P. 379 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 37801-37900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	bj-rvecrr 37801	The field of scalars of a real vector space is the field of real numbers. (Contributed by BJ, 6-Jan-2024.)
⊢ (𝑉 ∈ ℝ-Vec → (Scalar‘𝑉) = ℝfld)
Theorem	bj-isrvecd 37802	The predicate "is a real vector space". (Contributed by BJ, 6-Jan-2024.)
⊢ (𝜑 → (Scalar‘𝑉) = 𝐾)    ⇒   ⊢ (𝜑 → (𝑉 ∈ ℝ-Vec ↔ (𝑉 ∈ LMod ∧ 𝐾 = ℝfld)))
Theorem	bj-rvecvec 37803	Real vector spaces are vector spaces (elemental version). (Contributed by BJ, 6-Jan-2024.)
⊢ (𝑉 ∈ ℝ-Vec → 𝑉 ∈ LVec)
Theorem	bj-isrvec2 37804	The predicate "is a real vector space". (Contributed by BJ, 6-Jan-2024.)
⊢ (𝜑 → (Scalar‘𝑉) = 𝐾)    ⇒   ⊢ (𝜑 → (𝑉 ∈ ℝ-Vec ↔ (𝑉 ∈ LVec ∧ 𝐾 = ℝfld)))
Theorem	bj-rvecssvec 37805	Real vector spaces are vector spaces. (Contributed by BJ, 6-Jan-2024.)
⊢ ℝ-Vec ⊆ LVec
Theorem	bj-rveccmod 37806	Real vector spaces are subcomplex modules (elemental version). (Contributed by BJ, 6-Jan-2024.)
⊢ (𝑉 ∈ ℝ-Vec → 𝑉 ∈ ℂMod)
Theorem	bj-rvecsscmod 37807	Real vector spaces are subcomplex modules. (Contributed by BJ, 6-Jan-2024.)
⊢ ℝ-Vec ⊆ ℂMod
Theorem	bj-rvecsscvec 37808	Real vector spaces are subcomplex vector spaces. (Contributed by BJ, 6-Jan-2024.)
⊢ ℝ-Vec ⊆ ℂVec
Theorem	bj-rveccvec 37809	Real vector spaces are subcomplex vector spaces (elemental version). (Contributed by BJ, 6-Jan-2024.)
⊢ (𝑉 ∈ ℝ-Vec → 𝑉 ∈ ℂVec)
Theorem	bj-rvecssabl 37810	(The additive groups of) real vector spaces are commutative groups. (Contributed by BJ, 9-Jun-2019.)
⊢ ℝ-Vec ⊆ Abel
Theorem	bj-rvecabl 37811	(The additive groups of) real vector spaces are commutative groups (elemental version). (Contributed by BJ, 9-Jun-2019.)
⊢ (𝐴 ∈ ℝ-Vec → 𝐴 ∈ Abel)
21.19.8.2  Complex numbers (supplements) Some lemmas to ease algebraic manipulations.
Theorem	bj-subcom 37812	A consequence of commutativity of multiplication. (Contributed by BJ, 6-Jun-2019.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 · 𝐵) − (𝐵 · 𝐴)) = 0)
Theorem	bj-lineqi 37813	Solution of a (scalar) linear equation. (Contributed by BJ, 6-Jun-2019.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝑌 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → ((𝐴 · 𝑋) + 𝐵) = 𝑌)    ⇒   ⊢ (𝜑 → 𝑋 = ((𝑌 − 𝐵) / 𝐴))
21.19.8.3  Barycentric coordinates Lemmas about barycentric coordinates. For the moment, this is limited to the one-dimensional case (complex line), where existence and uniqueness of barycentric coordinates are proved by bj-bary1 37816 (which computes them). It would be nice to prove the two-dimensional case (is it easier to use ad hoc computations, or Cramer formulas?), in order to do some planar geometry.
Theorem	bj-bary1lem 37814	Lemma for bj-bary1 37816: expression for a barycenter of two points in one dimension (complex line). (Contributed by BJ, 6-Jun-2019.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    ⇒   ⊢ (𝜑 → 𝑋 = ((((𝐵 − 𝑋) / (𝐵 − 𝐴)) · 𝐴) + (((𝑋 − 𝐴) / (𝐵 − 𝐴)) · 𝐵)))
Theorem	bj-bary1lem1 37815	Lemma for bj-bary1 37816: computation of one of the two barycentric coordinates of a barycenter of two points in one dimension (complex line). (Contributed by BJ, 6-Jun-2019.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝑆 ∈ ℂ)    &   ⊢ (𝜑 → 𝑇 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝑋 = ((𝑆 · 𝐴) + (𝑇 · 𝐵)) ∧ (𝑆 + 𝑇) = 1) → 𝑇 = ((𝑋 − 𝐴) / (𝐵 − 𝐴))))
Theorem	bj-bary1 37816	Barycentric coordinates in one dimension (complex line). In the statement, 𝑋 is the barycenter of the two points 𝐴, 𝐵 with respective normalized coefficients 𝑆, 𝑇. (Contributed by BJ, 6-Jun-2019.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → 𝑆 ∈ ℂ)    &   ⊢ (𝜑 → 𝑇 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝑋 = ((𝑆 · 𝐴) + (𝑇 · 𝐵)) ∧ (𝑆 + 𝑇) = 1) ↔ (𝑆 = ((𝐵 − 𝑋) / (𝐵 − 𝐴)) ∧ 𝑇 = ((𝑋 − 𝐴) / (𝐵 − 𝐴)))))
21.19.9  Monoid of endomorphisms
Syntax	cend 37817	Token for the monoid of endomorphisms.
class End
Definition	df-bj-end 37818*	The monoid of endomorphisms on an object of a category. (Contributed by BJ, 4-Apr-2024.)
⊢ End = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ {⟨(Base‘ndx), (𝑥(Hom ‘𝑐)𝑥)⟩, ⟨(+g‘ndx), (⟨𝑥, 𝑥⟩(comp‘𝑐)𝑥)⟩}))
Theorem	bj-endval 37819	Value of the monoid of endomorphisms on an object of a category. (Contributed by BJ, 5-Apr-2024.) (Proof modification is discouraged.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))    ⇒   ⊢ (𝜑 → ((End ‘𝐶)‘𝑋) = {⟨(Base‘ndx), (𝑋(Hom ‘𝐶)𝑋)⟩, ⟨(+g‘ndx), (⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋)⟩})
Theorem	bj-endbase 37820	Base set of the monoid of endomorphisms on an object of a category. (Contributed by BJ, 5-Apr-2024.) (Proof modification is discouraged.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))    ⇒   ⊢ (𝜑 → (Base‘((End ‘𝐶)‘𝑋)) = (𝑋(Hom ‘𝐶)𝑋))
Theorem	bj-endcomp 37821	Composition law of the monoid of endomorphisms on an object of a category. (Contributed by BJ, 5-Apr-2024.) (Proof modification is discouraged.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))    ⇒   ⊢ (𝜑 → (+g‘((End ‘𝐶)‘𝑋)) = (⟨𝑋, 𝑋⟩(comp‘𝐶)𝑋))
Theorem	bj-endmnd 37822	The monoid of endomorphisms on an object of a category is a monoid. (Contributed by BJ, 5-Apr-2024.) (Proof modification is discouraged.)
⊢ (𝜑 → 𝐶 ∈ Cat)    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶))    ⇒   ⊢ (𝜑 → ((End ‘𝐶)‘𝑋) ∈ Mnd)
21.20  Mathbox for Jim Kingdon
21.20.1  Circle constant
Theorem	taupilem3 37823	Lemma for tau-related theorems. (Contributed by Jim Kingdon, 16-Feb-2019.)
⊢ (𝐴 ∈ (ℝ+ ∩ (◡cos “ {1})) ↔ (𝐴 ∈ ℝ+ ∧ (cos‘𝐴) = 1))
Theorem	taupilemrplb 37824*	A set of positive reals has (in the reals) a lower bound. (Contributed by Jim Kingdon, 19-Feb-2019.)
⊢ ∃𝑥 ∈ ℝ ∀𝑦 ∈ (ℝ+ ∩ 𝐴)𝑥 ≤ 𝑦
Theorem	taupilem1 37825	Lemma for taupi 37827. A positive real whose cosine is one is at least 2 · π. (Contributed by Jim Kingdon, 19-Feb-2019.)
⊢ ((𝐴 ∈ ℝ+ ∧ (cos‘𝐴) = 1) → (2 · π) ≤ 𝐴)
Theorem	taupilem2 37826	Lemma for taupi 37827. The smallest positive real whose cosine is one is at most 2 · π. (Contributed by Jim Kingdon, 19-Feb-2019.) (Revised by AV, 1-Oct-2020.)
⊢ τ ≤ (2 · π)
Theorem	taupi 37827	Relationship between τ and π. This can be seen as connecting the ratio of a circle's circumference to its radius and the ratio of a circle's circumference to its diameter. (Contributed by Jim Kingdon, 19-Feb-2019.) (Revised by AV, 1-Oct-2020.)
⊢ τ = (2 · π)
21.20.2  Number theory
Theorem	dfgcd3 37828*	Alternate definition of the gcd operator. (Contributed by Jim Kingdon, 31-Dec-2021.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) = (℩𝑑 ∈ ℕ0 ∀𝑧 ∈ ℤ (𝑧 ∥ 𝑑 ↔ (𝑧 ∥ 𝑀 ∧ 𝑧 ∥ 𝑁))))
21.20.3  Real numbers
Theorem	irrdifflemf 37829	Lemma for irrdiff 37830. The forward direction. (Contributed by Jim Kingdon, 20-May-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → ¬ 𝐴 ∈ ℚ)    &   ⊢ (𝜑 → 𝑄 ∈ ℚ)    &   ⊢ (𝜑 → 𝑅 ∈ ℚ)    &   ⊢ (𝜑 → 𝑄 ≠ 𝑅)    ⇒   ⊢ (𝜑 → (abs‘(𝐴 − 𝑄)) ≠ (abs‘(𝐴 − 𝑅)))
Theorem	irrdiff 37830*	The irrationals are exactly those reals that are a different distance from every rational. (Contributed by Jim Kingdon, 19-May-2024.)
⊢ (𝐴 ∈ ℝ → (¬ 𝐴 ∈ ℚ ↔ ∀𝑞 ∈ ℚ ∀𝑟 ∈ ℚ (𝑞 ≠ 𝑟 → (abs‘(𝐴 − 𝑞)) ≠ (abs‘(𝐴 − 𝑟)))))
Theorem	qdiff 37831*	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 37830 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 37830, see qdiffALT 37832. (Contributed by Jim Kingdon, 24-Apr-2026.)
⊢ (𝐴 ∈ ℝ → (𝐴 ∈ ℚ ↔ ∃𝑞 ∈ ℚ ∃𝑟 ∈ ℚ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))))
Theorem	qdiffALT 37832*	Alternate proof of qdiff 37831. This is a proof from irrdiff 37830 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)))))
Theorem	iccioo01 37833	The closed unit interval is equinumerous to the open unit interval. Based on a Mastodon post by Michael Kinyon. (Contributed by Jim Kingdon, 4-Jun-2024.)
⊢ (0[,]1) ≈ (0(,)1)
21.21  Mathbox for ML
21.21.1  Miscellaneous
Theorem	csbrecsg 37834	Move class substitution in and out of recs. (Contributed by ML, 25-Oct-2020.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌recs(𝐹) = recs(⦋𝐴 / 𝑥⦌𝐹))
Theorem	csbrdgg 37835	Move class substitution in and out of the recursive function generator. (Contributed by ML, 25-Oct-2020.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌rec(𝐹, 𝐼) = rec(⦋𝐴 / 𝑥⦌𝐹, ⦋𝐴 / 𝑥⦌𝐼))
Theorem	csboprabg 37836*	Move class substitution in and out of class abstractions of nested ordered pairs. (Contributed by ML, 25-Oct-2020.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ 𝜑} = {⟨⟨𝑦, 𝑧⟩, 𝑑⟩ ∣ [𝐴 / 𝑥]𝜑})
Theorem	csbmpo123 37837*	Move class substitution in and out of maps-to notation for operations. (Contributed by ML, 25-Oct-2020.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑦 ∈ 𝑌, 𝑧 ∈ 𝑍 ↦ 𝐷) = (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝑌, 𝑧 ∈ ⦋𝐴 / 𝑥⦌𝑍 ↦ ⦋𝐴 / 𝑥⦌𝐷))
Theorem	con1bii2 37838	A contraposition inference. (Contributed by ML, 18-Oct-2020.)
⊢ (¬ 𝜑 ↔ 𝜓)    ⇒   ⊢ (𝜑 ↔ ¬ 𝜓)
Theorem	con2bii2 37839	A contraposition inference. (Contributed by ML, 18-Oct-2020.)
⊢ (𝜑 ↔ ¬ 𝜓)    ⇒   ⊢ (¬ 𝜑 ↔ 𝜓)
Theorem	vtoclefex 37840*	Implicit substitution of a class for a setvar variable. (Contributed by ML, 17-Oct-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝑥 = 𝐴 → 𝜑)    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝜑)
Theorem	rnmptsn 37841*	The range of a function mapping to singletons. (Contributed by ML, 15-Jul-2020.)
⊢ ran (𝑥 ∈ 𝐴 ↦ {𝑥}) = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}
Theorem	f1omptsnlem 37842*	This is the core of the proof of f1omptsn 37843, but to avoid the distinct variables on the definitions, we split this proof into two. (Contributed by ML, 15-Jul-2020.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥})    &   ⊢ 𝑅 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}    ⇒   ⊢ 𝐹:𝐴–1-1-onto→𝑅
Theorem	f1omptsn 37843*	A function mapping to singletons is bijective onto a set of singletons. (Contributed by ML, 16-Jul-2020.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥})    &   ⊢ 𝑅 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}    ⇒   ⊢ 𝐹:𝐴–1-1-onto→𝑅
Theorem	mptsnunlem 37844*	This is the core of the proof of mptsnun 37845, but to avoid the distinct variables on the definitions, we split this proof into two. (Contributed by ML, 16-Jul-2020.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥})    &   ⊢ 𝑅 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}    ⇒   ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = ∪ (𝐹 “ 𝐵))
Theorem	mptsnun 37845*	A class 𝐵 is equal to the union of the class of all singletons of elements of 𝐵. (Contributed by ML, 16-Jul-2020.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥})    &   ⊢ 𝑅 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}    ⇒   ⊢ (𝐵 ⊆ 𝐴 → 𝐵 = ∪ (𝐹 “ 𝐵))
Theorem	dissneqlem 37846*	This is the core of the proof of dissneq 37847, but to avoid the distinct variables on the definitions, we split this proof into two. (Contributed by ML, 16-Jul-2020.)
⊢ 𝐶 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}    ⇒   ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴)) → 𝐵 = 𝒫 𝐴)
Theorem	dissneq 37847*	Any topology that contains every single-point set is the discrete topology. (Contributed by ML, 16-Jul-2020.)
⊢ 𝐶 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}}    ⇒   ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐵 ∈ (TopOn‘𝐴)) → 𝐵 = 𝒫 𝐴)
Theorem	exlimim 37848*	Closed form of exlimimd 37849. (Contributed by ML, 17-Jul-2020.)
⊢ ((∃𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)) → 𝜓)
Theorem	exlimimd 37849*	Existential elimination rule of natural deduction. (Contributed by ML, 17-Jul-2020.)
⊢ (𝜑 → ∃𝑥𝜓)    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	exellim 37850*	Closed form of exellimddv 37851. See also exlimim 37848 for a more general theorem. (Contributed by ML, 17-Jul-2020.)
⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) → 𝜑)
Theorem	exellimddv 37851*	Eliminate an antecedent when the antecedent is elementhood, deduction version. See exellim 37850 for the closed form, which requires the use of a universal quantifier. (Contributed by ML, 17-Jul-2020.)
⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	topdifinfindis 37852*	Part of Exercise 3 of [Munkres] p. 83. The topology of all subsets 𝑥 of 𝐴 such that the complement of 𝑥 in 𝐴 is infinite, or 𝑥 is the empty set, or 𝑥 is all of 𝐴, is the trivial topology when 𝐴 is finite. (Contributed by ML, 14-Jul-2020.)
⊢ 𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}    ⇒   ⊢ (𝐴 ∈ Fin → 𝑇 = {∅, 𝐴})
Theorem	topdifinffinlem 37853*	This is the core of the proof of topdifinffin 37854, but to avoid the distinct variables on the definition, we need to split this proof into two. (Contributed by ML, 17-Jul-2020.)
⊢ 𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}    ⇒   ⊢ (𝑇 ∈ (TopOn‘𝐴) → 𝐴 ∈ Fin)
Theorem	topdifinffin 37854*	Part of Exercise 3 of [Munkres] p. 83. The topology of all subsets 𝑥 of 𝐴 such that the complement of 𝑥 in 𝐴 is infinite, or 𝑥 is the empty set, or 𝑥 is all of 𝐴, is a topology only if 𝐴 is finite. (Contributed by ML, 17-Jul-2020.)
⊢ 𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}    ⇒   ⊢ (𝑇 ∈ (TopOn‘𝐴) → 𝐴 ∈ Fin)
Theorem	topdifinf 37855*	Part of Exercise 3 of [Munkres] p. 83. The topology of all subsets 𝑥 of 𝐴 such that the complement of 𝑥 in 𝐴 is infinite, or 𝑥 is the empty set, or 𝑥 is all of 𝐴, is a topology if and only if 𝐴 is finite, in which case it is the trivial topology. (Contributed by ML, 17-Jul-2020.)
⊢ 𝑇 = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}    ⇒   ⊢ ((𝑇 ∈ (TopOn‘𝐴) ↔ 𝐴 ∈ Fin) ∧ (𝑇 ∈ (TopOn‘𝐴) → 𝑇 = {∅, 𝐴}))
Theorem	topdifinfeq 37856*	Two different ways of defining the collection from Exercise 3 of [Munkres] p. 83. (Contributed by ML, 18-Jul-2020.)
⊢ {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ ((𝐴 ∖ 𝑥) = ∅ ∨ (𝐴 ∖ 𝑥) = 𝐴))} = {𝑥 ∈ 𝒫 𝐴 ∣ (¬ (𝐴 ∖ 𝑥) ∈ Fin ∨ (𝑥 = ∅ ∨ 𝑥 = 𝐴))}
Theorem	icorempo 37857*	Closed-below, open-above intervals of reals. (Contributed by ML, 26-Jul-2020.)
⊢ 𝐹 = ([,) ↾ (ℝ × ℝ))    ⇒   ⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {𝑧 ∈ ℝ ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
Theorem	icoreresf 37858	Closed-below, open-above intervals of reals map to subsets of reals. (Contributed by ML, 25-Jul-2020.)
⊢ ([,) ↾ (ℝ × ℝ)):(ℝ × ℝ)⟶𝒫 ℝ
Theorem	icoreval 37859*	Value of the closed-below, open-above interval function on reals. (Contributed by ML, 26-Jul-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,)𝐵) = {𝑧 ∈ ℝ ∣ (𝐴 ≤ 𝑧 ∧ 𝑧 < 𝐵)})
Theorem	icoreelrnab 37860*	Elementhood in the set of closed-below, open-above intervals of reals. (Contributed by ML, 27-Jul-2020.)
⊢ 𝐼 = ([,) “ (ℝ × ℝ))    ⇒   ⊢ (𝑋 ∈ 𝐼 ↔ ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑋 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)})
Theorem	isbasisrelowllem1 37861*	Lemma for isbasisrelowl 37864. (Contributed by ML, 27-Jul-2020.)
⊢ 𝐼 = ([,) “ (ℝ × ℝ))    ⇒   ⊢ ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})) ∧ (𝑎 ≤ 𝑐 ∧ 𝑏 ≤ 𝑑)) → (𝑥 ∩ 𝑦) ∈ 𝐼)
Theorem	isbasisrelowllem2 37862*	Lemma for isbasisrelowl 37864. (Contributed by ML, 27-Jul-2020.)
⊢ 𝐼 = ([,) “ (ℝ × ℝ))    ⇒   ⊢ ((((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑥 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) ∧ (𝑐 ∈ ℝ ∧ 𝑑 ∈ ℝ ∧ 𝑦 = {𝑧 ∈ ℝ ∣ (𝑐 ≤ 𝑧 ∧ 𝑧 < 𝑑)})) ∧ (𝑎 ≤ 𝑐 ∧ 𝑑 ≤ 𝑏)) → (𝑥 ∩ 𝑦) ∈ 𝐼)
Theorem	icoreclin 37863*	The set of closed-below, open-above intervals of reals is closed under finite intersection. (Contributed by ML, 27-Jul-2020.)
⊢ 𝐼 = ([,) “ (ℝ × ℝ))    ⇒   ⊢ ((𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∩ 𝑦) ∈ 𝐼)
Theorem	isbasisrelowl 37864	The set of all closed-below, open-above intervals of reals form a basis. (Contributed by ML, 27-Jul-2020.)
⊢ 𝐼 = ([,) “ (ℝ × ℝ))    ⇒   ⊢ 𝐼 ∈ TopBases
Theorem	icoreunrn 37865	The union of all closed-below, open-above intervals of reals is the set of reals. (Contributed by ML, 27-Jul-2020.)
⊢ 𝐼 = ([,) “ (ℝ × ℝ))    ⇒   ⊢ ℝ = ∪ 𝐼
Theorem	istoprelowl 37866	The set of all closed-below, open-above intervals of reals generate a topology on the reals. (Contributed by ML, 27-Jul-2020.)
⊢ 𝐼 = ([,) “ (ℝ × ℝ))    ⇒   ⊢ (topGen‘𝐼) ∈ (TopOn‘ℝ)
Theorem	icoreelrn 37867*	A class abstraction which is an element of the set of closed-below, open-above intervals of reals. (Contributed by ML, 1-Aug-2020.)
⊢ 𝐼 = ([,) “ (ℝ × ℝ))    ⇒   ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → {𝑧 ∈ ℝ ∣ (𝐴 ≤ 𝑧 ∧ 𝑧 < 𝐵)} ∈ 𝐼)
Theorem	iooelexlt 37868*	An element of an open interval is not its smallest element. (Contributed by ML, 2-Aug-2020.)
⊢ (𝑋 ∈ (𝐴(,)𝐵) → ∃𝑦 ∈ (𝐴(,)𝐵)𝑦 < 𝑋)
Theorem	relowlssretop 37869	The lower limit topology on the reals is finer than the standard topology. (Contributed by ML, 1-Aug-2020.)
⊢ 𝐼 = ([,) “ (ℝ × ℝ))    ⇒   ⊢ (topGen‘ran (,)) ⊆ (topGen‘𝐼)
Theorem	relowlpssretop 37870	The lower limit topology on the reals is strictly finer than the standard topology. (Contributed by ML, 2-Aug-2020.)
⊢ 𝐼 = ([,) “ (ℝ × ℝ))    ⇒   ⊢ (topGen‘ran (,)) ⊊ (topGen‘𝐼)
Theorem	sucneqond 37871	Inequality of an ordinal set with its successor. Does not use the axiom of regularity. (Contributed by ML, 18-Oct-2020.)
⊢ (𝜑 → 𝑋 = suc 𝑌)    &   ⊢ (𝜑 → 𝑌 ∈ On)    ⇒   ⊢ (𝜑 → 𝑋 ≠ 𝑌)
Theorem	sucneqoni 37872	Inequality of an ordinal set with its successor. Does not use the axiom of regularity. (Contributed by ML, 18-Oct-2020.)
⊢ 𝑋 = suc 𝑌    &   ⊢ 𝑌 ∈ On    ⇒   ⊢ 𝑋 ≠ 𝑌
Theorem	onsucuni3 37873	If an ordinal number has a predecessor, then it is successor of that predecessor. (Contributed by ML, 17-Oct-2020.)
⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → 𝐵 = suc ∪ 𝐵)
Theorem	1oequni2o 37874	The ordinal number 1o is the predecessor of the ordinal number 2o. (Contributed by ML, 19-Oct-2020.)
⊢ 1o = ∪ 2o
Theorem	rdgsucuni 37875	If an ordinal number has a predecessor, the value of the recursive definition generator at that number in terms of its predecessor. (Contributed by ML, 17-Oct-2020.)
⊢ ((𝐵 ∈ On ∧ 𝐵 ≠ ∅ ∧ ¬ Lim 𝐵) → (rec(𝐹, 𝐼)‘𝐵) = (𝐹‘(rec(𝐹, 𝐼)‘∪ 𝐵)))
Theorem	rdgeqoa 37876	If a recursive function with an initial value 𝐴 at step 𝑁 is equal to itself with an initial value 𝐵 at step 𝑀, then every finite number of successor steps will also be equal. (Contributed by ML, 21-Oct-2020.)
⊢ ((𝑁 ∈ On ∧ 𝑀 ∈ On ∧ 𝑋 ∈ ω) → ((rec(𝐹, 𝐴)‘𝑁) = (rec(𝐹, 𝐵)‘𝑀) → (rec(𝐹, 𝐴)‘(𝑁 +o 𝑋)) = (rec(𝐹, 𝐵)‘(𝑀 +o 𝑋))))
Theorem	elxp8 37877	Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp7 8009. (Contributed by ML, 19-Oct-2020.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ((1st ‘𝐴) ∈ 𝐵 ∧ 𝐴 ∈ (V × 𝐶)))
Theorem	cbveud 37878*	Deduction used to change bound variables in an existential uniqueness quantifier, using implicit substitution. (Contributed by ML, 27-Mar-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑦𝜒))
Theorem	cbvreud 37879*	Deduction used to change bound variables in a restricted existential uniqueness quantifier. (Contributed by ML, 27-Mar-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑦 ∈ 𝐴 𝜒))
Theorem	difunieq 37880	The difference of unions is a subset of the union of the difference. (Contributed by ML, 29-Mar-2021.)
⊢ (∪ 𝐴 ∖ ∪ 𝐵) ⊆ ∪ (𝐴 ∖ 𝐵)
Theorem	inunissunidif 37881	Theorem about subsets of the difference of unions. (Contributed by ML, 29-Mar-2021.)
⊢ ((𝐴 ∩ ∪ 𝐶) = ∅ → (𝐴 ⊆ ∪ 𝐵 ↔ 𝐴 ⊆ ∪ (𝐵 ∖ 𝐶)))
Theorem	rdgellim 37882	Elementhood in a recursive definition at a limit ordinal. (Contributed by ML, 30-Mar-2022.)
⊢ (((𝐵 ∈ On ∧ Lim 𝐵) ∧ 𝐶 ∈ 𝐵) → (𝑋 ∈ (rec(𝐹, 𝐴)‘𝐶) → 𝑋 ∈ (rec(𝐹, 𝐴)‘𝐵)))
Theorem	rdglimss 37883	A recursive definition at a limit ordinal is a superset of itself at any smaller ordinal. (Contributed by ML, 30-Mar-2022.)
⊢ (((𝐵 ∈ On ∧ Lim 𝐵) ∧ 𝐶 ∈ 𝐵) → (rec(𝐹, 𝐴)‘𝐶) ⊆ (rec(𝐹, 𝐴)‘𝐵))
Theorem	rdgssun 37884*	In a recursive definition where each step expands on the previous one using a union, every previous step is a subset of every later step. (Contributed by ML, 1-Apr-2022.)
⊢ 𝐹 = (𝑤 ∈ V ↦ (𝑤 ∪ 𝐵))    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((𝑋 ∈ On ∧ 𝑌 ∈ 𝑋) → (rec(𝐹, 𝐴)‘𝑌) ⊆ (rec(𝐹, 𝐴)‘𝑋))
Theorem	exrecfnlem 37885*	Lemma for exrecfn 37886. (Contributed by ML, 30-Mar-2022.)
⊢ 𝐹 = (𝑧 ∈ V ↦ (𝑧 ∪ ran (𝑦 ∈ 𝑧 ↦ 𝐵)))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦 𝐵 ∈ 𝑊) → ∃𝑥(𝐴 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝑥 𝐵 ∈ 𝑥))
Theorem	exrecfn 37886*	Theorem about the existence of infinite recursive sets. 𝑦 should usually be free in 𝐵. (Contributed by ML, 30-Mar-2022.)
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦 𝐵 ∈ 𝑊) → ∃𝑥(𝐴 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝑥 𝐵 ∈ 𝑥))
Theorem	exrecfnpw 37887*	For any base set, a set which contains the powerset of all of its own elements exists. (Contributed by ML, 30-Mar-2022.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥(𝐴 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝑥 𝒫 𝑦 ∈ 𝑥))
Theorem	finorwe 37888	If the Axiom of Infinity is denied, every total order is a well-order. The notion of a well-order cannot be usefully expressed without the Axiom of Infinity due to the inability to quantify over proper classes. (Contributed by ML, 5-Oct-2023.)
⊢ (¬ ω ∈ V → ( < Or 𝐴 → < We 𝐴))
21.21.2  Cartesian exponentiation
Syntax	cfinxp 37889	Extend the definition of a class to include Cartesian exponentiation.
class (𝑈↑↑𝑁)
Definition	df-finxp 37890*	Define Cartesian exponentiation on a class. Note that this definition is limited to finite exponents, since it is defined using nested ordered pairs. If tuples of infinite length are needed, or if they might be needed in the future, use df-ixp 8884 or df-map 8814 instead. The main advantage of this definition is that it integrates better with functions and relations. For example if 𝑅 is a subset of (𝐴↑↑2o), then df-br 5106 can be used on it, and df-fv 6533 can also be used, and so on. It's also worth keeping in mind that ((𝑈↑↑𝑀) × (𝑈↑↑𝑁)) is generally not equal to (𝑈↑↑(𝑀 +o 𝑁)). This definition is technical. Use finxp1o 37898 and finxpsuc 37904 for a more standard recursive experience. (Contributed by ML, 16-Oct-2020.)
⊢ (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))}
Theorem	dffinxpf 37891*	This theorem is the same as Definition df-finxp 37890, except that the large function is replaced by a class variable for brevity. (Contributed by ML, 24-Oct-2020.)
⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))    ⇒   ⊢ (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁))}
Theorem	finxpeq1 37892	Equality theorem for Cartesian exponentiation. (Contributed by ML, 19-Oct-2020.)
⊢ (𝑈 = 𝑉 → (𝑈↑↑𝑁) = (𝑉↑↑𝑁))
Theorem	finxpeq2 37893	Equality theorem for Cartesian exponentiation. (Contributed by ML, 19-Oct-2020.)
⊢ (𝑀 = 𝑁 → (𝑈↑↑𝑀) = (𝑈↑↑𝑁))
Theorem	csbfinxpg 37894*	Distribute proper substitution through Cartesian exponentiation. (Contributed by ML, 25-Oct-2020.)
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝑈↑↑𝑁) = (⦋𝐴 / 𝑥⦌𝑈↑↑⦋𝐴 / 𝑥⦌𝑁))
Theorem	finxpreclem1 37895*	Lemma for ↑↑ recursion theorems. (Contributed by ML, 17-Oct-2020.)
⊢ (𝑋 ∈ 𝑈 → ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩))
Theorem	finxpreclem2 37896*	Lemma for ↑↑ recursion theorems. (Contributed by ML, 17-Oct-2020.)
⊢ ((𝑋 ∈ V ∧ ¬ 𝑋 ∈ 𝑈) → ¬ ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))‘⟨1o, 𝑋⟩))
Theorem	finxp0 37897	The value of Cartesian exponentiation at zero. (Contributed by ML, 24-Oct-2020.)
⊢ (𝑈↑↑∅) = ∅
Theorem	finxp1o 37898	The value of Cartesian exponentiation at one. (Contributed by ML, 17-Oct-2020.)
⊢ (𝑈↑↑1o) = 𝑈
Theorem	finxpreclem3 37899*	Lemma for ↑↑ recursion theorems. (Contributed by ML, 20-Oct-2020.)
⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))    ⇒   ⊢ (((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑋 ∈ (V × 𝑈)) → ⟨∪ 𝑁, (1st ‘𝑋)⟩ = (𝐹‘⟨𝑁, 𝑋⟩))
Theorem	finxpreclem4 37900*	Lemma for ↑↑ recursion theorems. (Contributed by ML, 23-Oct-2020.)
⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))    ⇒   ⊢ (((𝑁 ∈ ω ∧ 2o ⊆ 𝑁) ∧ 𝑦 ∈ (V × 𝑈)) → (rec(𝐹, ⟨𝑁, 𝑦⟩)‘𝑁) = (rec(𝐹, ⟨∪ 𝑁, (1st ‘𝑦)⟩)‘∪ 𝑁))