𝑁 = (inv_g‘𝑅) & ⊢ 1 = (1_r‘𝑅) & ⊢ · = (._r‘𝑅) & ⊢ 𝑋 = (var₁‘𝑅) & ⊢ 𝐴 = (algSc‘𝑊) & ⊢ ↑ = (._g‘(mulGrp‘𝑅)) & ⊢ 𝐻 = (♯‘𝐼) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝑍:𝐼⟶𝐵) & ⊢ 𝐹 = (𝑀 Σ_g (𝑛 ∈ 𝐼 ↦ (𝑋 − (𝐴‘(𝑍‘𝑛))))) & ⊢ (𝜑 → 𝐾 ∈ (0...𝐻)) & ⊢ (𝜑 → 𝑌 ∈ 𝐼) & ⊢ 𝐽 = (𝐼 ∖ {𝑌}) & ⊢ (𝜑 → ∀𝑧 ∈ (𝐵 ↑_m 𝐽)∀𝑘 ∈ (0...(♯‘𝐽))((coe₁‘(𝑀 Σ_g (𝑛 ∈ 𝐽 ↦ (𝑋 − (𝐴‘(𝑧‘𝑛))))))‘((♯‘𝐽) − 𝑘)) = ((𝑘 ↑ (𝑁‘ 1 )) · (((𝐽 eval 𝑅)‘((𝐽eSymPoly𝑅)‘𝑘))‘𝑧))) & ⊢ (𝜑 → ((deg₁‘𝑅)‘(𝑀 Σ_g (𝑛 ∈ 𝐽 ↦ (𝑋 − (𝐴‘((𝑍 ↾ 𝐽)‘𝑛)))))) = (♯‘𝐽)) ⇒ ⊢ (𝜑 → ((coe₁‘𝐹)‘𝐾) = (((𝐻 − 𝐾) ↑ (𝑁‘ 1 )) · ((𝑄‘(𝐸‘(𝐻 − 𝐾)))‘𝑍)))

15-Feb-2026	vietadeg1 33949	The degree of a product of 𝐻 of linear polynomials of the form 𝑋 − 𝑍 is 𝐻. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝑊 = (Poly₁‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ − = (-_g‘𝑊) & ⊢ 𝑀 = (mulGrp‘𝑊) & ⊢ 𝑄 = (𝐼 eval 𝑅) & ⊢ 𝐸 = (𝐼eSymPoly𝑅) & ⊢ 𝑁 = (inv_g‘𝑅) & ⊢ 1 = (1_r‘𝑅) & ⊢ · = (._r‘𝑅) & ⊢ 𝑋 = (var₁‘𝑅) & ⊢ 𝐴 = (algSc‘𝑊) & ⊢ ↑ = (._g‘(mulGrp‘𝑅)) & ⊢ 𝐻 = (♯‘𝐼) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝑍:𝐼⟶𝐵) & ⊢ 𝐹 = (𝑀 Σ_g (𝑛 ∈ 𝐼 ↦ (𝑋 − (𝐴‘(𝑍‘𝑛))))) & ⊢ 𝐷 = (deg₁‘𝑅) ⇒ ⊢ (𝜑 → (𝐷‘𝐹) = 𝐻)

15-Feb-2026	esplyfvn 33948	Express the last elementary symmetric polynomial, evaluated at a given set of points 𝑍, in terms of the last elementary symmetric polynomial with one less variable. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+_g‘𝑅) & ⊢ · = (._r‘𝑅) & ⊢ 𝑄 = (𝐼 eval 𝑅) & ⊢ 𝑂 = (𝐽 eval 𝑅) & ⊢ 𝐸 = (𝐼eSymPoly𝑅) & ⊢ 𝐹 = (𝐽eSymPoly𝑅) & ⊢ 𝐻 = (♯‘𝐼) & ⊢ 𝐾 = (♯‘𝐽) & ⊢ 𝐽 = (𝐼 ∖ {𝑌}) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑌 ∈ 𝐼) & ⊢ (𝜑 → 𝑍:𝐼⟶𝐵) ⇒ ⊢ (𝜑 → ((𝑄‘(𝐸‘𝐻))‘𝑍) = ((𝑍‘𝑌) · ((𝑂‘(𝐹‘𝐾))‘(𝑍 ↾ 𝐽))))

15-Feb-2026	esplyindfv 33947	A recursive formula for the elementary symmetric polynomials, evaluated at a given set of points 𝑍. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ · = (._r‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑌 ∈ 𝐼) & ⊢ 𝐽 = (𝐼 ∖ {𝑌}) & ⊢ 𝐸 = (𝐽eSymPoly𝑅) & ⊢ (𝜑 → 𝐾 ∈ (0...(♯‘𝐽))) & ⊢ 𝐶 = {ℎ ∈ (ℕ₀ ↑_m 𝐽) ∣ ℎ finSupp 0} & ⊢ 𝐹 = ((𝐼eSymPoly𝑅)‘(𝐾 + 1)) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑄 = (𝐼 eval 𝑅) & ⊢ 𝑂 = (𝐽 eval 𝑅) & ⊢ + = (+_g‘𝑅) & ⊢ (𝜑 → 𝑍:𝐼⟶𝐵) ⇒ ⊢ (𝜑 → ((𝑄‘𝐹)‘𝑍) = (((𝑍‘𝑌) · ((𝑂‘(𝐸‘𝐾))‘(𝑍 ↾ 𝐽))) + ((𝑂‘(𝐸‘(𝐾 + 1)))‘(𝑍 ↾ 𝐽))))

15-Feb-2026	esplyfval0 33935	The 0-th elementary symmetric polynomial is the constant 1. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝑈 = (1_r‘(𝐼 mPoly 𝑅)) ⇒ ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘0) = 𝑈)

15-Feb-2026	evlextv 33913	Evaluating a variable-extended polynomial is the same as evaluating the polynomial in the original set of variables (in both cases, the additionial variable is ignored). (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝑄 = (𝐼 eval 𝑅) & ⊢ 𝑂 = (𝐽 eval 𝑅) & ⊢ 𝐽 = (𝐼 ∖ {𝑌}) & ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅)) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐸 = (𝐼extendVars𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝑀) & ⊢ (𝜑 → 𝐴:𝐼⟶𝐵) ⇒ ⊢ (𝜑 → ((𝑄‘((𝐸‘𝑌)‘𝐹))‘𝐴) = ((𝑂‘𝐹)‘(𝐴 ↾ 𝐽)))

15-Feb-2026	evlvarval 33912	Polynomial evaluation builder for a variable. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝑄 = (𝐼 eval 𝑆) & ⊢ 𝑃 = (𝐼 mPoly 𝑆) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ ∙ = (._r‘𝑃) & ⊢ · = (._r‘𝑆) & ⊢ (𝜑 → 𝐼 ∈ 𝑍) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑_m 𝐼)) & ⊢ 𝑉 = (𝐼 mVar 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) ⇒ ⊢ (𝜑 → ((𝑉‘𝑋) ∈ 𝐵 ∧ ((𝑄‘(𝑉‘𝑋))‘𝐴) = (𝐴‘𝑋)))

15-Feb-2026	evlscaval 33911	Polynomial evaluation for scalars. See evlsscaval 22258. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝑄 = (𝐼 eval 𝑅) & ⊢ 𝑊 = (𝐼 mPoly 𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐴 = (algSc‘𝑊) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝐿:𝐼⟶𝐵) ⇒ ⊢ (𝜑 → ((𝑄‘(𝐴‘𝑋))‘𝐿) = 𝑋)

15-Feb-2026	gsummoncoe1fz 33869	A coefficient of the polynomial represented as a sum of scaled monomials is the coefficient of the corresponding scaled monomial. See gsummoncoe1fzo 33868. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝑃 = (Poly₁‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑋 = (var₁‘𝑅) & ⊢ ↑ = (._g‘(mulGrp‘𝑃)) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ ∗ = ( ·_𝑠 ‘𝑃) & ⊢ (𝜑 → 𝐷 ∈ ℕ₀) & ⊢ (𝜑 → ∀𝑘 ∈ (0...𝐷)𝐴 ∈ 𝐾) & ⊢ (𝜑 → 𝐿 ∈ (0...𝐷)) & ⊢ (𝑘 = 𝐿 → 𝐴 = 𝐶) ⇒ ⊢ (𝜑 → ((coe₁‘(𝑃 Σ_g (𝑘 ∈ (0...𝐷) ↦ (𝐴 ∗ (𝑘 ↑ 𝑋)))))‘𝐿) = 𝐶)

15-Feb-2026	ply1coedeg 33860	Decompose a univariate polynomial 𝐾 as a sum of powers, up to its degree 𝐷. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝑃 = (Poly₁‘𝑅) & ⊢ 𝑋 = (var₁‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ · = ( ·_𝑠 ‘𝑃) & ⊢ 𝑀 = (mulGrp‘𝑃) & ⊢ ↑ = (._g‘𝑀) & ⊢ 𝐴 = (coe₁‘𝐾) & ⊢ 𝐷 = ((deg₁‘𝑅)‘𝐾) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐾 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝐾 = (𝑃 Σ_g (𝑘 ∈ (0...𝐷) ↦ ((𝐴‘𝑘) · (𝑘 ↑ 𝑋)))))

15-Feb-2026	deg1prod 33854	Degree of a product of polynomials. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝐷 = (deg₁‘𝑅) & ⊢ 𝑃 = (Poly₁‘𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝑀 = (mulGrp‘𝑃) & ⊢ 0 = (0_g‘𝑃) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝐹:𝐴⟶(𝐵 ∖ { 0 })) ⇒ ⊢ (𝜑 → (𝐷‘(𝑀 Σ_g 𝐹)) = Σ𝑘 ∈ 𝐴 (𝐷‘(𝐹‘𝑘)))

15-Feb-2026	assaassrd 33827	Right-associative property of an associative algebra, deduction version. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ · = ( ·_𝑠 ‘𝑊) & ⊢ × = (._r‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ AssAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌)))

15-Feb-2026	assaassd 33826	Left-associative property of an associative algebra, deduction version. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ · = ( ·_𝑠 ‘𝑊) & ⊢ × = (._r‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ AssAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)))

15-Feb-2026	domnprodeq0 33580	A product over a domain is zero exactly when one of the factors is zero. Generalization of domneq0 20794 for any number of factors. See also domnprodn0 33579. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0_g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → ((𝑀 Σ_g 𝐹) = 0 ↔ 0 ∈ ran 𝐹))

15-Feb-2026	ringm1expp1 33534	Ring exponentiation of minus one: Adding one to the exponent is the same as taking the additive inverse. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 1 = (1_r‘𝑅) & ⊢ 𝑁 = (inv_g‘𝑅) & ⊢ ↑ = (._g‘(mulGrp‘𝑅)) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐾 ∈ ℕ₀) ⇒ ⊢ (𝜑 → ((𝐾 + 1) ↑ (𝑁‘ 1 )) = (𝑁‘(𝐾 ↑ (𝑁‘ 1 ))))

15-Feb-2026	ringrngd 33530	A unital ring is a non-unital ring, deduction version. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → 𝑅 ∈ Rng)

15-Feb-2026	gsummulsubdishift2s 33372	Distribute a subtraction over an indexed sum, shift one of the resulting sums, and regroup terms. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+_g‘𝑅) & ⊢ − = (-_g‘𝑅) & ⊢ · = (._r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) & ⊢ (𝜑 → 𝑁 ∈ ℕ₀) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑁)) → 𝑉 ∈ 𝐵) & ⊢ (𝑖 = 0 → 𝑉 = 𝐺) & ⊢ (𝑖 = 𝑁 → 𝑉 = 𝐻) & ⊢ (𝑖 = 𝑘 → 𝑉 = 𝑃) & ⊢ (𝑖 = (𝑘 + 1) → 𝑉 = 𝑄) & ⊢ (𝜑 → 𝐸 = ((𝐺 · 𝐴) − (𝐻 · 𝐶))) & ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝐹 = ((𝑄 · 𝐴) − (𝑃 · 𝐶))) ⇒ ⊢ (𝜑 → ((𝑅 Σ_g (𝑘 ∈ (0...𝑁) ↦ 𝑃)) · (𝐴 − 𝐶)) = ((𝑅 Σ_g (𝑘 ∈ (0..^𝑁) ↦ 𝐹)) + 𝐸))

15-Feb-2026	gsummulsubdishift1s 33371	Distribute a subtraction over an indexed sum, shift one of the resulting sums, and regroup terms. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+_g‘𝑅) & ⊢ − = (-_g‘𝑅) & ⊢ · = (._r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) & ⊢ (𝜑 → 𝑁 ∈ ℕ₀) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑁)) → 𝑉 ∈ 𝐵) & ⊢ (𝑖 = 0 → 𝑉 = 𝐺) & ⊢ (𝑖 = 𝑁 → 𝑉 = 𝐻) & ⊢ (𝑖 = 𝑘 → 𝑉 = 𝑃) & ⊢ (𝑖 = (𝑘 + 1) → 𝑉 = 𝑄) & ⊢ (𝜑 → 𝐸 = ((𝐻 · 𝐴) − (𝐺 · 𝐶))) & ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝐹 = ((𝑃 · 𝐴) − (𝑄 · 𝐶))) ⇒ ⊢ (𝜑 → ((𝑅 Σ_g (𝑘 ∈ (0...𝑁) ↦ 𝑃)) · (𝐴 − 𝐶)) = ((𝑅 Σ_g (𝑘 ∈ (0..^𝑁) ↦ 𝐹)) + 𝐸))

15-Feb-2026	gsummulsubdishift2 33370	Distribute a subtraction over an indexed sum, shift one of the resulting sums, and regroup terms. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+_g‘𝑅) & ⊢ − = (-_g‘𝑅) & ⊢ · = (._r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) & ⊢ (𝜑 → 𝑁 ∈ ℕ₀) & ⊢ (𝜑 → 𝐷:(0...𝑁)⟶𝐵) & ⊢ (𝜑 → 𝐸 = (((𝐷‘0) · 𝐴) − ((𝐷‘𝑁) · 𝐶))) & ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝐹 = (((𝐷‘(𝑘 + 1)) · 𝐴) − ((𝐷‘𝑘) · 𝐶))) ⇒ ⊢ (𝜑 → ((𝑅 Σ_g 𝐷) · (𝐴 − 𝐶)) = ((𝑅 Σ_g (𝑘 ∈ (0..^𝑁) ↦ 𝐹)) + 𝐸))

15-Feb-2026	gsummulsubdishift1 33369	Distribute a subtraction over an indexed sum, shift one of the resulting sums, and regroup terms. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+_g‘𝑅) & ⊢ − = (-_g‘𝑅) & ⊢ · = (._r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) & ⊢ (𝜑 → 𝑁 ∈ ℕ₀) & ⊢ (𝜑 → 𝐷:(0...𝑁)⟶𝐵) & ⊢ (𝜑 → 𝐸 = (((𝐷‘𝑁) · 𝐴) − ((𝐷‘0) · 𝐶))) & ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝐹 = (((𝐷‘𝑘) · 𝐴) − ((𝐷‘(𝑘 + 1)) · 𝐶))) ⇒ ⊢ (𝜑 → ((𝑅 Σ_g 𝐷) · (𝐴 − 𝐶)) = ((𝑅 Σ_g (𝑘 ∈ (0..^𝑁) ↦ 𝐹)) + 𝐸))

15-Feb-2026	gsummptfzsplitla 33360	Split a group sum expressed as mapping with a finite set of sequential integers as domain into two parts, extracting a singleton from the left. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑌 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝑌 = 𝑋) ⇒ ⊢ (𝜑 → (𝐺 Σ_g (𝑘 ∈ (𝑀...𝑁) ↦ 𝑌)) = (𝑋 + (𝐺 Σ_g (𝑘 ∈ ((𝑀 + 1)...𝑁) ↦ 𝑌))))

15-Feb-2026	gsummptfzsplitra 33359	Split a group sum expressed as mapping with a finite set of sequential integers as domain into two parts, extracting a singleton from the right. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑌 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑘 = 𝑁) → 𝑌 = 𝑋) ⇒ ⊢ (𝜑 → (𝐺 Σ_g (𝑘 ∈ (𝑀...𝑁) ↦ 𝑌)) = ((𝐺 Σ_g (𝑘 ∈ (𝑀..^𝑁) ↦ 𝑌)) + 𝑋))

15-Feb-2026	gsummptp1 33358	Reindex a zero-based sum as a one-base sum. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CMnd) & ⊢ (𝜑 → 𝑁 ∈ ℕ₀) & ⊢ ((𝜑 ∧ 𝑙 ∈ (1...𝑁)) → 𝑌 ∈ 𝐵) & ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑙 = (𝑘 + 1)) → 𝑌 = 𝑋) ⇒ ⊢ (𝜑 → (𝑅 Σ_g (𝑘 ∈ (0..^𝑁) ↦ 𝑋)) = (𝑅 Σ_g (𝑙 ∈ (1...𝑁) ↦ 𝑌)))

15-Feb-2026	gsummptrev 33357	Revert ordering in a group sum. See also gsumwrev 19437. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝐵 = (Base‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ CMnd) & ⊢ (𝜑 → 𝑁 ∈ ℕ₀) & ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑋 ∈ 𝐵) & ⊢ (((𝜑 ∧ 𝑙 ∈ (0...𝑁)) ∧ 𝑘 = (𝑁 − 𝑙)) → 𝑋 = 𝑌) ⇒ ⊢ (𝜑 → (𝑀 Σ_g (𝑘 ∈ (0...𝑁) ↦ 𝑋)) = (𝑀 Σ_g (𝑙 ∈ (0...𝑁) ↦ 𝑌)))

15-Feb-2026	gsummptfsres 33355	Extend a finitely supported group sum by padding outside with zeroes. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ 𝑆)) → 𝑌 = 0 ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑌) finSupp 0 ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑆 ⊆ 𝐴) ⇒ ⊢ (𝜑 → (𝐺 Σ_g (𝑥 ∈ 𝐴 ↦ 𝑌)) = (𝐺 Σ_g (𝑥 ∈ 𝑆 ↦ 𝑌)))

15-Feb-2026	ablcomd 33346	An abelian group operation is commutative, deduction version. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Abel) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))

15-Feb-2026	grpinvinvd 33341	Double inverse law for groups. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑁 = (inv_g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑁‘(𝑁‘𝑋)) = 𝑋)

15-Feb-2026	indsn 33164	The indicator function of a singleton. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝑋 ∈ 𝑂) → ((𝟭‘𝑂)‘{𝑋}) = (𝑥 ∈ 𝑂 ↦ if(𝑥 = 𝑋, 1, 0)))

15-Feb-2026	nn0mnfxrd 33077	Nonnegative integers or minus infinity are extended real numbers. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ (𝜑 → 𝐴 ∈ (ℕ₀ ∪ {-∞})) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ^*)

15-Feb-2026	fresunsn 32951	Recover the original function from a point-added function. See also funresdfunsn 7189 and fsnunres 7188. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) = 𝑌) → ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, 𝑌⟩}) = 𝐹)

15-Feb-2026	dfmo 2568	Simplify definition df-mo 2567 by removing its provable hypothesis. (Contributed by Wolf Lammen, 15-Feb-2026.)
⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))

14-Feb-2026	dmqmap 39083	QMap preserves the domain. Confirms that QMap is defined exactly on the points where cosets [𝑥]𝑅 make sense (those in dom 𝑅). (Contributed by Peter Mazsa, 14-Feb-2026.)
⊢ (𝑅 ∈ 𝑉 → dom QMap 𝑅 = dom 𝑅)

14-Feb-2026	ecqmap 39079	QMap fibers are singletons of blocks. Makes QMap behave like a "block constructor function" on dom 𝑅. (Contributed by Peter Mazsa, 14-Feb-2026.)
⊢ (𝐴 ∈ dom 𝑅 → [𝐴] QMap 𝑅 = {[𝐴]𝑅})

14-Feb-2026	dfqmap3 39078	Alternate definition of the quotient map: QMap as ordered-pair class abstraction. Gives the raw set-builder characterization for extensional proofs, Rel proofs (relqmap 39082), and composition/intersection manipulations. (Contributed by Peter Mazsa, 14-Feb-2026.)
⊢ QMap 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ dom 𝑅 ∧ 𝑦 = [𝑥]𝑅)}

14-Feb-2026	dfqmap2 39077	Alternate definition of the quotient map: QMap in image-of-singleton form. (Contributed by Peter Mazsa, 14-Feb-2026.)
⊢ QMap 𝑅 = (𝑥 ∈ dom 𝑅 ↦ (𝑅 “ {𝑥}))

12-Feb-2026	disjqmap 39457	Disjointness of QMap equals unique generation of the quotient carrier. The cleaned, carrier-respecting version of disjqmap2 39456. This is the statement "each equivalence class has a unique representative" for the general coset carrier (dom 𝑅 / 𝑅). (Contributed by Peter Mazsa, 12-Feb-2026.)
⊢ (𝑅 ∈ 𝑉 → ( Disj QMap 𝑅 ↔ ∀𝑢 ∈ (dom 𝑅 / 𝑅)∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅))

12-Feb-2026	disjqmap2 39456	Disjointness of QMap equals ∃-generation. Pairs with disjqmap 39457 and raldmqseu 38995 to move between ∃ and ∃! depending on context. (Contributed by Peter Mazsa, 12-Feb-2026.)
⊢ (𝑅 ∈ 𝑉 → ( Disj QMap 𝑅 ↔ ∀𝑢∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅))

12-Feb-2026	qmapeldisjs 39455	When 𝑅 is a set (e.g., when it is an element of the class of relations df-rels 39070), the quotient map element of the class of disjoint relations and the disjoint relation predicate for quotient maps are the same. (Contributed by Peter Mazsa, 12-Feb-2026.)
⊢ (𝑅 ∈ 𝑉 → ( QMap 𝑅 ∈ Disjs ↔ Disj QMap 𝑅))

12-Feb-2026	rnqmap 39084	The range of the quotient map is the quotient carrier. It lets us replace quotient-carrier reasoning by map/range reasoning (and conversely) via df-qmap 39076 and dfqs2 8702. (Contributed by Peter Mazsa, 12-Feb-2026.)
⊢ ran QMap 𝑅 = (dom 𝑅 / 𝑅)

12-Feb-2026	relqmap 39082	Quotient map is a relation. Guarantees that QMap can be composed, restricted, and used in other relation infrastructure (e.g., membership in Disjs, Rels-based typing). (Contributed by Peter Mazsa, 12-Feb-2026.)
⊢ Rel QMap 𝑅

12-Feb-2026	qmapex 39081	Quotient map exists if 𝑅 exists. Type-safety: ensures QMap is a set under the standard "relation sethood" hypothesis. (Contributed by Peter Mazsa, 12-Feb-2026.)
⊢ (𝑅 ∈ 𝑉 → QMap 𝑅 ∈ V)

12-Feb-2026	df-qmap 39076	Define the quotient map (coset map), see also dfqmap2 39077 and dfqmap3 39078. QMap 𝑅 is the "send a generator / domain element to its 𝑅 -coset" map: it maps each 𝑥 ∈ dom 𝑅 to the block [𝑥]𝑅. Makes the quotient operation / structurally explicit as the range of a canonical map (see dfqs2 8702, rnqmap 39084). This is crucial for (i) modular "two-layer" characterizations (map layer + carrier layer) such as dfdisjs6 39572 / dfdisjs7 39573, (ii) transport of properties between a relation and its induced quotient-carrier (e.g. "elements are blocks" via rnqmap 39084), and (iii) expressing stability/invariance constraints as ordinary conditions on a graph (e.g. ran QMap 𝑟 ∈ ElDisjs, QMap 𝑟 ∈ Disjs). (Contributed by Peter Mazsa, 12-Feb-2026.)
⊢ QMap 𝑅 = (𝑥 ∈ dom 𝑅 ↦ [𝑥]𝑅)

11-Feb-2026	nfale2 42966	An inner existential quantifier's variable is bound. (Contributed by SN, 11-Feb-2026.)
⊢ Ⅎ𝑥∀𝑦∃𝑥𝜑

11-Feb-2026	nfe2 42965	An inner existential quantifier's variable is bound. (Contributed by SN, 11-Feb-2026.)
⊢ Ⅎ𝑥∃𝑦∃𝑥𝜑

11-Feb-2026	nfalh 42964	Version of nfal 2356 with an 'h' hypothesis, avoiding ax-12 2213. (Contributed by SN, 11-Feb-2026.)
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ Ⅎ𝑥∀𝑦𝜑

11-Feb-2026	eldisjsim3 39567	Disjs implies element-disjoint quotient carrier. Exports the carrier-disjointness property in the ElDisjs packaging. (Contributed by Peter Mazsa, 11-Feb-2026.)
⊢ (𝑅 ∈ Disjs → (dom 𝑅 / 𝑅) ∈ ElDisjs )

11-Feb-2026	disjsssrels 39566	The class of disjoint relations is a subclass of the class of relations. (Contributed by Peter Mazsa, 11-Feb-2026.)
⊢ Disjs ⊆ Rels

11-Feb-2026	eldisjsim2 39565	An element of the class of disjoint relations is an element of the class of relations. (Contributed by Peter Mazsa, 11-Feb-2026.)
⊢ (𝑅 ∈ Disjs → 𝑅 ∈ Rels )

11-Feb-2026	eldisjsim1 39564	An element of the class of disjoint relations is disjoint. (Contributed by Peter Mazsa, 11-Feb-2026.)
⊢ (𝑅 ∈ Disjs → Disj 𝑅)

11-Feb-2026	nfexa2 2212	An inner universal quantifier's variable is bound. (Contributed by SN, 11-Feb-2026.)
⊢ Ⅎ𝑥∃𝑦∀𝑥𝜑

11-Feb-2026	nfexhe 2211	Version of nfex 2357 with the existential dual to the 'h' hypothesis, avoiding ax-12 2213. (Contributed by SN, 11-Feb-2026.)
⊢ (∃𝑥𝜑 → 𝜑) ⇒ ⊢ Ⅎ𝑥∃𝑦𝜑

10-Feb-2026	eldisjdmqsim 39447	Shared output implies equal cosets (under ElDisj of quotient): if 𝑢 and 𝑣 both relate to the same 𝑥, then their cosets intersect, hence must coincide under quotient ElDisj. (Contributed by Peter Mazsa, 10-Feb-2026.)
⊢ (( ElDisj (dom 𝑅 / 𝑅) ∧ 𝑅 ∈ Rels ) → ((𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥) → [𝑢]𝑅 = [𝑣]𝑅))

10-Feb-2026	eldisjdmqsim2 39446	ElDisj of quotient implies coset-disjointness (domain form). Converts element-disjointness of the quotient carrier into a usable "cosets don't overlap unless equal" rule. (Contributed by Peter Mazsa, 10-Feb-2026.)
⊢ (( ElDisj (dom 𝑅 / 𝑅) ∧ 𝑅 ∈ Rels ) → ((𝑢 ∈ dom 𝑅 ∧ 𝑣 ∈ dom 𝑅) → (([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → [𝑢]𝑅 = [𝑣]𝑅)))

10-Feb-2026	enssdom 8974	Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998.) (Proof shortened by TM, 10-Feb-2026.)
⊢ ≈ ⊆ ≼

10-Feb-2026	f1oi 6861	A restriction of the identity relation is a one-to-one onto function. (Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) Avoid ax-12 2213. (Revised by TM, 10-Feb-2026.)
⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴

9-Feb-2026	rsp3eq 38997	From a restricted universal statement over 𝐴, specialize to an arbitrary element class, cf. rsp3 38996. (Contributed by Peter Mazsa, 9-Feb-2026.)
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ((𝑦 = 𝐵 ∧ 𝐵 ∈ 𝐴) → 𝜓))

9-Feb-2026	rsp3 38996	From a restricted universal statement over 𝐴, specialize to an arbitrary element 𝑦 ∈ 𝐴, cf. rsp 3253. (Contributed by Peter Mazsa, 9-Feb-2026.)
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (𝑦 ∈ 𝐴 → 𝜓))

6-Feb-2026	eldisjim3 39445	ElDisj elimination (two chosen elements). Standard specialization lemma: from ElDisj 𝐴 infer the disjointness condition for two specific elements. (Contributed by Peter Mazsa, 6-Feb-2026.)
⊢ ( ElDisj 𝐴 → ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → ((𝐵 ∩ 𝐶) ≠ ∅ → 𝐵 = 𝐶)))

6-Feb-2026	raldmqseu 38995	Equivalence between "exactly one" on the quotient carrier and "at most one" globally. Provides a type-safe way to talk about unique representatives either as ∃! on the intended carrier or as a global ∃* statement. (Contributed by Peter Mazsa, 6-Feb-2026.)
⊢ (𝑅 ∈ 𝑉 → (∀𝑢 ∈ (dom 𝑅 / 𝑅)∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅 ↔ ∀𝑢∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅))

6-Feb-2026	raldmqsmo 38993	On the quotient carrier, "at most one" and "exactly one" coincide for coset witnesses. (Contributed by Peter Mazsa, 6-Feb-2026.)
⊢ (∀𝑢 ∈ (dom 𝑅 / 𝑅)∃*𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅 ↔ ∀𝑢 ∈ (dom 𝑅 / 𝑅)∃!𝑡 ∈ dom 𝑅 𝑢 = [𝑡]𝑅)

5-Feb-2026	disjimeldisjdmqs 39563	Disj implies element-disjoint quotient carrier. Supplies the carrier-disjointness half of the Disjs pattern: under Disj 𝑅, the coset family is element-disjoint. (Contributed by Peter Mazsa, 5-Feb-2026.)
⊢ ( Disj 𝑅 → ElDisj (dom 𝑅 / 𝑅))

5-Feb-2026	disjimdmqseq 39439	Disjointness implies unique-generation of quotient blocks. Converts existence-quotient comprehension (see df-qs 8701) into a uniqueness-comprehension under disjointness; rewrites (dom 𝑅 / 𝑅) carriers as exactly the class of blocks with a unique representative. This is the "unique generator per block" content in a carrier-normal form. (Contributed by Peter Mazsa, 5-Feb-2026.)
⊢ ( Disj 𝑅 → (dom 𝑅 / 𝑅) = {𝑡 ∣ ∃!𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅})

5-Feb-2026	disjimrmoeqec 39438	Under Disj, every block has a unique generator (∃* form). If 𝑡 is a block in the quotient sense, then there is a uniquely determined 𝑢 in dom 𝑅 such that 𝑡 = [𝑢]𝑅. This is the existence+uniqueness engine behind Disjs and QMap characterizations: it is the "representative theorem" from which the ∃! forms are obtained. (Contributed by Peter Mazsa, 5-Feb-2026.)
⊢ ( Disj 𝑅 → ∃*𝑢 ∈ dom 𝑅 𝑡 = [𝑢]𝑅)

5-Feb-2026	dfsb 2098	Simplify definition df-sb 2097 by removing its provable hypothesis. (Contributed by Wolf Lammen, 5-Feb-2026.)
⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

4-Feb-2026	ss2rabd 4027	Subclass of a restricted class abstraction (deduction form). Saves ax-10 2176, ax-11 2192, ax-12 2213 over using ss2rab 4024 and sylibr 237. (Contributed by SN, 4-Feb-2026.)
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜒})

4-Feb-2026	sbt 2100	A substitution into a theorem yields a theorem. See sbtALT 2103 for a shorter proof requiring more axioms. See chvar 2427 and chvarv 2428 for versions using implicit substitution. (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 20-Jul-2018.) Revise df-sb 2097. (Revised by Steven Nguyen, 6-Jul-2023.) Revise df-sb 2097 again. (Revised by Wolf Lammen, 4-Feb-2026.)
⊢ 𝜑 ⇒ ⊢ [𝑡 / 𝑥]𝜑

4-Feb-2026	sbtlem 2099	In the case of sbt 2100, the hypothesis in df-sb 2097 is derivable from propositional axioms and ax-gen 1825 alone. The essential proof step is presented in this lemma. (Contributed by Wolf Lammen, 4-Feb-2026.)
⊢ 𝜑 ⇒ ⊢ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))

4-Feb-2026	df-sb 2097	Define proper substitution. For our notation, we use [𝑡 / 𝑥]𝜑 to mean "the wff that results from the proper substitution of 𝑡 for 𝑥 in the wff 𝜑". That is, 𝑡 properly replaces 𝑥. For example, [𝑡 / 𝑥]𝑧 ∈ 𝑥 is the same as 𝑧 ∈ 𝑡 (when 𝑥 and 𝑧 are distinct), as shown in elsb2 2160. Our notation was introduced in Haskell B. Curry's Foundations of Mathematical Logic (1977), p. 316 and is frequently used in textbooks of lambda calculus and combinatory logic. This notation improves the common but ambiguous notation, "𝜑(𝑡) is the wff that results when 𝑡 is properly substituted for 𝑥 in 𝜑(𝑥)". For example, if the original 𝜑(𝑥) is 𝑥 = 𝑡, then 𝜑(𝑡) is 𝑡 = 𝑡, from which we obtain that 𝜑(𝑥) is 𝑥 = 𝑥. So what exactly does 𝜑(𝑥) mean? Curry's notation solves this problem. A very similar notation, namely (𝑦 ∣ 𝑥)𝜑, was introduced in Bourbaki's Set Theory (Chapter 1, Description of Formal Mathematic, 1953). In most books, proper substitution has a somewhat complicated recursive definition with multiple cases based on the occurrences of free and bound variables in the wff. Instead, we use a single formula that is exactly equivalent and gives us a direct definition. We later prove that our definition has the properties we expect of proper substitution (see Theorems sbequ 2117, sbcom2 2207 and sbid2v 2541). Note that our definition is valid even when 𝑥 and 𝑡 are replaced with the same variable, as sbid 2291 shows. We achieve this by applying twice Tarski's definition sb6 2119 which is valid for disjoint variables, and introducing a dummy variable 𝑦 which isolates 𝑥 from 𝑡, as in dfsb7 2314 with respect to sb5 2311. We can also achieve this by having 𝑥 free in the first conjunct and bound in the second, as the alternate definition dfsb1 2513 shows. Another version that mixes free and bound variables is dfsb3 2526. When 𝑥 and 𝑡 are distinct, we can express proper substitution with the simpler expressions of sb5 2311 and sb6 2119. Note that the occurrences of a given variable in the definiens are either all bound (𝑥, 𝑦) or all free (𝑡). Also note that the definiens uses only primitive symbols. This double level definition will make several proofs using it appear as doubled. Alternately, one could often first prove as a lemma the same theorem with a disjoint variable condition on the substitute and the substituted variables, and then prove the original theorem by applying this lemma twice in a row. The hypothesis asserts that the definition is independent of the particular choice of the dummy variable 𝑦. Without this hypothesis, sbjust 2095 would be derivable from propositional axioms alone: one could apply the definiens for [𝑡 / 𝑥]𝜑 twice, using different dummy variables 𝑦 and 𝑧, and then invoke bitr3i 280 to establish their equivalence. This would jeopardize the independence of axioms, as demonstrated in an analoguous situation involving df-ss 3923 to prove ax-8 2145 (see in-ax8 36717). Prefer dfsb 2098 unless you can prove the hypothesis from fewer axioms in special cases, see sbt 2100. (Contributed by NM, 10-May-1993.) Revised from the original definition dfsb1 2513. (Revised by BJ, 22-Dec-2020.) Add the justification hypothesis. (Revised by Wolf Lammen, 4-Feb-2026.)
⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑧(𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) ⇒ ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

3-Feb-2026	disjimeceqbi 39436	Disj gives biconditional injectivity (domain-wise). Strengthens injectivity to an iff. (Contributed by Peter Mazsa, 3-Feb-2026.)
⊢ ( Disj 𝑅 → ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 ↔ 𝑢 = 𝑣))

3-Feb-2026	disjimeceqim 39434	Disj implies coset-equality injectivity (domain-wise). Extracts the practical consequence of Disj: the map 𝑢 ↦ [𝑢]𝑅 is injective on dom 𝑅. This is exactly the "canonicity" property used repeatedly when turning ∃* into ∃! and when reasoning about uniqueness of representatives. (Contributed by Peter Mazsa, 3-Feb-2026.)
⊢ ( Disj 𝑅 → ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 → 𝑢 = 𝑣))

3-Feb-2026	dfdisjALTV5a 39433	Alternate definition of the disjoint relation predicate. Disj 𝑅 means: different domain generators have disjoint cosets (unless the generators are equal), plus Rel 𝑅 for relation-typedness. This is the characterization that makes canonicity/uniqueness arguments modular. It is the starting point for the entire "Disj ↔ unique representative per block" pipeline that feeds into Disjs, see dfdisjs7 39573. (Contributed by Peter Mazsa, 3-Feb-2026.)
⊢ ( Disj 𝑅 ↔ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → 𝑢 = 𝑣) ∧ Rel 𝑅))

2-Feb-2026	ralrnmo 38991	On the range, "at most one" becomes "exactly one". (Contributed by Peter Mazsa, 27-Sep-2018.) (Revised by Peter Mazsa, 2-Feb-2026.)
⊢ (∀𝑥 ∈ ran 𝑅∃*𝑢 𝑢𝑅𝑥 ↔ ∀𝑥 ∈ ran 𝑅∃!𝑢 𝑢𝑅𝑥)

2-Feb-2026	ralmo 38990	"At most one" can be restricted to the range. (Contributed by Peter Mazsa, 2-Feb-2026.)
⊢ (∀𝑥∃𝑢 𝑢𝑅𝑥 ↔ ∀𝑥 ∈ ran 𝑅∃𝑢 𝑢𝑅𝑥)

2-Feb-2026	mptelee 29225	A condition for a mapping to be an element of a Euclidean space. (Contributed by Scott Fenton, 7-Jun-2013.) (Proof shortened by SN, 2-Feb-2026.)
⊢ (𝑁 ∈ ℕ → ((𝑘 ∈ (1...𝑁) ↦ (𝐴𝐹𝐵)) ∈ (𝔼‘𝑁) ↔ ∀𝑘 ∈ (1...𝑁)(𝐴𝐹𝐵) ∈ ℝ))

2-Feb-2026	moabex 5441	"At most one" existence implies a class abstraction exists. (Contributed by NM, 30-Dec-1996.) Avoid axioms. (Revised by SN, 2-Feb-2026.)
⊢ (∃*𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V)

2-Feb-2026	iunss 5010	Subset theorem for an indexed union. (Contributed by NM, 13-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) Avoid ax-10 2176, ax-12 2213. (Revised by SN, 2-Feb-2026.)
⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)

2-Feb-2026	iunssf 5008	Subset theorem for an indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.) Avoid ax-10 2176. (Revised by SN, 2-Feb-2026.)
⊢ Ⅎ𝑥𝐶 ⇒ ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)

1-Feb-2026	xp0 5763	The Cartesian product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by NM, 12-Apr-2004.) Avoid axioms. (Revised by TM, 1-Feb-2026.)
⊢ (𝐴 × ∅) = ∅

1-Feb-2026	uni0 4902	The union of the empty set is the empty set. Theorem 8.7 of [Quine] p. 54. (Contributed by NM, 16-Sep-1993.) Remove use of ax-nul 5270. (Revised by Eric Schmidt, 4-Apr-2007.) Avoid ax-11 2192. (Revised by TM, 1-Feb-2026.)
⊢ ∪ ∅ = ∅

1-Feb-2026	rabss2 4032	Subclass law for restricted abstraction. (Contributed by NM, 18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) Avoid axioms. (Revised by TM, 1-Feb-2026.)
⊢ (𝐴 ⊆ 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ {𝑥 ∈ 𝐵 ∣ 𝜑})

1-Feb-2026	ss2rabdv 4030	Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006.) Avoid axioms. (Revised by TM, 1-Feb-2026.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ⊆ {𝑥 ∈ 𝐴 ∣ 𝜒})

31-Jan-2026	cnv0 5871	The converse of the empty set. (Contributed by NM, 6-Apr-1998.) Remove dependency on ax-sep 5258, ax-nul 5270, ax-pr 5406. (Revised by KP, 25-Oct-2021.) Avoid ax-12 2213. (Revised by TM, 31-Jan-2026.)
⊢ ^◡∅ = ∅

30-Jan-2026	chnsuslle 47580	Length of a subsequence is bounded by the length of original chain. (Contributed by Ender Ting, 30-Jan-2026.)
⊢ (𝜑 → 𝑊 ∈ ( < Chain 𝐴)) & ⊢ (𝜑 → 𝐼 ∈ ( < Chain (0..^(♯‘𝑊)))) & ⊢ (𝜑 → < Po 𝐴) ⇒ ⊢ (𝜑 → (♯‘(𝑊 ∘ 𝐼)) ≤ (♯‘𝑊))

30-Jan-2026	dfsuccl4 39104	Alternate definition that incorporates the most desirable properties of the successor class. (Contributed by Peter Mazsa, 30-Jan-2026.)
⊢ Suc = {𝑛 ∣ ∃!𝑚 ∈ 𝑛 (𝑚 ⊆ 𝑛 ∧ suc 𝑚 = 𝑛)}

30-Jan-2026	dfsuccl3 39103	Alternate definition of the class of all successors. (Contributed by Peter Mazsa, 30-Jan-2026.)
⊢ Suc = {𝑛 ∣ ∃!𝑚 suc 𝑚 = 𝑛}

29-Jan-2026	nthrucw 47590	Some number sets form a chain of proper subsets. This is rephrasing nthruc 16309 as a statement about chains; the hypothesis sets the ordering relation to be "is a proper subset". The theorem talks about singleton 1, natural numbers, natural-or-zero numbers, integers, rational numbers, algebraic reals (the definition includes complex numbers as algebraic so intersection is taken), real numbers and complex numbers, which are proper subsets in order. (Contributed by Ender Ting, 29-Jan-2026.)
⊢ < = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ⊊ 𝑦} ⇒ ⊢ ⟨“{1}ℕℕ₀ℤℚ(𝔸 ∩ ℝ)ℝℂ”⟩ ∈ ( < Chain V)

29-Jan-2026	chner 47584	Any two elements are equivalent in a chain constructed on an equivalence relation. (Contributed by Ender Ting, 29-Jan-2026.)
⊢ (𝜑 → ∼ Er 𝐴) & ⊢ (𝜑 → 𝐶 ∈ ( ∼ Chain 𝐴)) & ⊢ (𝜑 → 𝐽 ∈ (0..^(♯‘𝐶))) & ⊢ (𝜑 → 𝐼 ∈ (0..^(♯‘𝐶))) ⇒ ⊢ (𝜑 → (𝐶‘𝐼) ∼ (𝐶‘𝐽))

29-Jan-2026	chnerlem3 47583	Lemma for chner 47584- trichotomy of integers within the word's domain. (Contributed by Ender Ting, 29-Jan-2026.)
⊢ (𝜑 → ∼ Er 𝐴) & ⊢ (𝜑 → 𝐶 ∈ ( ∼ Chain 𝐴)) & ⊢ (𝜑 → 𝐽 ∈ (0..^(♯‘𝐶))) & ⊢ (𝜑 → 𝐼 ∈ (0..^(♯‘𝐶))) ⇒ ⊢ (𝜑 → (𝐼 ∈ (0..^𝐽) ∨ 𝐽 ∈ (0..^𝐼) ∨ 𝐼 = 𝐽))

29-Jan-2026	chnerlem2 47582	Lemma for chner 47584 where the I-th element comes before the J-th. (Contributed by Ender Ting, 29-Jan-2026.)
⊢ (𝜑 → ∼ Er 𝐴) & ⊢ (𝜑 → 𝐶 ∈ ( ∼ Chain 𝐴)) & ⊢ (𝜑 → 𝐽 ∈ (0..^(♯‘𝐶))) ⇒ ⊢ ((𝜑 ∧ 𝐼 ∈ (0..^𝐽)) → (𝐶‘𝐼) ∼ (𝐶‘𝐽))

29-Jan-2026	chnerlem1 47581	In a chain constructed on an equivalence relation, the last element is equivalent to any. This theorem is a translation of chnub 18679 to equivalence relations. (Contributed by Ender Ting, 29-Jan-2026.)
⊢ (𝜑 → ∼ Er 𝐴) & ⊢ (𝜑 → 𝐶 ∈ ( ∼ Chain 𝐴)) & ⊢ (𝜑 → 𝐽 ∈ (0..^(♯‘𝐶))) ⇒ ⊢ (𝜑 → (𝐶‘𝐽) ∼ (lastS‘𝐶))

29-Jan-2026	dfsuccl2 39100	Alternate definition of the class of all successors. (Contributed by Peter Mazsa, 29-Jan-2026.)
⊢ Suc = {𝑛 ∣ ∃𝑚 suc 𝑚 = 𝑛}

29-Jan-2026	dfblockliftmap2 39091	Alternate definition of the block lift map. (Contributed by Peter Mazsa, 29-Jan-2026.)
⊢ (𝑅 BlockLiftMap 𝐴) = (𝑚 ∈ (𝐴 ∩ (dom 𝑅 ∖ {∅})) ↦ ([𝑚]𝑅 × 𝑚))

29-Jan-2026	dmxrncnvepres2 39063	Domain of the range product with restricted converse epsilon relation. (Contributed by Peter Mazsa, 29-Jan-2026.)
⊢ dom (𝑅 ⋉ (^◡ E ↾ 𝐴)) = (𝐴 ∩ (dom 𝑅 ∖ {∅}))

29-Jan-2026	frgr2wwlkeu 30659	For two different vertices in a friendship graph, there is exactly one third vertex being the middle vertex of a (simple) path/walk of length 2 between the two vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 12-May-2021.) (Proof shortened by AV, 4-Jan-2022.) (Revised by Ender Ting, 29-Jan-2026.)
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → ∃!𝑐 ∈ 𝑉 ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))

29-Jan-2026	usgr2wspthon 30298	A simple path of length 2 between two vertices corresponds to two adjacent edges in a simple graph. (Contributed by Alexander van der Vekens, 9-Mar-2018.) (Revised by AV, 17-May-2021.) (Revised by Ender Ting, 29-Jan-2026.)
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑇 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 ((𝑇 = ⟨“𝐴𝑏𝐶”⟩ ∧ 𝐴 ≠ 𝐶) ∧ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸))))

29-Jan-2026	usgr2wspthons3 30297	A simple path of length 2 between two vertices represented as length 3 string corresponds to two adjacent edges in a simple graph. (Contributed by Alexander van der Vekens, 8-Mar-2018.) (Revised by AV, 17-May-2021.) (Proof shortened by AV, 16-Mar-2022.) (Revised by Ender Ting, 29-Jan-2026.)
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ (𝐴 ≠ 𝐶 ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)))

29-Jan-2026	wpthswwlks2on 30294	For two different vertices, a walk of length 2 between these vertices is a simple path of length 2 between these vertices in a simple graph. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 16-Mar-2022.) (Revised by Ender Ting, 29-Jan-2026.)
⊢ ((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) → (𝐴(2 WSPathsNOn 𝐺)𝐵) = (𝐴(2 WWalksNOn 𝐺)𝐵))

29-Jan-2026	elwspths2onw 30293	A simple path of length 2 between two vertices (in a simple pseudograph) as length 3 string. This theorem avoids the Axiom of Choice for its proof, at the cost of requiring a simple graph; the more general version is elwspths2on 30292. (Contributed by Ender Ting, 29-Jan-2026.)
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ USPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶))))

29-Jan-2026	usgrwwlks2on 30288	A walk of length 2 between two vertices as word in a simple graph. This theorem is analogous to umgrwwlks2on 30289 except it talks about simple graphs and therefore does not require the Axiom of Choice for its proof. (Contributed by Ender Ting, 29-Jan-2026.)
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)))

29-Jan-2026	elreno2 28669	Alternate characterization of the surreal reals. Theorem 4.4(b) of [Gonshor] p. 39. (Contributed by Scott Fenton, 29-Jan-2026.)
⊢ (𝐴 ∈ ℝ_s ↔ (𝐴 ∈ No ∧ (∃𝑛 ∈ ℕ_s (( -u_s ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ ∀𝑥_𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∃𝑛 ∈ ℕ_s ( 1_s /_su 𝑛) ≤s (abs_s‘(𝐴 -_s 𝑥_𝑂)))))

29-Jan-2026	abssubs 28424	Swapping order of surreal subtraction doesn't change the absolute value. (Contributed by Scott Fenton, 29-Jan-2026.)
⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (abs_s‘(𝐴 -_s 𝐵)) = (abs_s‘(𝐵 -_s 𝐴)))

29-Jan-2026	lesubsd 28270	Swap subtrahends in a surreal inequality. (Contributed by Scott Fenton, 29-Jan-2026.)
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) ⇒ ⊢ (𝜑 → (𝐴 ≤s (𝐵 -_s 𝐶) ↔ 𝐶 ≤s (𝐵 -_s 𝐴)))

28-Jan-2026	dfsucmap4 39095	Alternate definition of the successor map. (Contributed by Peter Mazsa, 28-Jan-2026.)
⊢ SucMap = (𝑚 ∈ V ↦ suc 𝑚)

28-Jan-2026	dfsucmap2 39094	Alternate definition of the successor map. (Contributed by Peter Mazsa, 28-Jan-2026.)
⊢ SucMap = ( I AdjLiftMap dom I )

28-Jan-2026	dfsucmap3 39093	Alternate definition of the successor map. (Contributed by Peter Mazsa, 28-Jan-2026.)
⊢ SucMap = ( I AdjLiftMap V)

28-Jan-2026	blockadjliftmap 39088	A "two-stage" construction is obtained by first forming the block relation (𝑅 ⋉ ^◡ E ) and then adjoining elements as "BlockAdj". Combined, it uses the relation ((𝑅 ⋉ ^◡ E ) ∪ ^◡ E ). (Contributed by Peter Mazsa, 28-Jan-2026.)
⊢ ((𝑅 ⋉ ^◡ E ) AdjLiftMap 𝐴) = {⟨𝑚, 𝑛⟩ ∣ (𝑚 ∈ (𝐴 ∖ {∅}) ∧ 𝑛 = (𝑚 ∪ ([𝑚]𝑅 × 𝑚)))}

28-Jan-2026	dfadjliftmap2 39087	Alternate definition of the adjoined lift map. (Contributed by Peter Mazsa, 28-Jan-2026.)
⊢ (𝑅 AdjLiftMap 𝐴) = (𝑚 ∈ (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅}))) ↦ (𝑚 ∪ [𝑚]𝑅))

28-Jan-2026	ecuncnvepres 39025	The restricted union with converse epsilon relation coset of 𝐵. (Contributed by Peter Mazsa, 28-Jan-2026.)
⊢ (𝐵 ∈ 𝐴 → [𝐵]((𝑅 ∪ ^◡ E ) ↾ 𝐴) = (𝐵 ∪ [𝐵]𝑅))

28-Jan-2026	ecunres 39024	The restricted union coset of 𝐵. (Contributed by Peter Mazsa, 28-Jan-2026.)
⊢ (𝐵 ∈ 𝑉 → [𝐵]((𝑅 ∪ 𝑆) ↾ 𝐴) = ([𝐵](𝑅 ↾ 𝐴) ∪ [𝐵](𝑆 ↾ 𝐴)))

28-Jan-2026	ecun 39023	The union coset of 𝐴. (Contributed by Peter Mazsa, 28-Jan-2026.)
⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ∪ 𝑆) = ([𝐴]𝑅 ∪ [𝐴]𝑆))

28-Jan-2026	dmxrnuncnvepres 39022	Domain of the combined relation of two special relations, see blockadjliftmap 39088. (Contributed by Peter Mazsa, 28-Jan-2026.)
⊢ dom (((𝑅 ⋉ ^◡ E ) ∪ ^◡ E ) ↾ 𝐴) = (𝐴 ∖ {∅})

28-Jan-2026	dmuncnvepres 39021	Domain of the union with the converse epsilon, restricted. (Contributed by Peter Mazsa, 28-Jan-2026.)
⊢ dom ((𝑅 ∪ ^◡ E ) ↾ 𝐴) = (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅})))

28-Jan-2026	dmcnvepres 39020	Domain of the restricted converse epsilon relation. (Contributed by Peter Mazsa, 28-Jan-2026.)
⊢ dom (^◡ E ↾ 𝐴) = (𝐴 ∖ {∅})

28-Jan-2026	sps3wwlks2on 30287	A length 3 string which represents a walk of length 2 between two vertices. Concerns simple pseudographs, in contrast to s3wwlks2on 30286 and does not require the Axiom of Choice for its proof. (Contributed by Ender Ting, 28-Jan-2026.)
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ USPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2)))

27-Jan-2026	sucpre 39127	suc is a right-inverse of pre on Suc. This theorem states the partial inverse relation in the direction we most often need. (Contributed by Peter Mazsa, 27-Jan-2026.)
⊢ (𝑁 ∈ Suc → suc pre 𝑁 = 𝑁)

27-Jan-2026	eupre 39124	Unique predecessor exists on the successor class. (Contributed by Peter Mazsa, 27-Jan-2026.)
⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ Suc ↔ ∃!𝑚 𝑚 SucMap 𝑁))

27-Jan-2026	dfpre 39106	Alternate definition of the successor-predecessor. (Contributed by Peter Mazsa, 27-Jan-2026.)
⊢ pre 𝑁 = (℩𝑚𝑚 ∈ Pred( SucMap , V, 𝑁))

27-Jan-2026	df-pre 39105	Define the term-level successor-predecessor. It is the unique 𝑚 with suc 𝑚 = 𝑁 when such an 𝑚 exists; otherwise pre 𝑁 is the arbitrary default chosen by ℩. See its alternate definitions dfpre 39106, dfpre2 39107, dfpre3 39108 and dfpre4 39110. Our definition is a special case of the widely recognised general 𝑅 -predecessor class df-pred 6304 (the class of all elements 𝑚 of 𝐴 such that 𝑚𝑅𝑁, dfpred3g 6316, cf. also df-bnj14 35059) in several respects. Its most abstract property as a specialisation is that it has a unique existing value by default. This is in contrast to the general version. The uniqueness (conditional on existence) is implied by the property of this specific instance of the general case involving the successor map df-sucmap 39092 in place of 𝑅, so that 𝑚 SucMap 𝑁, cf. sucmapleftuniq 39120, which originates from suc11reg 9589. Existence ∃𝑚𝑚 SucMap 𝑁 holds exactly on 𝑁 ∈ ran SucMap, cf. elrng 5883. Note that dom SucMap = V (see dmsucmap 39098), so the equivalent definition dfpre 39106 uses (℩𝑚𝑚 ∈ Pred( SucMap , V, 𝑁)). (Contributed by Peter Mazsa, 27-Jan-2026.)
⊢ pre 𝑁 = (℩𝑚𝑚 ∈ Pred( SucMap , dom SucMap , 𝑁))

26-Jan-2026	dfpre4 39110	Alternate definition of the predecessor of the 𝑁 set. The ^◡ SucMap is just the "PreMap"; we did not define it because we do not expect to use it extensively in future (cf. the comments of df-sucmap 39092). (Contributed by Peter Mazsa, 26-Jan-2026.)
⊢ (𝑁 ∈ 𝑉 → pre 𝑁 = (℩𝑚𝑚 ∈ [𝑁]^◡ SucMap ))

26-Jan-2026	dfpred4 39109	Alternate definition of the predecessor class when 𝑁 is a set. (Contributed by Peter Mazsa, 26-Jan-2026.)
⊢ (𝑁 ∈ 𝑉 → Pred(𝑅, 𝐴, 𝑁) = [𝑁]^◡(𝑅 ↾ 𝐴))

25-Jan-2026	df-shiftstable 39112	Define shift-stability, a general "procedure" pattern for "the one-step backward shift/transport of 𝐹 along 𝑆", and then ∩ 𝐹 enforces "and it already holds here". Let 𝐹 be a relation encoding a property that depends on a "level" coordinate (for example, a feasibility condition indexed by a carrier, a grade, or a stage in a construction). Let 𝑆 be a shift relation between levels (for example, the successor map SucMap, or any other grading step). The composed relation (𝑆 ∘ 𝐹) transports 𝐹 one step along the shift: 𝑟(𝑆 ∘ 𝐹)𝑛 means there exists a predecessor level 𝑚 such that 𝑟𝐹𝑚 and 𝑚𝑆𝑛 (e.g., 𝑚 SucMap 𝑛). We do not introduce a separate notation for "Shift" because it is simply the standard relational composition df-co 5672. The intersection ((𝑆 ∘ 𝐹) ∩ 𝐹) is the locally shift-stable fragment of 𝐹: it consists exactly of those points where the property holds at some immediate predecessor that shifts to 𝑛 and also holds at level 𝑛. In other words, it isolates the part of 𝐹 that is already compatible with one-step tower coherence. This definition packages a common construction pattern used throughout the development: "constrain by one-step stability under a chosen shift, then additionally constrain by 𝐹". Iterating the operator (𝑋 ↦ ((𝑆 ∘ 𝑋) ∩ 𝑋) corresponds to multi-step/tower coherence; the one-step definition here is the economical kernel from which such "tower" readings can be developed when needed. (Contributed by Peter Mazsa, 25-Jan-2026.)
⊢ (𝑆 ShiftStable 𝐹) = ((𝑆 ∘ 𝐹) ∩ 𝐹)

25-Jan-2026	df-succl 39099	Define Suc as the class of all successors, i.e. the range of the successor map: 𝑛 ∈ Suc iff ∃𝑚suc 𝑚 = 𝑛 (see dfsuccl2 39100). By injectivity of suc (suc11reg 9589), every 𝑛 ∈ Suc has at most one predecessor, which is exactly what pre 𝑛 (df-pre 39105) names. Cf. dfsuccl3 39103 and dfsuccl4 39104. (Contributed by Peter Mazsa, 25-Jan-2026.)
⊢ Suc = ran SucMap

25-Jan-2026	df-sucmap 39092	Define the successor map, directly as the graph of the successor operation, using only elementary set theory (ordered-pair class abstraction). This avoids committing to any particular construction of the successor function/class from other operators (e.g. a union/composition presentation), while remaining provably equivalent to those presentations (cf. dfsucmap2 39094 and dfsucmap3 39093 vs. df-succf 36343 and dfsuccf2 36414). For maximum mappy shape, see dfsucmap4 39095. We also treat the successor relation as the default shift relation for grading/tower arguments (cf. df-shiftstable 39112). Because it is used pervasively in shift-lift infrastructure, we adopt the short name SucMap rather than the fully systematic "SucAdjLiftMap". You may also define the predecessor relation as the converse graph "PreMap" as ^◡ SucMap, which reverses successor edges ( cf. cnvopab 6139) and sends each successor to its (unique) predecessor when it exists. (Contributed by Peter Mazsa, 25-Jan-2026.)
⊢ SucMap = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}

25-Jan-2026	ecxrncnvep2 39040	The (𝑅 ⋉ ^◡ E )-coset of a set is the Cartesian product of its 𝑅-coset and the set. (Contributed by Peter Mazsa, 25-Jan-2026.)
⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ⋉ ^◡ E ) = ([𝐴]𝑅 × 𝐴))

25-Jan-2026	omprcomonb 35514	The class of all finite ordinals is a proper class iff all ordinal sets are finite. (Contributed by BTernaryTau, 25-Jan-2026.)
⊢ (¬ ω ∈ V ↔ ω = On)

25-Jan-2026	fineqvomonb 35513	All sets are finite iff all ordinal sets are finite. (Contributed by BTernaryTau, 25-Jan-2026.)
⊢ (Fin = V ↔ ω = On)

25-Jan-2026	r1omfv 35485	Value of the cumulative hierarchy of sets function at ω. (Contributed by BTernaryTau, 25-Jan-2026.)
⊢ (𝑅₁‘ω) = ∪ (𝑅₁ “ ω)

25-Jan-2026	r12 35469	Value of the cumulative hierarchy of sets function at 2_o. (Contributed by BTernaryTau, 25-Jan-2026.)
⊢ (𝑅₁‘2_o) = 2_o

25-Jan-2026	xoromon 35460	ω is either an ordinal set or the proper class of all ordinal sets, but not both. This is a stronger version of omon 7875. (Contributed by BTernaryTau, 25-Jan-2026.)
⊢ (ω ∈ On ⊻ ω = On)

25-Jan-2026	esplyind 33946	A recursive formula for the elementary symmetric polynomials. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝑊 = (𝐼 mPoly 𝑅) & ⊢ 𝑉 = (𝐼 mVar 𝑅) & ⊢ + = (+_g‘𝑊) & ⊢ · = (._r‘𝑊) & ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} & ⊢ 𝐺 = ((𝐼extendVars𝑅)‘𝑌) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑌 ∈ 𝐼) & ⊢ 𝐽 = (𝐼 ∖ {𝑌}) & ⊢ 𝐸 = (𝐽eSymPoly𝑅) & ⊢ (𝜑 → 𝐾 ∈ (1...(♯‘𝐼))) & ⊢ 𝐶 = {ℎ ∈ (ℕ₀ ↑_m 𝐽) ∣ ℎ finSupp 0} ⇒ ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) = (((𝑉‘𝑌) · (𝐺‘(𝐸‘(𝐾 − 1)))) + (𝐺‘(𝐸‘𝐾))))

25-Jan-2026	esplyfval3 33943	Alternate expression for the value of the 𝐾-th elementary symmetric polynomial. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐾 ∈ ℕ₀) & ⊢ 0 = (0_g‘𝑅) & ⊢ 1 = (1_r‘𝑅) ⇒ ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) = (𝑓 ∈ 𝐷 ↦ if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝐾), 1 , 0 )))

25-Jan-2026	esplyfval2 33936	When 𝐾 is out-of-bounds, the 𝐾-th elementary symmetric polynomial is zero. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐾 ∈ (ℕ₀ ∖ (0...(♯‘𝐼)))) & ⊢ 𝑍 = (0_g‘(𝐼 mPoly 𝑅)) ⇒ ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) = 𝑍)

25-Jan-2026	mplmulmvr 33910	Multiply a polynomial 𝐹 with a variable 𝑋 (i.e. with a monic monomial). (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝑃 = (𝐼 mPoly 𝑅) & ⊢ 𝑋 = ((𝐼 mVar 𝑅)‘𝑌) & ⊢ 𝑀 = (Base‘𝑃) & ⊢ · = (._r‘𝑃) & ⊢ 0 = (0_g‘𝑅) & ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} & ⊢ 𝐴 = ((𝟭‘𝐼)‘{𝑌}) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝐼) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐹 ∈ 𝑀) ⇒ ⊢ (𝜑 → (𝑋 · 𝐹) = (𝑏 ∈ 𝐷 ↦ if((𝑏‘𝑌) = 0, 0 , (𝐹‘(𝑏 ∘_f − 𝐴)))))

25-Jan-2026	mvrvalind 33909	Value of the generating elements of the power series structure, expressed using the indicator function. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝑉 = (𝐼 mVar 𝑅) & ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} & ⊢ 0 = (0_g‘𝑅) & ⊢ 1 = (1_r‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ 𝑌) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ 𝐴 = ((𝟭‘𝐼)‘{𝑋}) ⇒ ⊢ (𝜑 → ((𝑉‘𝑋)‘𝐹) = if(𝐹 = 𝐴, 1 , 0 ))

25-Jan-2026	extvfvalf 33908	The "variable extension" function maps polynomials with variables indexed in 𝐽 to polynomials with variables indexed in 𝐼. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} & ⊢ 0 = (0_g‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐽 = (𝐼 ∖ {𝐴}) & ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅)) & ⊢ (𝜑 → 𝐴 ∈ 𝐼) & ⊢ 𝑁 = (Base‘(𝐼 mPoly 𝑅)) ⇒ ⊢ (𝜑 → ((𝐼extendVars𝑅)‘𝐴):𝑀⟶𝑁)

25-Jan-2026	extvfvcl 33907	Closure for the "variable extension" function evaluated for converting a given polynomial 𝐹 by adding a variable with index 𝐴. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} & ⊢ 0 = (0_g‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐽 = (𝐼 ∖ {𝐴}) & ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅)) & ⊢ (𝜑 → 𝐴 ∈ 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝑀) & ⊢ 𝑁 = (Base‘(𝐼 mPoly 𝑅)) ⇒ ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) ∈ 𝑁)

25-Jan-2026	extvfvvcl 33906	Closure for the "variable extension" function evaluated for converting a given polynomial 𝐹 by adding a variable with index 𝐴. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} & ⊢ 0 = (0_g‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐽 = (𝐼 ∖ {𝐴}) & ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅)) & ⊢ (𝜑 → 𝐴 ∈ 𝐼) & ⊢ (𝜑 → 𝐹 ∈ 𝑀) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) ⇒ ⊢ (𝜑 → ((((𝐼extendVars𝑅)‘𝐴)‘𝐹)‘𝑋) ∈ 𝐵)

25-Jan-2026	extvfvv 33905	The "variable extension" function evaluated for converting a given polynomial 𝐹 by adding a variable with index 𝐴. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} & ⊢ 0 = (0_g‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ (𝜑 → 𝐴 ∈ 𝐼) & ⊢ 𝐽 = (𝐼 ∖ {𝐴}) & ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅)) & ⊢ (𝜑 → 𝐹 ∈ 𝑀) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) ⇒ ⊢ (𝜑 → ((((𝐼extendVars𝑅)‘𝐴)‘𝐹)‘𝑋) = if((𝑋‘𝐴) = 0, (𝐹‘(𝑋 ↾ 𝐽)), 0 ))

25-Jan-2026	extvfv 33904	The "variable extension" function evaluated for converting a given polynomial 𝐹 by adding a variable with index 𝐴. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} & ⊢ 0 = (0_g‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ (𝜑 → 𝐴 ∈ 𝐼) & ⊢ 𝐽 = (𝐼 ∖ {𝐴}) & ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅)) & ⊢ (𝜑 → 𝐹 ∈ 𝑀) ⇒ ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) = (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )))

25-Jan-2026	extvfval 33903	The "variable extension" function evaluated for adding a variable with index 𝐴. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} & ⊢ 0 = (0_g‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ (𝜑 → 𝐴 ∈ 𝐼) & ⊢ 𝐽 = (𝐼 ∖ {𝐴}) & ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅)) ⇒ ⊢ (𝜑 → ((𝐼extendVars𝑅)‘𝐴) = (𝑓 ∈ 𝑀 ↦ (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝑓‘(𝑥 ↾ 𝐽)), 0 ))))

25-Jan-2026	extvval 33902	Value of the "variable extension" function. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} & ⊢ 0 = (0_g‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ 𝐽 = (𝐼 ∖ {𝑎}) & ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅)) ⇒ ⊢ (𝜑 → (𝐼extendVars𝑅) = (𝑎 ∈ 𝐼 ↦ (𝑓 ∈ 𝑀 ↦ (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝑎) = 0, (𝑓‘(𝑥 ↾ (𝐼 ∖ {𝑎}))), 0 )))))

25-Jan-2026	nn0diffz0 33120	Upper set of the nonnegative integers. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ (𝑁 ∈ ℕ₀ → (ℕ₀ ∖ (0...𝑁)) = (ℤ_≥‘(𝑁 + 1)))

25-Jan-2026	rnressnsn 33003	The range of a restriction to a singleton is a singleton. See dmressnsn 6024. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ran (𝐹 ↾ {𝐴}) = {(𝐹‘𝐴)})

25-Jan-2026	partfun2 33002	Rewrite a function defined by parts, using a mapping and an if construct, into a union of functions on disjoint domains. See also partfun 6684 and ifmpt2v 7514. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ 𝜑} ⇒ ⊢ (𝑥 ∈ 𝐴 ↦ if(𝜑, 𝐵, 𝐶)) = ((𝑥 ∈ 𝐷 ↦ 𝐵) ∪ (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 𝐶))

25-Jan-2026	indconst1 12232	Indicator of the whole set. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ (𝑂 ∈ 𝑉 → ((𝟭‘𝑂)‘𝑂) = (𝑂 × {1}))

25-Jan-2026	indconst0 12231	Indicator of the empty set. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ (𝑂 ∈ 𝑉 → ((𝟭‘𝑂)‘∅) = (𝑂 × {0}))

25-Jan-2026	tz6.12-2 6870	Function value when 𝐹 is not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) Avoid ax-10 2176, ax-11 2192, ax-12 2213. (Revised by TM, 25-Jan-2026.)
⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∅)

24-Jan-2026	r1omhfb 35489	The class of all hereditarily finite sets is the only class with the property that all sets are members of it iff they are finite and all of their elements are members of it. (Contributed by BTernaryTau, 24-Jan-2026.)
⊢ (𝐻 = ∪ (𝑅₁ “ ω) ↔ ∀𝑥(𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)))

24-Jan-2026	trssfir1om 35488	If every element in a transitive class is finite, then every element is also hereditarily finite. (Contributed by BTernaryTau, 24-Jan-2026.)
⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ Fin) → 𝐴 ⊆ ∪ (𝑅₁ “ ω))

24-Jan-2026	r11 35468	Value of the cumulative hierarchy of sets function at 1_o. (Contributed by BTernaryTau, 24-Jan-2026.)
⊢ (𝑅₁‘1_o) = 1_o

24-Jan-2026	rnco 6255	The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) Avoid ax-11 2192. (Revised by TM, 24-Jan-2026.)
⊢ ran (𝐴 ∘ 𝐵) = ran (𝐴 ↾ ran 𝐵)

24-Jan-2026	dm0rn0 5916	An empty domain is equivalent to an empty range. (Contributed by NM, 21-May-1998.) Avoid ax-10 2176, ax-11 2192, ax-12 2213. (Revised by TM, 24-Jan-2026.)
⊢ (dom 𝐴 = ∅ ↔ ran 𝐴 = ∅)

24-Jan-2026	eqabcbw 2837	Version of eqabcb 2903 using implicit substitution, which requires fewer axioms. (Contributed by TM, 24-Jan-2026.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ ∀𝑦(𝜓 ↔ 𝑦 ∈ 𝐴))

24-Jan-2026	excomw 2076	Weak version of excom 2197 and biconditional form of excomimw 2074. Uses only Tarski's FOL axiom schemes. (Contributed by TM, 24-Jan-2026.)
⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜒)) ⇒ ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑)

22-Jan-2026	chnsubseq 47579	An order-preserving subsequence of an ordered chain is itself a chain. (Contributed by Ender Ting, 22-Jan-2026.)
⊢ (𝜑 → 𝑊 ∈ ( < Chain 𝐴)) & ⊢ (𝜑 → 𝐼 ∈ ( < Chain (0..^(♯‘𝑊)))) & ⊢ (𝜑 → < Po 𝐴) ⇒ ⊢ (𝜑 → (𝑊 ∘ 𝐼) ∈ ( < Chain 𝐴))

22-Jan-2026	chnsubseqwl 47578	A subsequence of a chain has the same length as its indexing sequence. (Contributed by Ender Ting, 22-Jan-2026.)
⊢ (𝜑 → 𝑊 ∈ ( < Chain 𝐴)) & ⊢ (𝜑 → 𝐼 ∈ ( < Chain (0..^(♯‘𝑊)))) ⇒ ⊢ (𝜑 → (♯‘(𝑊 ∘ 𝐼)) = (♯‘𝐼))

22-Jan-2026	chnsubseqword 47577	A subsequence of a chain is a word. (Contributed by Ender Ting, 22-Jan-2026.)
⊢ (𝜑 → 𝑊 ∈ ( < Chain 𝐴)) & ⊢ (𝜑 → 𝐼 ∈ ( < Chain (0..^(♯‘𝑊)))) ⇒ ⊢ (𝜑 → (𝑊 ∘ 𝐼) ∈ Word 𝐴)

22-Jan-2026	r1filim 35479	A finite set appears in the cumulative hierarchy prior to a limit ordinal iff all of its elements appear in the cumulative hierarchy prior to that limit ordinal. (Contributed by BTernaryTau, 22-Jan-2026.)
⊢ ((𝐴 ∈ Fin ∧ Lim 𝐵) → (𝐴 ∈ ∪ (𝑅₁ “ 𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅₁ “ 𝐵)))

22-Jan-2026	rankfilimb 35477	The rank of a finite well-founded set is less than a limit ordinal iff the ranks of all of its elements are less than that limit ordinal. (Contributed by BTernaryTau, 22-Jan-2026.)
⊢ ((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅₁ “ On) ∧ Lim 𝐵) → ((rank‘𝐴) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵))

21-Jan-2026	r1omhfbregs 35531	The class of all hereditarily finite sets is the only class with the property that all sets are members of it iff they are finite and all of their elements are members of it. This version of r1omhfb 35489 replaces setinds2 9721 with setinds2regs 35525 and trssfir1om 35488 with trssfir1omregs 35530. (Contributed by BTernaryTau, 21-Jan-2026.)
⊢ (𝐻 = ∪ (𝑅₁ “ ω) ↔ ∀𝑥(𝑥 ∈ 𝐻 ↔ (𝑥 ∈ Fin ∧ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐻)))

20-Jan-2026	trssfir1omregs 35530	If every element in a transitive class is finite, then every element is also hereditarily finite. This version of trssfir1om 35488 replaces setinds2 9721 with setinds2regs 35525. (Contributed by BTernaryTau, 20-Jan-2026.)
⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ Fin) → 𝐴 ⊆ ∪ (𝑅₁ “ ω))

20-Jan-2026	df-extv 33901	Define the "variable extension" function. The function ((𝐼extendVars𝑅)‘𝐴) converts polynomials with variables indexed by (𝐼 ∖ {𝐴}) into polynomials indexed by 𝐼, and therefore maps elements of ((𝐼 ∖ {𝐴}) mPoly 𝑅) onto (𝐼 mPoly 𝑅). (Contributed by Thierry Arnoux, 20-Jan-2026.)
⊢ extendVars = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑎 ∈ 𝑖 ↦ (𝑓 ∈ (Base‘((𝑖 ∖ {𝑎}) mPoly 𝑟)) ↦ (𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝑖) ∣ ℎ finSupp 0} ↦ if((𝑥‘𝑎) = 0, (𝑓‘(𝑥 ↾ (𝑖 ∖ {𝑎}))), (0_g‘𝑟))))))

20-Jan-2026	chnfibg 18693	Given a partial order, the set of chains is finite iff the alphabet is finite. (Contributed by Ender Ting, 20-Jan-2026.)
⊢ ( < Po 𝐴 → (𝐴 ∈ Fin ↔ ( < Chain 𝐴) ∈ Fin))

20-Jan-2026	chninf 18692	There is an infinite number of chains for any infinite alphabet and any relation. For instance, all the singletons of alphabet characters match. (Contributed by Ender Ting, 20-Jan-2026.)
⊢ (𝐴 ∉ Fin → ( < Chain 𝐴) ∉ Fin)

20-Jan-2026	chnfi 18691	There is a finite number of chains over finite domain, as long as the relation orders it. (Contributed by Ender Ting, 20-Jan-2026.)
⊢ ((𝐴 ∈ Fin ∧ < Po 𝐴) → ( < Chain 𝐴) ∈ Fin)

20-Jan-2026	chnpolfz 18690	Provided that chain's relation is a partial order, the chain length is restricted to a specific integer range. (Contributed by Ender Ting, 20-Jan-2026.)
⊢ (𝜑 → < Po 𝐴) & ⊢ (𝜑 → 𝐵 ∈ ( < Chain 𝐴)) & ⊢ (𝜑 → 𝐴 ∈ Fin) ⇒ ⊢ (𝜑 → (♯‘𝐵) ∈ (0...(♯‘𝐴)))

20-Jan-2026	chnpolleha 18689	A chain under relation which orders the alphabet has at most alphabet's size elements in it. (Contributed by Ender Ting, 20-Jan-2026.)
⊢ (𝜑 → < Po 𝐴) & ⊢ (𝜑 → 𝐵 ∈ ( < Chain 𝐴)) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → (♯‘𝐵) ≤ (♯‘𝐴))

20-Jan-2026	chnpoadomd 18688	A chain under relation which orders the alphabet cannot have more elements than the alphabet itself. (Contributed by Ender Ting, 20-Jan-2026.)
⊢ (𝜑 → < Po 𝐴) & ⊢ (𝜑 → 𝐵 ∈ ( < Chain 𝐴)) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → (0..^(♯‘𝐵)) ≼ 𝐴)

20-Jan-2026	chnpof1 18687	A chain under relation which orders the alphabet is a one-to-one function from its domain to alphabet. (Contributed by Ender Ting, 20-Jan-2026.)
⊢ (𝜑 → < Po 𝐴) & ⊢ (𝜑 → 𝐵 ∈ ( < Chain 𝐴)) ⇒ ⊢ (𝜑 → 𝐵:(0..^(♯‘𝐵))–1-1→𝐴)

20-Jan-2026	chnf 18686	A chain is a zero-based finite sequence with a recoverable upper limit. (Contributed by Ender Ting, 20-Jan-2026.)
⊢ (𝐵 ∈ ( < Chain 𝐴) → 𝐵:(0..^(♯‘𝐵))⟶𝐴)

20-Jan-2026	chnrev 18684	Reverse of a chain is chain under the converse relation and same domain. (Contributed by Ender Ting, 20-Jan-2026.)
⊢ (𝐵 ∈ ( < Chain 𝐴) → (reverse‘𝐵) ∈ (^◡ < Chain 𝐴))

20-Jan-2026	chnccat 18683	Concatenate two chains. (Contributed by Ender Ting, 20-Jan-2026.)
⊢ (𝜑 → 𝑇 ∈ ( < Chain 𝐴)) & ⊢ (𝜑 → 𝑈 ∈ ( < Chain 𝐴)) & ⊢ (𝜑 → (𝑇 = ∅ ∨ 𝑈 = ∅ ∨ (lastS‘𝑇) < (𝑈‘0))) ⇒ ⊢ (𝜑 → (𝑇 ++ 𝑈) ∈ ( < Chain 𝐴))

20-Jan-2026	chnrdss 18674	Subset theorem for chains. (Contributed by Ender Ting, 20-Jan-2026.)
⊢ (( < ⊆ 𝑅 ∧ 𝐴 ⊆ 𝐵) → ( < Chain 𝐴) ⊆ (𝑅 Chain 𝐵))

20-Jan-2026	chndss 18673	Chains with an alphabet are also chains with any superset alphabet. (Contributed by Ender Ting, 20-Jan-2026.)
⊢ (𝐴 ⊆ 𝐵 → ( < Chain 𝐴) ⊆ ( < Chain 𝐵))

20-Jan-2026	chnrss 18672	Chains under a relation are also chains under any superset relation. (Contributed by Ender Ting, 20-Jan-2026.)
⊢ ( < ⊆ 𝑅 → ( < Chain 𝐴) ⊆ (𝑅 Chain 𝐴))

20-Jan-2026	nfchnd 18668	Bound-variable hypothesis builder for chain collection constructor. (Contributed by Ender Ting, 20-Jan-2026.)
⊢ (𝜑 → Ⅎ𝑥 < ) & ⊢ (𝜑 → Ⅎ𝑥𝐴) ⇒ ⊢ (𝜑 → Ⅎ𝑥( < Chain 𝐴))

19-Jan-2026	r1omhf 35481	A set is hereditarily finite iff it is finite and all of its elements are hereditarily finite. (Contributed by BTernaryTau, 19-Jan-2026.)
⊢ (𝐴 ∈ ∪ (𝑅₁ “ ω) ↔ (𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅₁ “ ω)))

19-Jan-2026	r1filimi 35478	If all elements in a finite set appear in the cumulative hierarchy prior to a limit ordinal, then that set also appears in the cumulative hierarchy prior to the limit ordinal. (Contributed by BTernaryTau, 19-Jan-2026.)
⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅₁ “ 𝐵) ∧ Lim 𝐵) → 𝐴 ∈ ∪ (𝑅₁ “ 𝐵))

19-Jan-2026	rankfilimbi 35476	If all elements in a finite well-founded set have a rank less than a limit ordinal, then the rank of that set is also less than the limit ordinal. (Contributed by BTernaryTau, 19-Jan-2026.)
⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅₁ “ On)) ∧ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 ∧ Lim 𝐵)) → (rank‘𝐴) ∈ 𝐵)

19-Jan-2026	rankval4b 35474	The rank of a set is the supremum of the successors of the ranks of its members. Exercise 9.1 of [Jech] p. 72. Also a special case of Theorem 7V(b) of [Enderton] p. 204. This variant of rankval4 9840 does not use Regularity, and so requires the assumption that 𝐴 is in the range of 𝑅₁. (Contributed by BTernaryTau, 19-Jan-2026.)
⊢ (𝐴 ∈ ∪ (𝑅₁ “ On) → (rank‘𝐴) = ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))

19-Jan-2026	rankval2b 35473	Value of an alternate definition of the rank function. Definition of [BellMachover] p. 478. This variant of rankval2 9791 does not use Regularity, and so requires the assumption that 𝐴 is in the range of 𝑅₁. (Contributed by BTernaryTau, 19-Jan-2026.)
⊢ (𝐴 ∈ ∪ (𝑅₁ “ On) → (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (𝑅₁‘𝑥)})

19-Jan-2026	r1wf 35470	Each stage in the cumulative hierarchy is well-founded. (Contributed by BTernaryTau, 19-Jan-2026.)
⊢ (𝑅₁‘𝐴) ∈ ∪ (𝑅₁ “ On)

18-Jan-2026	esplysply 33942	The 𝐾-th elementary symmetric polynomial is symmetric. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐾 ∈ (0...(♯‘𝐼))) ⇒ ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) ∈ (𝐼SymPoly𝑅))

18-Jan-2026	esplyfv 33941	Coefficient for the 𝐾-th elementary symmetric polynomial and a bag of variables 𝐹: the coefficient is 1 for the bags of exactly 𝐾 variables, having exponent at most 1. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐾 ∈ (0...(♯‘𝐼))) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ 0 = (0_g‘𝑅) & ⊢ 1 = (1_r‘𝑅) ⇒ ⊢ (𝜑 → (((𝐼eSymPoly𝑅)‘𝐾)‘𝐹) = if((ran 𝐹 ⊆ {0, 1} ∧ (♯‘(𝐹 supp 0)) = 𝐾), 1 , 0 ))

18-Jan-2026	esplyfv1 33940	Coefficient for the 𝐾-th elementary symmetric polynomial and a bag of variables 𝐹 where variables are not raised to a power. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐾 ∈ (0...(♯‘𝐼))) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ 0 = (0_g‘𝑅) & ⊢ 1 = (1_r‘𝑅) & ⊢ (𝜑 → ran 𝐹 ⊆ {0, 1}) ⇒ ⊢ (𝜑 → (((𝐼eSymPoly𝑅)‘𝐾)‘𝐹) = if((♯‘(𝐹 supp 0)) = 𝐾, 1 , 0 ))

18-Jan-2026	esplymhp 33939	The 𝐾-th elementary symmetric polynomial is homogeneous of degree 𝐾. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐾 ∈ ℕ₀) & ⊢ 𝐻 = (𝐼 mHomP 𝑅) ⇒ ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) ∈ (𝐻‘𝐾))

18-Jan-2026	esplympl 33938	Elementary symmetric polynomials are polynomials. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐾 ∈ ℕ₀) & ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅)) ⇒ ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) ∈ 𝑀)

18-Jan-2026	esplylem 33937	Lemma for esplyfv 33941 and others. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐾 ∈ ℕ₀) ⇒ ⊢ (𝜑 → ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}) ⊆ 𝐷)

18-Jan-2026	esplyfval 33934	The 𝐾-th elementary polynomial for a given index 𝐼 of variables and base ring 𝑅. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ (𝜑 → 𝐾 ∈ ℕ₀) ⇒ ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) = ((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))))

18-Jan-2026	esplyval 33933	The elementary polynomials for a given index 𝐼 of variables and base ring 𝑅. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐼eSymPoly𝑅) = (𝑘 ∈ ℕ₀ ↦ ((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝑘})))))

18-Jan-2026	issply 33932	Conditions for being a symmetric polynomial. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ 𝑆 = (SymGrp‘𝐼) & ⊢ 𝑃 = (Base‘𝑆) & ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅)) & ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) & ⊢ (𝜑 → 𝐹 ∈ 𝑀) & ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ 𝑥 ∈ 𝐷) → (𝐹‘(𝑥 ∘ 𝑝)) = (𝐹‘𝑥)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝐼SymPoly𝑅))

18-Jan-2026	df-esply 33929	Define elementary symmetric polynomials. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ eSymPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑘 ∈ ℕ₀ ↦ ((ℤRHom‘𝑟) ∘ ((𝟭‘{ℎ ∈ (ℕ₀ ↑_m 𝑖) ∣ ℎ finSupp 0})‘((𝟭‘𝑖) “ {𝑐 ∈ 𝒫 𝑖 ∣ (♯‘𝑐) = 𝑘})))))

18-Jan-2026	gsumind 33646	The group sum of an indicator function of the set 𝐴 gives the size of 𝐴. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ⊆ 𝑂) & ⊢ (𝜑 → 𝐴 ∈ Fin) ⇒ ⊢ (𝜑 → (ℂ_fld Σ_g ((𝟭‘𝑂)‘𝐴)) = (♯‘𝐴))

18-Jan-2026	indfsid 33170	Conditions for a function to be an indicator function. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝑂⟶{0, 1}) ⇒ ⊢ (𝜑 → 𝐹 = ((𝟭‘𝑂)‘(𝐹 supp 0)))

18-Jan-2026	indfsd 33169	The indicator function of a finite set has finite support. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ⊆ 𝑂) & ⊢ (𝜑 → 𝐴 ∈ Fin) ⇒ ⊢ (𝜑 → ((𝟭‘𝑂)‘𝐴) finSupp 0)

18-Jan-2026	hashimaf1 33136	Taking the image of a set by a one-to-one function does not affect size. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵) & ⊢ (𝜑 → 𝐶 ⊆ 𝐴) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → (♯‘(𝐹 “ 𝐶)) = (♯‘𝐶))

18-Jan-2026	pw2cut2 28636	Cut expression for powers of two. Theorem 12 of [Conway] p. 12-13. (Contributed by Scott Fenton, 18-Jan-2026.)
⊢ ((𝐴 ∈ ℤ_s ∧ 𝑁 ∈ ℕ_0s) → (𝐴 /_su (2_s↑_s𝑁)) = ({((𝐴 -_s 1_s ) /_su (2_s↑_s𝑁))} \|s {((𝐴 +_s 1_s ) /_su (2_s↑_s𝑁))}))

18-Jan-2026	pw2ltsdiv1d 28626	Surreal less-than relationship for division by a power of two. (Contributed by Scott Fenton, 18-Jan-2026.)
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝑁 ∈ ℕ_0s) ⇒ ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ (𝐴 /_su (2_s↑_s𝑁)) <s (𝐵 /_su (2_s↑_s𝑁))))

18-Jan-2026	sltssnb 27943	Surreal set less-than of two singletons. (Contributed by Scott Fenton, 18-Jan-2026.)
⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) ⇒ ⊢ (𝜑 → ({𝐴} <<s {𝐵} ↔ 𝐴 <s 𝐵))

17-Jan-2026	ex-chn2 18695	Example: sequence <" ZZ NN QQ "> is a valid chain under the equinumerosity relation in universal domain. (Contributed by Ender Ting, 17-Jan-2026.)
⊢ ⟨“ℤℕℚ”⟩ ∈ ( ≈ Chain V)

17-Jan-2026	ex-chn1 18694	Example: a doubleton of twos is a valid chain under the identity relation and domain of integers. (Contributed by Ender Ting, 17-Jan-2026.)
⊢ ⟨“22”⟩ ∈ ( I Chain ℤ)

17-Jan-2026	chnflenfi 18685	There is a finite number of chains with fixed length over finite alphabet. Trivially holds for invalid lengths as there're no matching sequences. (Contributed by Ender Ting, 5-Jan-2025.) (Revised by Ender Ting, 17-Jan-2026.)
⊢ (𝐴 ∈ Fin → {𝑎 ∈ ( < Chain 𝐴) ∣ (♯‘𝑎) = 𝑇} ∈ Fin)

17-Jan-2026	nulchn 18676	Empty set is an increasing chain for every range and every relation. (Contributed by Ender Ting, 19-Nov-2024.) (Revised by Ender Ting, 17-Jan-2026.)
⊢ ∅ ∈ ( < Chain 𝐴)

17-Jan-2026	chnexg 18675	Chains with a set given for range form a set. (Contributed by Ender Ting, 21-Nov-2024.) (Revised by Ender Ting, 17-Jan-2026.)
⊢ (𝐴 ∈ 𝑉 → ( < Chain 𝐴) ∈ V)

17-Jan-2026	chneq12 18671	Equality theorem for chains. (Contributed by Ender Ting, 17-Jan-2026.)
⊢ (( < = 𝑅 ∧ 𝐴 = 𝐵) → ( < Chain 𝐴) = (𝑅 Chain 𝐵))

17-Jan-2026	chneq2 18670	Equality theorem for chains. (Contributed by Ender Ting, 17-Jan-2026.)
⊢ (𝐴 = 𝐵 → ( < Chain 𝐴) = ( < Chain 𝐵))

17-Jan-2026	chneq1 18669	Equality theorem for chains. (Contributed by Ender Ting, 17-Jan-2026.)
⊢ ( < = 𝑅 → ( < Chain 𝐴) = (𝑅 Chain 𝐴))

15-Jan-2026	r1ssel 35482	A set is a subset of the value of the cumulative hierarchy of sets function iff it is an element of the value at the successor. (Contributed by BTernaryTau, 15-Jan-2026.)
⊢ (𝐵 ∈ On → (𝐴 ⊆ (𝑅₁‘𝐵) ↔ 𝐴 ∈ (𝑅₁‘suc 𝐵)))

15-Jan-2026	fissorduni 35461	The union (supremum) of a finite set of ordinals less than a nonzero ordinal class is an element of that ordinal class. (Contributed by BTernaryTau, 15-Jan-2026.)
⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ (Ord 𝐵 ∧ 𝐵 ≠ ∅)) → ∪ 𝐴 ∈ 𝐵)

15-Jan-2026	splysubrg 33931	The symmetric polynomials form a subring of the ring of polynomials. (Contributed by Thierry Arnoux, 15-Jan-2026.)
⊢ 𝑆 = (SymGrp‘𝐼) & ⊢ 𝑃 = (Base‘𝑆) & ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅)) & ⊢ 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑)))) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → (𝐼SymPoly𝑅) ∈ (SubRing‘(𝐼 mPoly 𝑅)))

15-Jan-2026	mplvrpmrhm 33918	The action of permuting variables in a multivariate polynomial is a ring homomorphism. (Contributed by Thierry Arnoux, 15-Jan-2026.)
⊢ 𝑆 = (SymGrp‘𝐼) & ⊢ 𝑃 = (Base‘𝑆) & ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅)) & ⊢ 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑)))) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ 𝐹 = (𝑓 ∈ 𝑀 ↦ (𝐷𝐴𝑓)) & ⊢ 𝑊 = (𝐼 mPoly 𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝑊 RingHom 𝑊))

15-Jan-2026	cocnvf1o 33055	Composing with the inverse of a bijection. (Contributed by Thierry Arnoux, 15-Jan-2026.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐻:𝐴–1-1-onto→𝐴) ⇒ ⊢ (𝜑 → (𝐹 = (𝐺 ∘ 𝐻) ↔ 𝐺 = (𝐹 ∘ ^◡𝐻)))

15-Jan-2026	ofrco 32936	Function relation between function compositions. (Contributed by Thierry Arnoux, 15-Jan-2026.)
⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐺 Fn 𝐴) & ⊢ (𝜑 → 𝐻:𝐶⟶𝐴) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑊) & ⊢ (𝜑 → 𝐹 ∘_r 𝑅𝐺) ⇒ ⊢ (𝜑 → (𝐹 ∘ 𝐻) ∘_r 𝑅(𝐺 ∘ 𝐻))

15-Jan-2026	fnfvor 32935	Relation between two functions implies the same relation for the function value at a given 𝑋. See also fnfvof 7693. (Contributed by Thierry Arnoux, 15-Jan-2026.)
⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐺 Fn 𝐴) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 ∘_r 𝑅𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝐹‘𝑋)𝑅(𝐺‘𝑋))

15-Jan-2026	elrabrd 3654	Deduction version of elrab 3651, just like elrabd 3653, but backwards direction. (Contributed by Thierry Arnoux, 15-Jan-2026.)
⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → 𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜓}) ⇒ ⊢ (𝜑 → 𝜒)

12-Jan-2026	preel 39130	Predecessor is a subset of its successor. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ (𝑁 ∈ Suc → pre 𝑁 ∈ 𝑁)

12-Jan-2026	press 39129	Predecessor is a subset of its successor. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ (𝑁 ∈ Suc → pre 𝑁 ⊆ 𝑁)

12-Jan-2026	presuc 39128	pre is a left-inverse of suc. This theorem gives a clean rewrite rule that eliminates pre on explicit successors. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ (𝑀 ∈ 𝑉 → pre suc 𝑀 = 𝑀)

12-Jan-2026	preuniqval 39126	Uniqueness/canonicity of pre. presucmap 39125 gives one witness; this theorem gives it is the only one. It turns any predecessor proof into an equality with pre 𝑁. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ (𝑁 ∈ ran SucMap → ∀𝑚(𝑚 SucMap 𝑁 → 𝑚 = pre 𝑁))

12-Jan-2026	presucmap 39125	pre is really a predecessor (when it should be). This correctness theorem for pre makes it usable in proofs without unfolding ℩. This theorem gives one witness; preuniqval 39126 gives it is the only one. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ (𝑁 ∈ ran SucMap → pre 𝑁 SucMap 𝑁)

12-Jan-2026	eupre2 39123	Unique predecessor exists on the range of the successor map. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ ran SucMap ↔ ∃!𝑚 𝑚 SucMap 𝑁))

12-Jan-2026	preex 39122	The successor-predecessor exists. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ pre 𝑁 ∈ V

12-Jan-2026	exeupre 39121	Whenever a predecessor exists, it exists alone. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ (𝑁 ∈ 𝑉 → (∃𝑚 𝑚 SucMap 𝑁 ↔ ∃!𝑚 𝑚 SucMap 𝑁))

12-Jan-2026	dfpre3 39108	Alternate definition of the successor-predecessor. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ (𝑁 ∈ 𝑉 → pre 𝑁 = (℩𝑚 suc 𝑚 = 𝑁))

12-Jan-2026	dfpre2 39107	Alternate definition of the successor-predecessor. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ (𝑁 ∈ 𝑉 → pre 𝑁 = (℩𝑚𝑚 SucMap 𝑁))

12-Jan-2026	exeupre2 39102	Whenever a predecessor exists, it exists alone. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ (∃𝑚 suc 𝑚 = 𝑁 ↔ ∃!𝑚 suc 𝑚 = 𝑁)

12-Jan-2026	mopre 39101	There is at most one predecessor of 𝑁. (Contributed by Peter Mazsa, 12-Jan-2026.)
⊢ ∃*𝑚 suc 𝑚 = 𝑁

12-Jan-2026	fineqvnttrclse 35518	A counterexample demonstrating that ttrclse 9697 does not hold when all sets are finite. (Contributed by BTernaryTau, 12-Jan-2026.)
⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 = suc 𝑦)} & ⊢ 𝐴 = ω ⇒ ⊢ (Fin = V → (𝑅 Se 𝐴 ∧ ¬ t++(𝑅 ↾ 𝐴) Se 𝐴))

12-Jan-2026	fineqvnttrclselem3 35517	Lemma for fineqvnttrclse 35518. (Contributed by BTernaryTau, 12-Jan-2026.)
⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 = suc 𝑦)} & ⊢ 𝐴 = ω & ⊢ 𝐹 = (𝑣 ∈ suc suc 𝑁 ↦ ∪ {𝑑 ∈ On ∣ (𝑣 +_o 𝑑) = 𝐵}) ⇒ ⊢ ((𝐵 ∈ (ω ∖ 1_o) ∧ 𝑁 ∈ 𝐵) → ∀𝑎 ∈ suc 𝑁(𝐹‘𝑎)𝑅(𝐹‘suc 𝑎))

12-Jan-2026	fineqvnttrclselem2 35516	Lemma for fineqvnttrclse 35518. (Contributed by BTernaryTau, 12-Jan-2026.)
⊢ 𝐹 = (𝑣 ∈ suc suc 𝑁 ↦ ∪ {𝑑 ∈ On ∣ (𝑣 +_o 𝑑) = 𝐵}) ⇒ ⊢ ((𝐵 ∈ (ω ∖ 1_o) ∧ 𝑁 ∈ 𝐵 ∧ 𝐴 ∈ suc suc 𝑁) → (𝐴 +_o (𝐹‘𝐴)) = 𝐵)

12-Jan-2026	fineqvnttrclselem1 35515	Lemma for fineqvnttrclse 35518. (Contributed by BTernaryTau, 12-Jan-2026.)
⊢ (𝐵 ∈ (ω ∖ 1_o) → ∪ {𝑑 ∈ On ∣ (𝐴 +_o 𝑑) = 𝐵} ∈ ω)

11-Jan-2026	splyval 33930	The symmetric polynomials for a given index 𝐼 of variables and base ring 𝑅. These are the fixed points of the action 𝐴 which permutes variables. (Contributed by Thierry Arnoux, 11-Jan-2026.)
⊢ 𝑆 = (SymGrp‘𝐼) & ⊢ 𝑃 = (Base‘𝑆) & ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅)) & ⊢ 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑)))) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐼SymPoly𝑅) = (𝑀FixPts𝐴))

11-Jan-2026	df-sply 33928	Define symmetric polynomials. See splyval 33930 for a more readable expression. (Contributed by Thierry Arnoux, 11-Jan-2026.)
⊢ SymPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((Base‘(𝑖 mPoly 𝑟))FixPts(𝑑 ∈ (Base‘(SymGrp‘𝑖)), 𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ↦ (𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝑖) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))))

11-Jan-2026	mplvrpmmhm 33917	The action of permuting variables in a multivariate polynomial is a monoid homomorphism. (Contributed by Thierry Arnoux, 11-Jan-2026.)
⊢ 𝑆 = (SymGrp‘𝐼) & ⊢ 𝑃 = (Base‘𝑆) & ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅)) & ⊢ 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑)))) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ 𝐹 = (𝑓 ∈ 𝑀 ↦ (𝐷𝐴𝑓)) & ⊢ 𝑊 = (𝐼 mPoly 𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝑊 MndHom 𝑊))

11-Jan-2026	mplvrpmlem 33914	Lemma for mplvrpmga 33916 and others. (Contributed by Thierry Arnoux, 11-Jan-2026.)
⊢ 𝑆 = (SymGrp‘𝐼) & ⊢ 𝑃 = (Base‘𝑆) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0}) ⇒ ⊢ (𝜑 → (𝑋 ∘ 𝐷) ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0})

11-Jan-2026	constcof 32947	Composition with a constant function. See also fcoconst 7132. (Contributed by Thierry Arnoux, 11-Jan-2026.)
⊢ (𝜑 → 𝐹:𝑋⟶𝐼) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝐼 × {𝑌}) ∘ 𝐹) = (𝑋 × {𝑌}))

10-Jan-2026	finextalg 34069	A finite field extension is algebraic. Proposition 1.1 of [Lang], p. 224. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ (𝜑 → 𝐸/_FinExt𝐹) ⇒ ⊢ (𝜑 → 𝐸/_AlgExt𝐹)

10-Jan-2026	bralgext 34068	Express the fact that a field extension 𝐸 / 𝐹 is algebraic. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ 𝐵 = (Base‘𝐸) & ⊢ 𝐶 = (Base‘𝐹) & ⊢ (𝜑 → 𝐸 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐸/_AlgExt𝐹 ↔ (𝐸/_FldExt𝐹 ∧ (𝐸 IntgRing 𝐶) = 𝐵)))

10-Jan-2026	extdgfialg 34065	A finite field extension 𝐸 / 𝐹 is algebraic. Part of the proof of Proposition 1.1 of [Lang], p. 224. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ 𝐵 = (Base‘𝐸) & ⊢ 𝐷 = (dim‘((subringAlg ‘𝐸)‘𝐹)) & ⊢ (𝜑 → 𝐸 ∈ Field) & ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸)) & ⊢ (𝜑 → 𝐷 ∈ ℕ₀) ⇒ ⊢ (𝜑 → (𝐸 IntgRing 𝐹) = 𝐵)

10-Jan-2026	extdgfialglem2 34064	Lemma for extdgfialg 34065. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ 𝐵 = (Base‘𝐸) & ⊢ 𝐷 = (dim‘((subringAlg ‘𝐸)‘𝐹)) & ⊢ (𝜑 → 𝐸 ∈ Field) & ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸)) & ⊢ (𝜑 → 𝐷 ∈ ℕ₀) & ⊢ 𝑍 = (0_g‘𝐸) & ⊢ · = (._r‘𝐸) & ⊢ 𝐺 = (𝑛 ∈ (0...𝐷) ↦ (𝑛(._g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝐴:(0...𝐷)⟶𝐹) & ⊢ (𝜑 → 𝐴 finSupp 𝑍) & ⊢ (𝜑 → (𝐸 Σ_g (𝐴 ∘_f · 𝐺)) = 𝑍) & ⊢ (𝜑 → 𝐴 ≠ ((0...𝐷) × {𝑍})) ⇒ ⊢ (𝜑 → 𝑋 ∈ (𝐸 IntgRing 𝐹))

10-Jan-2026	extdgfialglem1 34063	Lemma for extdgfialg 34065. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ 𝐵 = (Base‘𝐸) & ⊢ 𝐷 = (dim‘((subringAlg ‘𝐸)‘𝐹)) & ⊢ (𝜑 → 𝐸 ∈ Field) & ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸)) & ⊢ (𝜑 → 𝐷 ∈ ℕ₀) & ⊢ 𝑍 = (0_g‘𝐸) & ⊢ · = (._r‘𝐸) & ⊢ 𝐺 = (𝑛 ∈ (0...𝐷) ↦ (𝑛(._g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → ∃𝑎 ∈ (𝐹 ↑_m (0...𝐷))(𝑎 finSupp 𝑍 ∧ ((𝐸 Σ_g (𝑎 ∘_f · 𝐺)) = 𝑍 ∧ 𝑎 ≠ ((0...𝐷) × {𝑍}))))

10-Jan-2026	finextfldext 34035	A finite field extension is a field extension. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ (𝜑 → 𝐸/_FinExt𝐹) ⇒ ⊢ (𝜑 → 𝐸/_FldExt𝐹)

10-Jan-2026	srapwov 33960	The "power" operation on a subring algebra. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ 𝐴 = ((subringAlg ‘𝑊)‘𝑆) & ⊢ (𝜑 → 𝑊 ∈ Ring) & ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) ⇒ ⊢ (𝜑 → (._g‘(mulGrp‘𝑊)) = (._g‘(mulGrp‘𝐴)))

10-Jan-2026	mplvrpmga 33916	The action of permuting variables in a multivariate polynomial is a group action. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ 𝑆 = (SymGrp‘𝐼) & ⊢ 𝑃 = (Base‘𝑆) & ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅)) & ⊢ 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑)))) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐴 ∈ (𝑆 GrpAct 𝑀))

10-Jan-2026	mplvrpmfgalem 33915	Permuting variables in a multivariate polynomial conserves finite support. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ 𝑆 = (SymGrp‘𝐼) & ⊢ 𝑃 = (Base‘𝑆) & ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅)) & ⊢ 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑)))) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ 0 = (0_g‘𝑅) & ⊢ (𝜑 → 𝐹 ∈ 𝑀) & ⊢ (𝜑 → 𝑄 ∈ 𝑃) ⇒ ⊢ (𝜑 → (𝑄𝐴𝐹) finSupp 0 )

10-Jan-2026	psrbasfsupp 33882	Rewrite a finite support for nonnegative integers : For functions mapping a set 𝐼 to the nonnegative integers, having finite support can also be written as having a finite preimage of the positive integers. The latter expression is used for example in psrbas 22065, but with the former expression, theorems about finite support can be used more directly. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ 𝐷 = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ 𝑓 finSupp 0} ⇒ ⊢ 𝐷 = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}

10-Jan-2026	evls1monply1 33850	Subring evaluation of a scaled monomial. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ 𝑄 = (𝑆 evalSub₁ 𝑅) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝑊 = (Poly₁‘𝑈) & ⊢ 𝑈 = (𝑆 ↾_s 𝑅) & ⊢ 𝑋 = (var₁‘𝑈) & ⊢ ↑ = (._g‘(mulGrp‘𝑊)) & ⊢ ∧ = (._g‘(mulGrp‘𝑆)) & ⊢ ∗ = ( ·_𝑠 ‘𝑊) & ⊢ · = (._r‘𝑆) & ⊢ (𝜑 → 𝑆 ∈ CRing) & ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) & ⊢ (𝜑 → 𝐴 ∈ 𝑅) & ⊢ (𝜑 → 𝑁 ∈ ℕ₀) & ⊢ (𝜑 → 𝑌 ∈ 𝐾) ⇒ ⊢ (𝜑 → ((𝑄‘(𝐴 ∗ (𝑁 ↑ 𝑋)))‘𝑌) = (𝐴 · (𝑁 ∧ 𝑌)))

10-Jan-2026	fcobijfs2 33048	Composing finitely supported functions with a bijection yields a bijection between sets of finitely supported functions. See also fcobijfs 33047 and mapfien 9369. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ (𝜑 → 𝐺:𝑅–1-1-onto→𝑆) & ⊢ (𝜑 → 𝑅 ∈ 𝑈) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑇 ∈ 𝑊) & ⊢ (𝜑 → 𝑂 ∈ 𝑇) & ⊢ 𝑋 = {𝑔 ∈ (𝑇 ↑_m 𝑆) ∣ 𝑔 finSupp 𝑂} & ⊢ 𝑌 = {ℎ ∈ (𝑇 ↑_m 𝑅) ∣ ℎ finSupp 𝑂} ⇒ ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝑓 ∘ 𝐺)):𝑋–1-1-onto→𝑌)

10-Jan-2026	f1oeq3dd 32955	Equality deduction for one-to-one onto functions. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐵)

10-Jan-2026	fconst7v 32946	An alternative way to express a constant function. (Contributed by Glauco Siliprandi, 5-Feb-2022.) Removed hyphotheses as suggested by SN (Revised by Thierry Arnoux, 10-Jan-2026.)
⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵) ⇒ ⊢ (𝜑 → 𝐹 = (𝐴 × {𝐵}))

10-Jan-2026	breq2dd 5129	Equality deduction for a binary relation. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶𝑅𝐴) ⇒ ⊢ (𝜑 → 𝐶𝑅𝐵)

10-Jan-2026	breq1dd 5128	Equality deduction for a binary relation. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐴𝑅𝐶) ⇒ ⊢ (𝜑 → 𝐵𝑅𝐶)

8-Jan-2026	sucmapleftuniq 39120	Left uniqueness of the successor mapping. (Contributed by Peter Mazsa, 8-Jan-2026.)
⊢ ((𝐿 ∈ 𝑉 ∧ 𝑀 ∈ 𝑊 ∧ 𝑁 ∈ 𝑋) → ((𝐿 SucMap 𝑁 ∧ 𝑀 SucMap 𝑁) → 𝐿 = 𝑀))

7-Jan-2026	sucmapsuc 39119	A set is succeeded by its successor. (Contributed by Peter Mazsa, 7-Jan-2026.)
⊢ (𝑀 ∈ 𝑉 → 𝑀 SucMap suc 𝑀)

7-Jan-2026	dmsucmap 39098	The domain of the successor map is the universe. (Contributed by Peter Mazsa, 7-Jan-2026.)
⊢ dom SucMap = V

7-Jan-2026	relsucmap 39097	The successor map is a relation. (Contributed by Peter Mazsa, 7-Jan-2026.)
⊢ Rel SucMap

6-Jan-2026	brsucmap 39096	Binary relation form of the successor map, general version. (Contributed by Peter Mazsa, 6-Jan-2026.)
⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → (𝑀 SucMap 𝑁 ↔ suc 𝑀 = 𝑁))

6-Jan-2026	dfsuccf2 36414	Alternate definition of Scott Fenton's version of Succ, cf. df-sucmap 39092. (Contributed by Peter Mazsa, 6-Jan-2026.)
⊢ Succ = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}

1-Jan-2026	rightge0 27995	A surreal is non-negative iff all its right options are positive. (Contributed by Scott Fenton, 1-Jan-2026.)
⊢ (𝜑 → 𝐴 <<s 𝐵) & ⊢ (𝜑 → 𝑋 = (𝐴 \|s 𝐵)) ⇒ ⊢ (𝜑 → ( 0_s ≤s 𝑋 ↔ ∀𝑥_𝑅 ∈ 𝐵 0_s <s 𝑥_𝑅))

31-Dec-2025	tz9.1regs 35528	Every set has a transitive closure (the smallest transitive extension). This version of tz9.1 9699 depends on ax-regs 35520 instead of ax-reg 9555 and ax-inf2 9611. This suggests a possible answer to the third question posed in tz9.1 9699, namely that the missing property is that countably infinite classes must obey regularity. In ZF set theory we can prove this by showing that countably infinite classes are sets and thus ax-reg 9555 applies to them directly, but in a finitist context it seems that an axiom like ax-regs 35520 is required since countably infinite classes are proper classes. A related candidate for the missing property is the non-existence of infinite descending ∈-chains, proven as noinfep 9630 using ax-reg 9555 and ax-inf2 9611 and as noinfepregs 35527 using ax-regs 35520. If all sets are finite, then the existence of such a chain implies there is a set which does not have a transitive closure, as shown in fineqvinfep 35519. (Contributed by BTernaryTau, 31-Dec-2025.)
⊢ 𝐴 ∈ V ⇒ ⊢ ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦))

31-Dec-2025	setinds2regs 35525	Principle of set induction (or E-induction). If a property passes from all elements of 𝑥 to 𝑥 itself, then it holds for all 𝑥. (Contributed by BTernaryTau, 31-Dec-2025.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (∀𝑦 ∈ 𝑥 𝜓 → 𝜑) ⇒ ⊢ 𝜑

31-Dec-2025	nelaneqOLD 9566	Obsolete version of nelaneq 9565 as of 22-Apr-2026. (Proposed by BJ, 18-Jun-2022.) (Contributed by AV, 18-Jun-2022.) (Proof shortened by TM, 31-Dec-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐴 = 𝐵)

31-Dec-2025	zfregcl 9557	The Axiom of Regularity with class variables. (Contributed by NM, 5-Aug-1994.) Replace sethood hypothesis with sethood antecedent. (Revised by BJ, 27-Apr-2021.) Avoid ax-10 2176 and ax-12 2213. (Revised by TM, 31-Dec-2025.)
⊢ (𝐴 ∈ 𝑉 → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝐴))

31-Dec-2025	dmcosseq 5970	Domain of a composition. (Contributed by NM, 28-May-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Avoid ax-11 2192. (Revised by BTernaryTau, 23-Jun-2025.) Avoid ax-10 2176 and ax-12 2213. (Revised by TM, 31-Dec-2025.)
⊢ (ran 𝐵 ⊆ dom 𝐴 → dom (𝐴 ∘ 𝐵) = dom 𝐵)

31-Dec-2025	dmcoss 5967	Domain of a composition. Theorem 21 of [Suppes] p. 63. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Avoid ax-10 2176 and ax-12 2213. (Revised by TM, 31-Dec-2025.)
⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵

30-Dec-2025	grlimedgnedg 48879	In general, the image of an edge of a graph by a local isomprphism is not an edge of the other graph, proven by an example (see gpg5edgnedg 48878). This theorem proves that the analogon (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ 𝐾 ∈ 𝐼)) → (𝐹 “ 𝐾) ∈ 𝐸) of grimedgi 48684 for ordinarily isomorphic graphs does not hold in general. (Contributed by AV, 30-Dec-2025.)
⊢ ∃𝑔 ∈ USGraph ∃ℎ ∈ USGraph ∃𝑓 ∈ (𝑔 GraphLocIso ℎ)∃𝑎 ∈ (Vtx‘𝑔)∃𝑏 ∈ (Vtx‘𝑔)({𝑎, 𝑏} ∈ (Edg‘𝑔) ∧ {(𝑓‘𝑎), (𝑓‘𝑏)} ∉ (Edg‘ℎ))

30-Dec-2025	grimedgi 48684	Graph isomorphisms map edges onto the corresponding edges. (Contributed by AV, 30-Dec-2025.)
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (Edg‘𝐺) & ⊢ 𝐸 = (Edg‘𝐻) ⇒ ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → (𝐾 ∈ 𝐼 → (𝐹 “ 𝐾) ∈ 𝐸))

30-Dec-2025	fineqvr1ombregs 35532	All sets are finite iff all sets are hereditarily finite. (Contributed by BTernaryTau, 30-Dec-2025.)
⊢ (Fin = V ↔ ∪ (𝑅₁ “ ω) = V)

30-Dec-2025	unir1regs 35529	The cumulative hierarchy of sets covers the universe. This version of unir1 9786 replaces setind 9717 with setindregs 35524. (Contributed by BTernaryTau, 30-Dec-2025.)
⊢ ∪ (𝑅₁ “ On) = V

30-Dec-2025	setindregs 35524	Set (epsilon) induction. This version of setind 9717 replaces zfregs 9702 with axregszf 35523. (Contributed by BTernaryTau, 30-Dec-2025.)
⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → 𝐴 = V)

30-Dec-2025	axregszf 35523	Derivation of zfregs 9702 using ax-regs 35520. (Contributed by BTernaryTau, 30-Dec-2025.)
⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)

30-Dec-2025	axregscl 35522	A version of ax-regs 35520 with a class variable instead of a wff variable. Axiom D in Gödel, The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory (1940), p. 6. (Contributed by BTernaryTau, 30-Dec-2025.)
⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝐴)))

30-Dec-2025	axreg 35521	Derivation of ax-reg 9555 from ax-regs 35520 and Tarski's FOL axiom schemes. This demonstrates the sense in which ax-regs 35520 is a stronger version of ax-reg 9555. (Contributed by BTernaryTau, 30-Dec-2025.)
⊢ (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)))

30-Dec-2025	fineqvomon 35512	If all sets are finite, then the class of all natural numbers equals the proper class of all ordinal numbers. (Contributed by BTernaryTau, 30-Dec-2025.)
⊢ (Fin = V → ω = On)

30-Dec-2025	r1omfi 35480	Hereditarily finite sets are finite sets. (Contributed by BTernaryTau, 30-Dec-2025.)
⊢ ∪ (𝑅₁ “ ω) ⊆ Fin

30-Dec-2025	r1elcl 35472	Each set of the cumulative hierarchy is closed under membership. (Contributed by BTernaryTau, 30-Dec-2025.)
⊢ ((𝐴 ∈ (𝑅₁‘𝐵) ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ (𝑅₁‘𝐵))

Older news:

(29-Jul-2020) Mario Carneiro presented MM0 at the CICM conference. See this Google Group post which includes a YouTube link.

(20-Jul-2020) Rohan Ridenour found 5 shorter D-proofs in our Shortest known proofs... file. In particular, he reduced *4.39 from 901 to 609 steps. A note on the Metamath Solitaire page mentions a tool that he worked with.

(19-Jul-2020) David A. Wheeler posted a video (https://youtu.be/3R27Qx69jHc) on how to (re)prove Schwabh�user 4.6 for the Metamath Proof Explorer. See also his older videos.

(19-Jul-2020) In version 0.184 of the metamath program, "verify markup" now checks that mathboxes are independent i.e. do not cross-reference each other. To turn off this check, use "/mathbox_skip"

(30-Jun-2020) In version 0.183 of the metamath program, (1) "verify markup" now has checking for (i) underscores in labels, (ii) that *ALT and *OLD theorems have both discouragement tags, and (iii) that lines don't have trailing spaces. (2) "save proof.../rewrap" no longer left-aligns $p/$a comments that contain the string "<HTML>"; see this note.

(5-Apr-2020) Glauco Siliprandi added a new proof to the 100 theorem list, e is Transcendental etransc, bringing the Metamath total to 74.

(12-Feb-2020) A bug in the 'minimize' command of metamath.exe versions 0.179 (29-Nov-2019) and 0.180 (10-Dec-2019) may incorrectly bring in the use of new axioms. Version 0.181 fixes it.

(20-Jan-2020) David A. Wheeler created a video called Walkthrough of the tutorial in mmj2. See the Google Group announcement for more details. (All of his videos are listed on the Other Metamath-Related Topics page.)

(18-Jan-2020) The FOMM 2020 talks are on youtube now. Mario Carneiro's talk is Metamath Zero, or: How to Verify a Verifier. Since they are washed out in the video, the PDF slides are available separately.

(14-Dec-2019) Glauco Siliprandi added a new proof to the 100 theorem list, Fourier series convergence fourier, bringing the Metamath total to 73.

(25-Nov-2019) Alexander van der Vekens added a new proof to the 100 theorem list, The Cayley-Hamilton Theorem cayleyhamilton, bringing the Metamath total to 72.

(25-Oct-2019) Mario Carneiro's paper "Metamath Zero: The Cartesian Theorem Prover" (submitted to CPP 2020) is now available on arXiv: https://arxiv.org/abs/1910.10703. There is a related discussion on Hacker News.

(30-Sep-2019) Mario Carneiro's talk about MM0 at ITP 2019 is available on YouTube: x86 verification from scratch (24 minutes). Google Group discussion: Metamath Zero.

(29-Sep-2019) David Wheeler created a fascinating Gource video that animates the construction of set.mm, available on YouTube: Metamath set.mm contributions viewed with Gource through 2019-09-26 (4 minutes). Google Group discussion: Gource video of set.mm contributions.

(24-Sep-2019) nLab added a page for Metamath. It mentions Stefan O'Rear's Busy Beaver work using the set.mm axiomatization (and fails to mention Mario's definitional soundness checker)

(1-Sep-2019) Xuanji Li published a Visual Studio Code extension to support metamath syntax highlighting.

(10-Aug-2019) (revised 21-Sep-2019) Version 0.178 of the metamath program has the following changes: (1) "minimize_with" will now prevent dependence on new $a statements unless the new qualifier "/allow_new_axioms" is specified. For routine usage, it is suggested that you use "minimize_with * /allow_new_axioms * /no_new_axioms_from ax-*" instead of just "minimize_with *". See "help minimize_with" and this Google Group post. Also note that the qualifier "/allow_growth" has been renamed to "/may_grow". (2) "/no_versioning" was added to "write theorem_list".

(8-Jul-2019) Jon Pennant announced the creation of a Metamath search engine. Try it and feel free to comment on it at https://groups.google.com/d/msg/metamath/cTeU5AzUksI/5GesBfDaCwAJ.

(16-May-2019) Set.mm now has a major new section on elementary geometry. This begins with definitions that implement Tarski's axioms of geometry (including concepts such as congruence and betweenness). This uses set.mm's extensible structures, making them easier to use for many circumstances. The section then connects Tarski geometry with geometry in Euclidean places. Most of the work in this section is due to Thierry Arnoux, with earlier work by Mario Carneiro and Scott Fenton. [Reported by DAW.]

(9-May-2019) We are sad to report that long-time contributor Alan Sare passed away on Mar. 23. There is some more information at the top of his mathbox (click on "Mathbox for Alan Sare") and his obituary. We extend our condolences to his family.

(10-Mar-2019) Jon Pennant and Mario Carneiro added a new proof to the 100 theorem list, Heron's formula heron, bringing the Metamath total to 71.

(22-Feb-2019) Alexander van der Vekens added a new proof to the 100 theorem list, Cramer's rule cramer, bringing the Metamath total to 70.

(6-Feb-2019) David A. Wheeler has made significant improvements and updates to the Metamath book. Any comments, errors found, or suggestions are welcome and should be turned into an issue or pull request at https://github.com/metamath/metamath-book (or sent to me if you prefer).

(26-Dec-2018) I added Appendix 8 to the MPE Home Page that cross-references new and old axiom numbers.

(20-Dec-2018) The axioms have been renumbered according to this Google Groups post.

(24-Nov-2018) Thierry Arnoux created a new page on topological structures. The page along with its SVG files are maintained on GitHub.

(11-Oct-2018) Alexander van der Vekens added a new proof to the 100 theorem list, the Friendship Theorem friendship, bringing the Metamath total to 69.

(1-Oct-2018) Naip Moro has written gramm, a Metamath proof verifier written in Antlr4/Java.

(16-Sep-2018) The definition df-riota has been simplified so that it evaluates to the empty set instead of an Undef value. This change affects a significant part of set.mm.

(2-Sep-2018) Thierry Arnoux added a new proof to the 100 theorem list, Euler's partition theorem eulerpart, bringing the Metamath total to 68.

(1-Sep-2018) The Kate editor now has Metamath syntax highlighting built in. (Communicated by Wolf Lammen.)

(15-Aug-2018) The Intuitionistic Logic Explorer now has a Most Recent Proofs page.

(4-Aug-2018) Version 0.163 of the metamath program now indicates (with an asterisk) which Table of Contents headers have associated comments.

(10-May-2018) George Szpiro, journalist and author of several books on popular mathematics such as Poincare's Prize and Numbers Rule, used a genetic algorithm to find shorter D-proofs of "*3.37" and "meredith" in our Shortest known proofs... file.

(19-Apr-2018) The EMetamath Eclipse plugin has undergone many improvements since its initial release as the change log indicates. Thierry uses it as his main proof assistant and writes, "I added support for mmj2's auto-transformations, which allows it to infer several steps when building proofs. This added a lot of comfort for writing proofs.... I can now switch back and forth between the proof assistant and editing the Metamath file.... I think no other proof assistant has this feature."

(11-Apr-2018) Benoît Jubin solved an open problem about the "Axiom of Twoness," showing that it is necessary for completeness. See item 14 on the "Open problems and miscellany" page.

(25-Mar-2018) Giovanni Mascellani has announced mmpp, a new proof editing environment for the Metamath language.

(27-Feb-2018) Bill Hale has released an app for the Apple iPad and desktop computer that allows you to browse Metamath theorems and their proofs.

(17-Jan-2018) Dylan Houlihan has kindly provided a new mirror site. He has also provided an rsync server; type "rsync uk.metamath.org::" in a bash shell to check its status (it should return "metamath metamath").

(15-Jan-2018) The metamath program, version 0.157, has been updated to implement the file inclusion conventions described in the 21-Dec-2017 entry of mmnotes.txt.

(11-Dec-2017) I added a paragraph, suggested by Gérard Lang, to the distinct variable description here.

(10-Dec-2017) Per FL's request, his mathbox will be removed from set.mm. If you wish to export any of his theorems, today's version (master commit 1024a3a) is the last one that will contain it.

(11-Nov-2017) Alan Sare updated his completeusersproof program.

(3-Oct-2017) Sean B. Palmer created a web page that runs the metamath program under emulated Linux in JavaScript. He also wrote some programs to work with our shortest known proofs of the PM propositional calculus theorems.

(28-Sep-2017) Ivan Kuckir wrote a tutorial blog entry, Introduction to Metamath, that summarizes the language syntax. (It may have been written some time ago, but I was not aware of it before.)

(26-Sep-2017) The default directory for the Metamath Proof Explorer (MPE) has been changed from the GIF version (mpegif) to the Unicode version (mpeuni) throughout the site. Please let me know if you find broken links or other issues.

(24-Sep-2017) Saveliy Skresanov added a new proof to the 100 theorem list, Ceva's Theorem cevath, bringing the Metamath total to 67.

(3-Sep-2017) Brendan Leahy added a new proof to the 100 theorem list, Area of a Circle areacirc, bringing the Metamath total to 66.

(7-Aug-2017) Mario Carneiro added a new proof to the 100 theorem list, Principle of Inclusion/Exclusion incexc, bringing the Metamath total to 65.

(1-Jul-2017) Glauco Siliprandi added a new proof to the 100 theorem list, Stirling's Formula stirling, bringing the Metamath total to 64. Related theorems include 2 versions of Wallis' formula for π (wallispi and wallispi2).

(7-May-2017) Thierry Arnoux added a new proof to the 100 theorem list, Betrand's Ballot Problem ballotth, bringing the Metamath total to 63.

(20-Apr-2017) Glauco Siliprandi added a new proof in the supplementary list on the 100 theorem list, Stone-Weierstrass Theorem stowei.

(28-Feb-2017) David Moews added a new proof to the 100 theorem list, Product of Segments of Chords chordthm, bringing the Metamath total to 62.

(1-Jan-2017) Saveliy Skresanov added a new proof to the 100 theorem list, Isosceles triangle theorem isosctr, bringing the Metamath total to 61.

(1-Jan-2017) Mario Carneiro added 2 new proofs to the 100 theorem list, L'Hôpital's Rule lhop and Taylor's Theorem taylth, bringing the Metamath total to 60.

(28-Dec-2016) David A. Wheeler is putting together a page on Metamath (specifically set.mm) conventions. Comments are welcome on the Google Group thread.

(24-Dec-2016) Mario Carneiro introduced the abbreviation "F/ x ph" (symbols: turned F, x, phi) in df-nf to represent the "effectively not free" idiom "A. x ( ph -> A. x ph )". Theorem nf2 shows a version without nested quantifiers.

(22-Dec-2016) Naip Moro has developed a Metamath database for G. Spencer-Brown's Laws of Form. You can follow the Google Group discussion here.

(20-Dec-2016) In metamath program version 0.137, 'verify markup *' now checks that ax-XXX $a matches axXXX $p when the latter exists, per the discussion at https://groups.google.com/d/msg/metamath/Vtz3CKGmXnI/Fxq3j1I_EQAJ.

(24-Nov-2016) Mingl Yuan has kindly provided a mirror site in Beijing, China. He has also provided an rsync server; type "rsync cn.metamath.org::" in a bash shell to check its status (it should return "metamath metamath").

(14-Aug-2016) All HTML pages on this site should now be mobile-friendly and pass the Mobile-Friendly Test. If you find one that does not, let me know.

(14-Aug-2016) Daniel Whalen wrote a paper describing the use of using deep learning to prove 14% of test theorems taken from set.mm: Holophrasm: a neural Automated Theorem Prover for higher-order logic. The associated program is called Holophrasm.

(14-Aug-2016) David A. Wheeler created a video called Metamath Proof Explorer: A Modern Principia Mathematica

(12-Aug-2016) A Gitter chat room has been created for Metamath.

(9-Aug-2016) Mario Carneiro wrote a Metamath proof verifier in the Scala language as part of the ongoing Metamath -> MMT import project

(9-Aug-2016) David A. Wheeler created a GitHub project called metamath-test (last execution run) to check that different verifiers both pass good databases and detect errors in defective ones.

(4-Aug-2016) Mario gave two presentations at CICM 2016.

(17-Jul-2016) Thierry Arnoux has written EMetamath, a Metamath plugin for the Eclipse IDE.

(16-Jul-2016) Mario recovered Chris Capel's collapsible proof demo.

(13-Jul-2016) FL sent me an updated version of PDF (LaTeX source) developed with Lamport's pf2 package. See the 23-Apr-2012 entry below.

(12-Jul-2016) David A. Wheeler produced a new video for mmj2 called "Creating functions in Metamath". It shows a more efficient approach than his previous recent video "Creating functions in Metamath" (old) but it can be of interest to see both approaches.

(10-Jul-2016) Metamath program version 0.132 changes the command 'show restricted' to 'show discouraged' and adds a new command, 'set discouragement'. See the mmnotes.txt entry of 11-May-2016 (updated 10-Jul-2016).

(12-Jun-2016) Dan Getz has written Metamath.jl, a Metamath proof verifier written in the Julia language.

(10-Jun-2016) If you are using metamath program versions 0.128, 0.129, or 0.130, please update to version 0.131. (In the bad versions, 'minimize_with' ignores distinct variable violations.)

(1-Jun-2016) Mario Carneiro added new proofs to the 100 theorem list, the Prime Number Theorem pnt and the Perfect Number Theorem perfect, bringing the Metamath total to 58.

(12-May-2016) Mario Carneiro added a new proof to the 100 theorem list, Dirichlet's theorem dirith, bringing the Metamath total to 56. (Added 17-May-2016) An informal exposition of the proof can be found at http://metamath-blog.blogspot.com/2016/05/dirichlets-theorem.html

(10-Mar-2016) Metamath program version 0.125 adds a new qualifier, /fast, to 'save proof'. See the mmnotes.txt entry of 10-Mar-2016.

(6-Mar-2016) The most recent set.mm has a large update converting variables from letters to symbols. See this Google Groups post.

(16-Feb-2016) Mario Carneiro's new paper "Models for Metamath" can be found here and on arxiv.org.

(6-Feb-2016) There are now 22 math symbols that can be used as variable names. See mmascii.html near the 50th table row, starting with "./\".

(29-Jan-2016) Metamath program version 0.123 adds /packed and /explicit qualifiers to 'save proof' and 'show proof'. See this Google Groups post.

(13-Jan-2016) The Unicode math symbols now provide for external CSS and use the XITS web font. Thanks to David A. Wheeler, Mario Carneiro, Cris Perdue, Jason Orendorff, and Frédéric Liné for discussions on this topic. Two commands, htmlcss and htmlfont, were added to the $t comment in set.mm and are recognized by Metamath program version 0.122.

(21-Dec-2015) Axiom ax-12, now renamed ax-12o, was replaced by a new shorter equivalent, ax-12. The equivalence is provided by theorems ax12o and ax12.

(13-Dec-2015) A new section on the theory of classes was added to the MPE Home Page. Thanks to Gérard Lang for suggesting this section and improvements to it.

(17-Nov-2015) Metamath program version 0.121: 'verify markup' was added to check comment markup consistency; see 'help verify markup'. You are encouraged to make sure 'verify markup */f' has no warnings prior to mathbox submissions. The date consistency rules are given in this Google Groups post.

(23-Sep-2015) Drahflow wrote, "I am currently working on yet another proof assistant, main reason being: I understand stuff best if I code it. If anyone is interested: https://github.com/Drahflow/Igor (but in my own programming language, so expect a complicated build process :P)"

(23-Aug-2015) Ivan Kuckir created MM Tool, a Metamath proof verifier and editor written in JavaScript that runs in a browser.

(25-Jul-2015) Axiom ax-10 is shown to be redundant by theorem ax10 , so it was removed from the predicate calculus axiom list.

(19-Jul-2015) Mario Carneiro gave two talks related to Metamath at CICM 2015, which are linked to at Other Metamath-Related Topics.

(18-Jul-2015) The metamath program has been updated to version 0.118. 'show trace_back' now has a '/to' qualifier to show the path back to a specific axiom such as ax-ac. See 'help show trace_back'.

(12-Jul-2015) I added the HOL Explorer for Mario Carneiro's hol.mm database. Although the home page needs to be filled out, the proofs can be accessed.

(11-Jul-2015) I started a new page, Other Metamath-Related Topics, that will hold miscellaneous material that doesn't fit well elsewhere (or is hard to find on this site). Suggestions welcome.

(23-Jun-2015) Metamath's mascot, Penny the cat (2007 photo), passed away today. She was 18 years old.

(21-Jun-2015) Mario Carneiro added 3 new proofs to the 100 theorem list: All Primes (1 mod 4) Equal the Sum of Two Squares 2sq, The Law of Quadratic Reciprocity lgsquad and the AM-GM theorem amgm, bringing the Metamath total to 55.

(13-Jun-2015) Stefan O'Rear's smm, written in JavaScript, can now be used as a standalone proof verifier. This brings the total number of independent Metamath verifiers to 8, written in just as many languages (C, Java. JavaScript, Python, Haskell, Lua, C#, C++).

(12-Jun-2015) David A. Wheeler added 2 new proofs to the 100 theorem list: The Law of Cosines lawcos and Ptolemy's Theorem ptolemy, bringing the Metamath total to 52.

(30-May-2015) The metamath program has been updated to version 0.117. (1) David A. Wheeler provided an enhancement to speed up the 'improve' command by 28%; see README.TXT for more information. (2) In web pages with proofs, local hyperlinks on step hypotheses no longer clip the Expression cell at the top of the page.

(9-May-2015) Stefan O'Rear has created an archive of older set.mm releases back to 1998: https://github.com/sorear/set.mm-history/.

(7-May-2015) The set.mm dated 7-May-2015 is a major revision, updated by Mario, that incorporates the new ordered pair definition df-op that was agreed upon. There were 700 changes, listed at the top of set.mm. Mathbox users are advised to update their local mathboxes. As usual, if any mathbox user has trouble incorporating these changes into their mathbox in progress, Mario or I will be glad to do them for you.

(7-May-2015) Mario has added 4 new theorems to the 100 theorem list: Ramsey's Theorem ramsey, The Solution of a Cubic cubic, The Solution of the General Quartic Equation quart, and The Birthday Problem birthday. In the Supplementary List, Stefan O'Rear added the Hilbert Basis Theorem hbt.

(28-Apr-2015) A while ago, Mario Carneiro wrote up a proof of the unambiguity of set.mm's grammar, which has now been added to this site: grammar-ambiguity.txt.

(22-Apr-2015) The metamath program has been updated to version 0.114. In MM-PA, 'show new_proof/unknown' now shows the relative offset (-1, -2,...) used for 'assign' arguments, suggested by Stefan O'Rear.

(20-Apr-2015) I retrieved an old version of the missing "Metamath 100" page from archive.org and updated it to what I think is the current state: mm_100.html. Anyone who wants to edit it can email updates to this page to me.

(19-Apr-2015) The metamath program has been updated to version 0.113, mostly with patches provided by Stefan O'Rear. (1) 'show statement %' (or any command allowing label wildcards) will select statements whose proofs were changed in current session. ('help search' will show all wildcard matching rules.) (2) 'show statement =' will select the statement being proved in MM-PA. (3) The proof date stamp is now created only if the proof is complete.

(18-Apr-2015) There is now a section for Scott Fenton's NF database: New Foundations Explorer.

(16-Apr-2015) Mario describes his recent additions to set.mm at https://groups.google.com/forum/#!topic/metamath/VAGNmzFkHCs. It include 2 new additions to the Formalizing 100 Theorems list, Leibniz' series for pi (leibpi) and the Konigsberg Bridge problem (konigsberg)

(10-Mar-2015) Mario Carneiro has written a paper, "Arithmetic in Metamath, Case Study: Bertrand's Postulate," for CICM 2015. A preprint is available at arXiv:1503.02349.

(23-Feb-2015) Scott Fenton has created a Metamath formalization of NF set theory: https://github.com/sctfn/metamath-nf/. For more information, see the Metamath Google Group posting.

(28-Jan-2015) Mario Carneiro added Wilson's Theorem (wilth), Ascending or Descending Sequences (erdsze, erdsze2), and Derangements Formula (derangfmla, subfaclim), bringing the Metamath total for Formalizing 100 Theorems to 44.

(19-Jan-2015) Mario Carneiro added Sylow's Theorem (sylow1, sylow2, sylow2b, sylow3), bringing the Metamath total for Formalizing 100 Theorems to 41.

(9-Jan-2015) The hypothesis order of mpbi*an* was changed. See the Notes entry of 9-Jan-2015.

(1-Jan-2015) Mario Carneiro has written a paper, "Conversion of HOL Light proofs into Metamath," that has been submitted to the Journal of Formalized Reasoning. A preprint is available on arxiv.org.

(22-Nov-2014) Stefan O'Rear added the Solutions to Pell's Equation (rmxycomplete) and Liouville's Theorem and the Construction of Transcendental Numbers (aaliou), bringing the Metamath total for Formalizing 100 Theorems to 40.

(22-Nov-2014) The metamath program has been updated with version 0.111. (1) Label wildcards now have a label range indicator "~" so that e.g. you can show or search all of the statements in a mathbox. See 'help search'. (Stefan O'Rear added this to the program.) (2) A qualifier was added to 'minimize_with' to prevent the use of any axioms not already used in the proof e.g. 'minimize_with * /no_new_axioms_from ax-*' will prevent the use of ax-ac if the proof doesn't already use it. See 'help minimize_with'.

(10-Oct-2014) Mario Carneiro has encoded the axiomatic basis for the HOL theorem prover into a Metamath source file, hol.mm.

(24-Sep-2014) Mario Carneiro added the Sum of the Angles of a Triangle (ang180), bringing the Metamath total for Formalizing 100 Theorems to 38.

(15-Sep-2014) Mario Carneiro added the Fundamental Theorem of Algebra (fta), bringing the Metamath total for Formalizing 100 Theorems to 37.

(3-Sep-2014) Mario Carneiro added the Fundamental Theorem of Integral Calculus (ftc1, ftc2). This brings the Metamath total for Formalizing 100 Theorems to 35. (added 14-Sep-2014) Along the way, he added the Mean Value Theorem (mvth), bringing the total to 36.

(16-Aug-2014) Mario Carneiro started a Metamath blog at http://metamath-blog.blogspot.com/.

(10-Aug-2014) Mario Carneiro added Erdős's proof of the divergence of the inverse prime series (prmrec). This brings the Metamath total for Formalizing 100 Theorems to 34.

(31-Jul-2014) Mario Carneiro added proofs for Euler's Summation of 1 + (1/2)^2 + (1/3)^2 + .... (basel) and The Factor and Remainder Theorems (facth, plyrem). This brings the Metamath total for Formalizing 100 Theorems to 33.

(16-Jul-2014) Mario Carneiro added proofs for Four Squares Theorem (4sq), Formula for the Number of Combinations (hashbc), and Divisibility by 3 Rule (3dvds). This brings the Metamath total for Formalizing 100 Theorems to 31.

(11-Jul-2014) Mario Carneiro added proofs for Divergence of the Harmonic Series (harmonic), Order of a Subgroup (lagsubg), and Lebesgue Measure and Integration (itgcl). This brings the Metamath total for Formalizing 100 Theorems to 28.

(7-Jul-2014) Mario Carneiro presented a talk, "Natural Deduction in the Metamath Proof Language," at the 6PCM conference. Slides Audio

(25-Jun-2014) In version 0.108 of the metamath program, the 'minimize_with' command is now more automated. It now considers compressed proof length; it scans the statements in forward and reverse order and chooses the best; and it avoids $d conflicts. The '/no_distinct', '/brief', and '/reverse' qualifiers are obsolete, and '/verbose' no longer lists all statements scanned but gives more details about decision criteria.

(12-Jun-2014) To improve naming uniformity, theorems about operation values now use the abbreviation "ov". For example, df-opr, opreq1, oprabval5, and oprvres are now called df-ov, oveq1, ov5, and ovres respectively.

(11-Jun-2014) Mario Carneiro finished a major revision of set.mm. His notes are under the 11-Jun-2014 entry in the Notes

(4-Jun-2014) Mario Carneiro provided instructions and screenshots for syntax highlighting for the jEdit editor for use with Metamath and mmj2 source files.

(19-May-2014) Mario Carneiro added a feature to mmj2, in the build at https://github.com/digama0/mmj2/raw/dev-build/mmj2jar/mmj2.jar, which tests all but 5 definitions in set.mm for soundness. You can turn on the test by adding
SetMMDefinitionsCheckWithExclusions,ax-*,df-bi,df-clab,df-cleq,df-clel,df-sbc
to your RunParms.txt file.

(17-May-2014) A number of labels were changed in set.mm, listed at the top of set.mm as usual. Note in particular that the heavily-used visset, elisseti, syl11anc, syl111anc were changed respectively to vex, elexi, syl2anc, syl3anc.

(16-May-2014) Scott Fenton formalized a proof for "Sum of kth powers": fsumkthpow. This brings the Metamath total for Formalizing 100 Theorems to 25.

(9-May-2014) I (Norm Megill) presented an overview of Metamath at the "Formalization of mathematics in proof assistants" workshop at the Institut Henri Poincar� in Paris. The slides for this talk are here.

(22-Jun-2014) Version 0.107 of the metamath program adds a "PART" indention level to the Statement List table of contents, adds 'show proof ... /size' to show source file bytes used, and adds 'show elapsed_time'. The last one is helpful for measuring the run time of long commands. See 'help write theorem_list', 'help show proof', and 'help show elapsed_time' for more information.

(2-May-2014) Scott Fenton formalized a proof of Sum of the Reciprocals of the Triangular Numbers: trirecip. This brings the Metamath total for Formalizing 100 Theorems to 24.

(19-Apr-2014) Scott Fenton formalized a proof of the Formula for Pythagorean Triples: pythagtrip. This brings the Metamath total for Formalizing 100 Theorems to 23.

(11-Apr-2014) David A. Wheeler produced a much-needed and well-done video for mmj2, called "Introduction to Metamath & mmj2". Thanks, David!

(15-Mar-2014) Mario Carneiro formalized a proof of Bertrand's postulate: bpos. This brings the Metamath total for Formalizing 100 Theorems to 22.

(18-Feb-2014) Mario Carneiro proved that complex number axiom ax-cnex is redundant (theorem cnex). See also Real and Complex Numbers.

(11-Feb-2014) David A. Wheeler has created a theorem compilation that tracks those theorems in Freek Wiedijk's Formalizing 100 Theorems list that have been proved in set.mm. If you find a error or omission in this list, let me know so it can be corrected. (Update 1-Mar-2014: Mario has added eulerth and bezout to the list.)

(4-Feb-2014) Mario Carneiro writes:

The latest commit on the mmj2 development branch introduced an exciting new feature, namely syntax highlighting for mmp files in the main window. (You can pick up the latest mmj2.jar at https://github.com/digama0/mmj2/blob/develop/mmj2jar/mmj2.jar .) The reason I am asking for your help at this stage is to help with design for the syntax tokenizer, which is responsible for breaking down the input into various tokens with names like "comment", "set", and "stephypref", which are then colored according to the user's preference. As users of mmj2 and metamath, what types of highlighting would be useful to you?
One limitation of the tokenizer is that since (for performance reasons) it can be started at any line in the file, highly contextual coloring, like highlighting step references that don't exist previously in the file, is difficult to do. Similarly, true parsing of the formulas using the grammar is possible but likely to be unmanageably slow. But things like checking theorem labels against the database is quite simple to do under the current setup.
That said, how can this new feature be optimized to help you when writing proofs?

(13-Jan-2014) Mathbox users: the *19.21a*, *19.23a* series of theorems have been renamed to *alrim*, *exlim*. You can update your mathbox with a global replacement of string '19.21a' with 'alrim' and '19.23a' with 'exlim'.

(5-Jan-2014) If you downloaded mmj2 in the past 3 days, please update it with the current version, which fixes a bug introduced by the recent changes that made it unable to read in most of the proofs in the textarea properly.

(4-Jan-2014) I added a list of "Allowed substitutions" under the "Distinct variable groups" list on the theorem web pages, for example axsep. This is an experimental feature and comments are welcome.

(3-Jan-2014) Version 0.102 of the metamath program produces more space-efficient compressed proofs (still compatible with the specification in Appendix B of the Metamath book) using an algorithm suggested by Mario Carneiro. See 'help save proof' in the program. Also, mmj2 now generates proofs in the new format. The new mmj2 also has a mandatory update that fixes a bug related to the new format; you must update your mmj2 copy to use it with the latest set.mm.

(23-Dec-2013) Mario Carneiro has updated many older definitions to use the maps-to notation. If you have difficulty updating your local mathbox, contact him or me for assistance.

(1-Nov-2013) 'undo' and 'redo' commands were added to the Proof Assistant in metamath program version 0.07.99. See 'help undo' in the program.

(8-Oct-2013) Today's Notes entry describes some proof repair techniques.

(5-Oct-2013) Today's Notes entry explains some recent extensible structure improvements.

(8-Sep-2013) Mario Carneiro has revised the square root and sequence generator definitions. See today's Notes entry.

(3-Aug-2013) Mario Carneiro writes: "I finally found enough time to create a GitHub repository for development at https://github.com/digama0/mmj2. A permalink to the latest version plus source (akin to mmj2.zip) is https://github.com/digama0/mmj2/zipball/, and the jar file on its own (mmj2.jar) is at https://github.com/digama0/mmj2/blob/master/mmj2jar/mmj2.jar?raw=true. Unfortunately there is no easy way to automatically generate mmj2jar.zip, but this is available as part of the zip distribution for mmj2.zip. History tracking will be handled by the repository now. Do you have old versions of the mmj2 directory? I could add them as historical commits if you do."

(18-Jun-2013) Mario Carneiro has done a major revision and cleanup of the construction of real and complex numbers. In particular, rather than using equivalence classes as is customary for the construction of the temporary rationals, he used only "reduced fractions", so that the use of the axiom of infinity is avoided until it becomes necessary for the construction of the temporary reals.

(18-May-2013) Mario Carneiro has added the ability to produce compressed proofs to mmj2. This is not an official release but can be downloaded here if you want to try it: mmj2.jar. If you have any feedback, send it to me (NM), and I will forward it to Mario. (Disclaimer: this release has not been endorsed by Mel O'Cat. If anyone has been in contact with him, please let me know.)

(29-Mar-2013) Charles Greathouse reduced the size of our PNG symbol images using the pngout program.

(8-Mar-2013) Wolf Lammen has reorganized the theorems in the "Logical negation" section of set.mm into a more orderly, less scattered arrangement.

(27-Feb-2013) Scott Fenton has done a large cleanup of set.mm, eliminating *OLD references in 144 proofs. See the Notes entry for 27-Feb-2013.

(21-Feb-2013) *ATTENTION MATHBOX USERS* The order of hypotheses of many syl* theorems were changed, per a suggestion of Mario Carneiro. You need to update your local mathbox copy for compatibility with the new set.mm, or I can do it for you if you wish. See the Notes entry for 21-Feb-2013.

(16-Feb-2013) Scott Fenton shortened the direct-from-axiom proofs of *3.1, *3.43, *4.4, *4.41, *4.5, *4.76, *4.83, *5.33, *5.35, *5.36, and meredith in the "Shortest known proofs of the propositional calculus theorems from Principia Mathematica" (pmproofs.txt).

(27-Jan-2013) Scott Fenton writes, "I've updated Ralph Levien's mmverify.py. It's now a Python 3 program, and supports compressed proofs and file inclusion statements. This adds about fifty lines to the original program. Enjoy!"

(10-Jan-2013) A new mathbox was added for Mario Carneiro, who has contributed a number of cardinality theorems without invoking the Axiom of Choice. This is nice work, and I will be using some of these (those suffixed with "NEW") to replace the existing ones in the main part of set.mm that currently invoke AC unnecessarily.

(4-Jan-2013) As mentioned in the 19-Jun-2012 item below, Eric Schmidt discovered that the complex number axioms axaddcom (now addcom) and ax0id (now addid1) are redundant (schmidt-cnaxioms.pdf, .tex). In addition, ax1id (now mulid1) can be weakened to ax1rid. Scott Fenton has now formalized this work, so that now there are 23 instead of 25 axioms for real and complex numbers in set.mm. The Axioms for Complex Numbers page has been updated with these results. An interesting part of the proof, showing how commutativity of addition follows from other laws, is in addcomi.

(27-Nov-2012) The frequently-used theorems "an1s", "an1rs", "ancom13s", "ancom31s" were renamed to "an12s", "an32s", "an13s", "an31s" to conform to the convention for an12 etc.

(4-Nov-2012) The changes proposed in the Notes, renaming Grp to GrpOp etc., have been incorporated into set.mm. See the list of changes at the top of set.mm. If you want me to update your mathbox with these changes, send it to me along with the version of set.mm that it works with.

(20-Sep-2012) Mel O'Cat updated https://us.metamath.org/ocat/mmj2/TESTmmj2jar.zip. See the README.TXT for a description of the new features.

(21-Aug-2012) Mel O'Cat has uploaded SearchOptionsMockup9.zip, a mockup for the new search screen in mmj2. See the README.txt file for instructions. He will welcome feedback via x178g243 at yahoo.com.

(19-Jun-2012) Eric Schmidt has discovered that in our axioms for complex numbers, axaddcom and ax0id are redundant. (At some point these need to be formalized for set.mm.) He has written up these and some other nice results, including some independence results for the axioms, in schmidt-cnaxioms.pdf (schmidt-cnaxioms.tex).

(23-Apr-2012) Frédéric Liné sent me a PDF (LaTeX source) developed with Lamport's pf2 package. He wrote: "I think it works well with Metamath since the proofs are in a tree form. I use it to have a sketch of a proof. I get this way a better understanding of the proof and I can cut down its size. For instance, inpreima5 was reduced by 50% when I wrote the corresponding proof with pf2."

(5-Mar-2012) I added links to Wikiproofs and its recent changes in the "Wikis" list at the top of this page.

(12-Jan-2012) Thanks to William Hoza who sent me a ZFC T-shirt, and thanks to the ZFC models (courtesy of the Inaccessible Cardinals agency).

Front	Back	Detail

(24-Nov-2011) In metamath program version 0.07.71, the 'minimize_with' command by default now scans from bottom to top instead of top to bottom, since empirically this often (although not always) results in a shorter proof. A top to bottom scan can be specified with a new qualifier '/reverse'. You can try both methods (starting from the same original proof, of course) and pick the shorter proof.

(15-Oct-2011) From Mel O'Cat:
I just uploaded mmj2.zip containing the 1-Nov-2011 (20111101) release: https://us.metamath.org/ocat/mmj2/mmj2.zip https://us.metamath.org/ocat/mmj2/mmj2.md5
A few last minute tweaks:
1. I now bless double-click starting of mmj2.bat (MacMMJ2.command in Mac OS-X)! See mmj2\QuickStart.html
2. Much improved support of Mac OS-X systems. See mmj2\QuickStart.html
3. I tweaked the Command Line Argument Options report to
a) print every time;
b) print as much as possible even if there are errors in the command line arguments -- and the last line printed corresponds to the argument in error;
c) removed Y/N argument on the command line to enable/disable the report. this simplifies things.
4) Documentation revised, including the PATutorial.
See CHGLOG.TXT for list of all changes. Good luck. And thanks for all of your help!

(15-Sep-2011) MATHBOX USERS: I made a large number of label name changes to set.mm to improve naming consistency. There is a script at the top of the current set.mm that you can use to update your mathbox or older set.mm. Or if you wish, I can do the update on your next mathbox submission - in that case, please include a .zip of the set.mm version you used.

(30-Aug-2011) Scott Fenton shortened the direct-from-axiom proofs of *3.33, *3.45, *4.36, and meredith in the "Shortest known proofs of the propositional calculus theorems from Principia Mathematica" (pmproofs.txt).

(21-Aug-2011) A post on reddit generated 60,000 hits (and a TOS violation notice from my provider...),

(18-Aug-2011) The Metamath Google Group has a discussion of my canonical conjunctions proposal. Any feedback directly to me (Norm Megill) is also welcome.

(4-Jul-2011) John Baker has provided (metamath_kindle.zip) "a modified version of [the] metamath.tex [Metamath] book source that is formatted for the Kindle. If you compile the document the resulting PDF can be loaded into into a Kindle and easily read." (Update: the PDF file is now included also.)

(3-Jul-2011) Nested 'submit' calls are now allowed, in metamath program version 0.07.68. Thus you can create or modify a command file (script) from within a command file then 'submit' it. While 'submit' cannot pass arguments (nor are there plans to add this feature), you can 'substitute' strings in the 'submit' target file before calling it in order to emulate this.

(28-Jun-2011)The metamath program version 0.07.64 adds the '/include_mathboxes' qualifier to 'minimize_with'; by default, 'minimize_with *' will now skip checking user mathboxes. Since mathboxes should be independent from each other, this will help prevent accidental cross-"contamination". Also, '/rewrap' was added to 'write source' to automatically wrap $a and $p comments so as to conform to the current formatting conventions used in set.mm. This means you no longer have to be concerned about line length < 80 etc.

(19-Jun-2011) ATTENTION MATHBOX USERS: The wff variables et, ze, si, and rh are now global. This change was made primarily to resolve some conflicts between mathboxes, but it will also let you avoid having to constantly redeclare these locally in the future. Unfortunately, this change can affect the $f hypothesis order, which can cause proofs referencing theorems that use these variables to fail. All mathbox proofs currently in set.mm have been corrected for this, and you should refresh your local copy for further development of your mathbox. You can correct your proofs that are not in set.mm as follows. Only the proofs that fail under the current set.mm (using version 0.07.62 or later of the metamath program) need to be modified.

To fix a proof that references earlier theorems using et, ze, si, and rh, do the following (using a hypothetical theorem 'abc' as an example): 'prove abc' (ignore error messages), 'delete floating', 'initialize all', 'unify all/interactive', 'improve all', 'save new_proof/compressed'. If your proof uses dummy variables, these must be reassigned manually.

To fix a proof that uses et, ze, si, and rh as local variables, make sure the proof is saved in 'compressed' format. Then delete the local declarations ($v and $f statements) and follow the same steps above to correct the proof.

I apologize for the inconvenience. If you have trouble fixing your proofs, you can contact me for assistance.

Note: Versions of the metamath program before 0.07.62 did not flag an error when global variables were redeclared locally, as it should have according to the spec. This caused these spec violations to go unnoticed in some older set.mm versions. The new error messages are in fact just informational and can be ignored when working with older set.mm versions.

(7-Jun-2011) The metamath program version 0.07.60 fixes a bug with the 'minimize_with' command found by Andrew Salmon.

(12-May-2010) Andrew Salmon shortened many proofs, shown above. For comparison, I have temporarily kept the old version, which is suffixed with OLD, such as oridmOLD for oridm.

(9-Dec-2010) Eric Schmidt has written a Metamath proof verifier in C++, called checkmm.cpp.

(3-Oct-2010) The following changes were made to the tokens in set.mm. The subset and proper subset symbol changes to C_ and C. were made to prevent defeating the parenthesis matching in Emacs. Other changes were made so that all letters a-z and A-Z are now available for variable names. One-letter constants such as _V, _e, and _i are now shown on the web pages with Roman instead of italic font, to disambiguate italic variable names. The new convention is that a prefix of _ indicates Roman font and a prefix of ~ indicates a script (curly) font. Thanks to Stefan Allan and Frédéric Liné for discussions leading to this change.

Old	New	Description
C.	_C	binomial coefficient
E	_E	epsilon relation
e	_e	Euler's constant
I	_I	identity relation
i	_i	imaginary unit
V	_V	universal class
(_	C_	subset
(.	C.	proper subset
P~	~P	power class
H~	~H	Hilbert space

(25-Sep-2010) The metamath program (version 0.07.54) now implements the current Metamath spec, so footnote 2 on p. 92 of the Metamath book can be ignored.

(24-Sep-2010) The metamath program (version 0.07.53) fixes bug 2106, reported by Michal Burger.

(14-Sep-2010) The metamath program (version 0.07.52) has a revamped LaTeX output with 'show statement xxx /tex', which produces the combined statement, description, and proof similar to the web page generation. Also, 'show proof xxx /lemmon/renumber' now matches the web page step numbers. ('show proof xxx/renumber' still has the indented form conforming to the actual RPN proof, with slightly different numbering.)

(9-Sep-2010) The metamath program (version 0.07.51) was updated with a modification by Stefan Allan that adds hyperlinks the the Ref column of proofs.

(12-Jun-2010) Scott Fenton contributed a D-proof (directly from axioms) of Meredith's single axiom (see the end of pmproofs.txt). A description of Meredith's axiom can be found in theorem meredith.

(11-Jun-2010) A new Metamath mirror was added in Austria, courtesy of Kinder-Enduro.

(28-Feb-2010) Raph Levien's Ghilbert project now has a new Ghilbert site and a Google Group.

(26-Jan-2010) Dmitri Vlasov writes, "I admire the simplicity and power of the metamath language, but still I see its great disadvantage - the proofs in metamath are completely non-manageable by humans without proof assistants. Therefore I decided to develop another language, which would be a higher-level superstructure language towards metamath, and which will support human-readable/writable proofs directly, without proof assistants. I call this language mdl (acronym for 'mathematics development language')." The latest version of Dmitri's translators from metamath to mdl and back can be downloaded from http://mathdevlanguage.sourceforge.net/. Currently only Linux is supported, but Dmitri says is should not be difficult to port it to other platforms that have a g++ compiler.

(11-Sep-2009) The metamath program (version 0.07.48) has been updated to enforce the whitespace requirement of the current spec.

(10-Sep-2009) Matthew Leitch has written an nice article, "How to write mathematics clearly", that briefly mentions Metamath. Overall it makes some excellent points. (I have written to him about a few things I disagree with.)

(28-May-2009) AsteroidMeta is back on-line. Note the URL change.

(12-May-2009) Charles Greathouse wrote a Greasemonkey script to reformat the axiom list on Metamath web site proof pages. This is a beta version; he will appreciate feedback.

(11-May-2009) Stefan Allan modified the metamath program to add the command "show statement xxx /mnemonics", which produces the output file Mnemosyne.txt for use with the Mnemosyne project. The current Metamath program download incorporates this command. Instructions: Create the file mnemosyne.txt with e.g. "show statement ax-* /mnemonics". In the Mnemosyne program, load the file by choosing File->Import then file format "Q and A on separate lines". Notes: (1) Don't try to load all of set.mm, it will crash the program due to a bug in Mnemosyne. (2) On my computer, the arrows in ax-1 don't display. Stefan reports that they do on his computer. (Both are Windows XP.)

(3-May-2009) Steven Baldasty wrote a Metamath syntax highlighting file for the gedit editor. Screenshot.

(1-May-2009) Users on a gaming forum discuss our 2+2=4 proof. Notable comments include "Ew math!" and "Whoever wrote this has absolutely no life."

(12-Mar-2009) Chris Capel has created a Javascript theorem viewer demo that (1) shows substitutions and (2) allows expanding and collapsing proof steps. You are invited to take a look and give him feedback at his Metablog.

(28-Feb-2009) Chris Capel has written a Metamath proof verifier in C#, available at http://pdf23ds.net/bzr/MathEditor/Verifier/Verifier.cs and weighing in at 550 lines. Also, that same URL without the file on it is a Bazaar repository.

(2-Dec-2008) A new section was added to the Deduction Theorem page, called Logic, Metalogic, Metametalogic, and Metametametalogic.

(24-Aug-2008) (From ocat): The 1-Aug-2008 version of mmj2 is ready (mmj2.zip), size = 1,534,041 bytes. This version contains the Theorem Loader enhancement which provides a "sandboxing" capability for user theorems and dynamic update of new theorems to the Metamath database already loaded in memory by mmj2. Also, the new "mmj2 Service" feature enables calling mmj2 as a subroutine, or having mmj2 call your program, and provides access to the mmj2 data structures and objects loaded in memory (i.e. get started writing those Jython programs!) See also mmj2 on AsteroidMeta.

(23-May-2008) Gérard Lang pointed me to Bob Solovay's note on AC and strongly inaccessible cardinals. One of the eventual goals for set.mm is to prove the Axiom of Choice from Grothendieck's axiom, like Mizar does, and this note may be helpful for anyone wanting to attempt that. Separately, I also came across a history of the size reduction of grothprim (viewable in Firefox and some versions of Internet Explorer).

(14-Apr-2008) A "/join" qualifier was added to the "search" command in the metamath program (version 0.07.37). This qualifier will join the $e hypotheses to the $a or $p for searching, so that math tokens in the $e's can be matched as well. For example, "search *com* +v" produces no results, but "search *com* +v /join" yields commutative laws involving vector addition. Thanks to Stefan Allan for suggesting this idea.

(8-Apr-2008) The 8,000th theorem, hlrel, was added to the Metamath Proof Explorer part of the database.

(2-Mar-2008) I added a small section to the end of the Deduction Theorem page.

(17-Feb-2008) ocat has uploaded the "1-Mar-2008" mmj2: mmj2.zip. See the description.

(16-Jan-2008) O'Cat has written mmj2 Proof Assistant Quick Tips.

(30-Dec-2007) "How to build a library of formalized mathematics".

(22-Dec-2007) The Metamath Proof Explorer was included in the top 30 science resources for 2007 by the University at Albany Science Library.

(17-Dec-2007) Metamath's Wikipedia entry says, "This article may require cleanup to meet Wikipedia's quality standards" (see its discussion page). Volunteers are welcome. :) (In the interest of objectivity, I don't edit this entry.)

(20-Nov-2007) Jeff Hoffman created nicod.mm and posted it to the Google Metamath Group.

(19-Nov-2007) Reinder Verlinde suggested adding tooltips to the hyperlinks on the proof pages, which I did for proof step hyperlinks. Discussion.

(5-Nov-2007) A Usenet challenge. :)

(4-Aug-2007) I added a "Request for comments on proposed 'maps to' notation" at the bottom of the AsteroidMeta set.mm discussion page.

(21-Jun-2007) A preprint (PDF file) describing Kurt Maes' axiom of choice with 5 quantifiers, proved in set.mm as ackm.

(20-Jun-2007) The 7,000th theorem, ifpr, was added to the Metamath Proof Explorer part of the database.

(29-Apr-2007) Blog mentions of Metamath: here and here.

(21-Mar-2007) Paul Chapman is working on a new proof browser, which has highlighting that allows you to see the referenced theorem before and after the substitution was made. Here is a screenshot of theorem 0nn0 and a screenshot of theorem 2p2e4.

(15-Mar-2007) A picture of Penny the cat guarding the us.metamath.org server and making the rounds.

(16-Feb-2007) For convenience, the program "drule.c" (pronounced "D-rule", not "drool") mentioned in pmproofs.txt can now be downloaded (drule.c) without having to ask me for it. The same disclaimer applies: even though this program works and has no known bugs, it was not intended for general release. Read the comments at the top of the program for instructions.

(28-Jan-2007) Jason Orendorff set up a new mailing list for Metamath: http://groups.google.com/group/metamath.

(20-Jan-2007) Bob Solovay provided a revised version of his Metamath database for Peano arithmetic, peano.mm.

(2-Jan-2007) Raph Levien has set up a wiki called Barghest for the Ghilbert language and software.

(26-Dec-2006) I posted an explanation of theorem ecoprass on Usenet.

(2-Dec-2006) Berislav Žarnić translated the Metamath Solitaire applet to Croatian.

(26-Nov-2006) Dan Getz has created an RSS feed for new theorems as they appear on this page.

(6-Nov-2006) The first 3 paragraphs in Appendix 2: Note on the Axioms were rewritten to clarify the connection between Tarski's axiom system and Metamath.

(31-Oct-2006) ocat asked for a do-over due to a bug in mmj2 -- if you downloaded the mmj2.zip version dated 10/28/2006, then download the new version dated 10/30.

(29-Oct-2006) ocat has announced that the long-awaited 1-Nov-2006 release of mmj2 is available now.
     The new "Unify+Get Hints" is quite useful, and any proof can be generated as follows. With "?" in the Hyp field and Ref field blank, select "Unify+Get Hints". Select a hint from the list and put it in the Ref field. Edit any $n dummy variables to become the desired wffs. Rinse and repeat for the new proof steps generated, until the proof is done.
     The new tutorial, mmj2PATutorial.bat, explains this in detail. One way to reduce or avoid dummy $n's is to fill in the Hyp field with a comma-separated list of any known hypothesis matches to earlier proof steps, keeping a "?" in the list to indicate that the remaining hypotheses are unknown. Then "Unify+Get Hints" can be applied. The tutorial page \mmj2\data\mmp\PATutorial\Page405.mmp has an example.
     Don't forget that the eimm export/import program lets you go back and forth between the mmj2 and the metamath program proof assistants, without exiting from either one, to exploit the best features of each as required.

(21-Oct-2006) Martin Kiselkov has written a Metamath proof verifier in the Lua scripting language, called verify.lua. While it is not practical as an everyday verifier - he writes that it takes about 40 minutes to verify set.mm on a a Pentium 4 - it could be useful to someone learning Lua or Metamath, and importantly it provides another independent way of verifying the correctness of Metamath proofs. His code looks like it is nicely structured and very readable. He is currently working on a faster version in C++.

(19-Oct-2006) New AsteroidMeta page by Raph, Distinctors_vs_binders.

(13-Oct-2006) I put a simple Metamath browser on my PDA (Palm Tungsten E) so that I don't have to lug around my laptop. Here is a screenshot. It isn't polished, but I'll provide the file + instructions if anyone wants it.

(3-Oct-2006) A blog entry, Principia for Reverse Mathematics.

(28-Sep-2006) A blog entry, Metamath responds.

(26-Sep-2006) A blog entry, Metamath isn't hygienic.

(11-Aug-2006) A blog entry, Metamath and the Peano Induction Axiom.

(26-Jul-2006) A new open problem in predicate calculus was added.

(18-Jun-2006) The 6,000th theorem, mt4d, was added to the Metamath Proof Explorer part of the database.

(9-May-2006) Luca Ciciriello has upgraded the t2mf program, which is a C program used to create the MIDI files on the Metamath Music Page, so that it works on MacOS X. This is a nice accomplishment, since the original program was written before C was standardized by ANSI and will not compile on modern compilers.
Unfortunately, the original program source states no copyright terms. The main author, Tim Thompson, has kindly agreed to release his code to public domain, but two other authors have also contributed to the code, and so far I have been unable to contact them for copyright clearance. Therefore I cannot offer the MacOS X version for public download on this site until this is resolved. Update 10-May-2006: Another author, M. Czeiszperger, has released his contribution to public domain.
If you are interested in Luca's modified source code, please contact me directly.

(18-Apr-2006) Incomplete proofs in progress can now be interchanged between the Metamath program's CLI Proof Assistant and mmj2's GUI Proof Assistant, using a new export-import program called eimm. This can be done without exiting either proof assistant, so that the strengths of each approach can be exploited during proof development. See "Use Case 5a" and "Use Case 5b" at mmj2ProofAssistantFeedback.

(28-Mar-2006) Scott Fenton updated his second version of Metamath Solitaire (the one that uses external axioms). He writes: "I've switched to making it a standalone program, as it seems silly to have an applet that can't be run in a web browser. Check the README file for further info." The download is mmsol-0.5.tar.gz.

(27-Mar-2006) Scott Fenton has updated the Metamath Solitaire Java applet to Java 1.5: (1) QSort has been stripped out: its functionality is in the Collections class that Sun ships; (2) all Vectors have been replaced by ArrayLists; (3) generic types have been tossed in wherever they fit: this cuts back drastically on casting; and (4) any warnings Eclipse spouted out have been dealt with. I haven't yet updated it officially, because I don't know if it will work with Microsoft's JVM in older versions of Internet Explorer. The current official version is compiled with Java 1.3, because it won't work with Microsoft's JVM if it is compiled with Java 1.4. (As distasteful as that seems, I will get complaints from users if it doesn't work with Microsoft's JVM.) If anyone can verify that Scott's new version runs on Microsoft's JVM, I would be grateful. Scott's new version is mm.java-1.5.gz; after uncompressing it, rename it to mm.java, use it to replace the existing mm.java file in the Metamath Solitaire download, and recompile according to instructions in the mm.java comments.
Scott has also created a second version, mmsol-0.2.tar.gz, that reads the axioms from ASCII files, instead of having the axioms hard-coded in the program. This can be very useful if you want to play with custom axioms, and you can also add a collection of starting theorems as "axioms" to work from. However, it must be run from the local directory with appletviewer, since the default Java security model doesn't allow reading files from a browser. It works with the JDK 5 Update 6 Java download.
To compile (from Windows Command Prompt): C:\Program Files\Java\jdk1.5.0_06\bin\javac.exe mm.java
To run (from Windows Command Prompt): C:\Program Files\Java\jdk1.5.0_06\bin\appletviewer.exe mms.html

(21-Jan-2006) Juha Arpiainen proved the independence of axiom ax-11 from the others. This was published as an open problem in my 1995 paper (Remark 9.5 on PDF page 17). See Item 9a on the Workshop Miscellany for his seven-line proof. See also the Asteroid Meta metamathMathQuestions page under the heading "Axiom of variable substitution: ax-11". Congratulations, Juha!

(20-Oct-2005) Juha Arpiainen is working on a proof verifier in Common Lisp called Bourbaki. Its proof language has its roots in Metamath, with the goal of providing a more powerful syntax and definitional soundness checking. See its documentation and related discussion.

(17-Oct-2005) Marnix Klooster has written a Metamath proof verifier in Haskell, called Hmm. Also see his Announcement. The complete program (Hmm.hs, HmmImpl.hs, and HmmVerify.hs) has only 444 lines of code, excluding comments and blank lines. It verifies compressed as well as regular proofs; moreover, it transparently verifies both per-spec compressed proofs and the flawed format he uncovered (see comment below of 16-Oct-05).

(16-Oct-2005) Marnix Klooster noticed that for large proofs, the compressed proof format did not match the spec in the book. His algorithm to correct the problem has been put into the Metamath program (version 0.07.6). The program still verifies older proofs with the incorrect format, but the user will be nagged to update them with 'save proof *'. In set.mm, 285 out of 6376 proofs are affected. (The incorrect format did not affect proof correctness or verification, since the compression and decompression algorithms matched each other.)

(13-Sep-2005) Scott Fenton found an interesting axiom, ax46, which could be used to replace both ax-4 and ax-6.

(29-Jul-2005) Metamath was selected as site of the week by American Scientist Online.

(8-Jul-2005) Roy Longton has contributed 53 new theorems to the Quantum Logic Explorer. You can see them in the Theorem List starting at lem3.3.3lem1. He writes, "If you want, you can post an open challenge to see if anyone can find shorter proofs of the theorems I submitted."

(10-May-2005) A Usenet post I posted about the infinite prime proof; another one about indexed unions.

(3-May-2005) The theorem divexpt is the 5,000th theorem added to the Metamath Proof Explorer database.

(12-Apr-2005) Raph Levien solved the open problem in item 16 on the Workshop Miscellany page and as a corollary proved that axiom ax-9 is independent from the other axioms of predicate calculus and equality. This is the first such independence proof so far; a goal is to prove all of them independent (or to derive any redundant ones from the others).

(8-Mar-2005) I added a paragraph above our complex number axioms table, summarizing the construction and indicating where Dedekind cuts are defined. Thanks to Andrew Buhr for comments on this.

(16-Feb-2005) The Metamath Music Page is mentioned as a reference or resource for a university course called Math, Mind, and Music. .

(28-Jan-2005) Steven Cullinane parodied the Metamath Music Page in his blog.

(18-Jan-2005) Waldek Hebisch upgraded the Metamath program to run on the AMD64 64-bit processor.

(17-Jan-2005) A symbol list summary was added to the beginning of the Hilbert Space Explorer Home Page. Thanks to Mladen Pavicic for suggesting this.

(6-Jan-2005) Someone assembled an amazon.com list of some of the books in the Metamath Proof Explorer Bibliography.

(4-Jan-2005) The definition of ordinal exponentiation was decided on after this Usenet discussion.

(19-Dec-2004) A bit of trivia: my Erdös number is 2, as you can see from this list.

(20-Oct-2004) I started this Usenet discussion about the "reals are uncountable" proof (127 comments; last one on Nov. 12).

(12-Oct-2004) gch-kn shows the equivalence of the Generalized Continuum Hypothesis and Prof. Nambiar's Axiom of Combinatorial Sets. This proof answers his Open Problem 2 (PDF file).

(5-Aug-2004) I gave a talk on "Hilbert Lattice Equations" at the Argonne workshop.

(25-Jul-2004) The theorem nthruz is the 4,000th theorem added to the Metamath Proof Explorer database.

(27-May-2004) Josiah Burroughs contributed the proofs u1lemn1b, u1lem3var1, oi3oa3lem1, and oi3oa3 to the Quantum Logic Explorer database ql.mm.

(23-May-2004) Some minor typos found by Josh Purinton were corrected in the Metamath book. In addition, Josh simplified the definition of the closure of a pre-statement of a formal system in Appendix C.

(5-May-2004) Gregory Bush has found shorter proofs for 67 of the 193 propositional calculus theorems listed in Principia Mathematica, thus establishing 67 new records. (This was challenge #4 on the open problems page.)

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