P. 326 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 32501-32600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dmdcompli 32501	A condition equivalent to the dual modular pair property. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    ⇒   ⊢ (𝐴 𝑀ℋ* 𝐵 ↔ (𝐴 ∩ (⊥‘(𝐴 ∩ 𝐵))) 𝑀ℋ* (𝐵 ∩ (⊥‘(𝐴 ∩ 𝐵))))
Theorem	mddmdin0i 32502*	If dual modular implies modular whenever meet is zero, then dual modular implies modular for arbitrary lattice elements. This theorem is needed for the remark after Lemma 7 of [Holland] p. 1524 to hold. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
⊢ 𝐴 ∈ Cℋ    &   ⊢ 𝐵 ∈ Cℋ    &   ⊢ ∀𝑥 ∈ Cℋ ∀𝑦 ∈ Cℋ ((𝑥 𝑀ℋ* 𝑦 ∧ (𝑥 ∩ 𝑦) = 0ℋ) → 𝑥 𝑀ℋ 𝑦)    ⇒   ⊢ (𝐴 𝑀ℋ* 𝐵 → 𝐴 𝑀ℋ 𝐵)
Theorem	cdjreui 32503*	A member of the sum of disjoint subspaces has a unique decomposition. Part of Lemma 5 of [Holland] p. 1520. (Contributed by NM, 20-May-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    ⇒   ⊢ ((𝐶 ∈ (𝐴 +ℋ 𝐵) ∧ (𝐴 ∩ 𝐵) = 0ℋ) → ∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥 +ℎ 𝑦))
Theorem	cdj1i 32504*	Two ways to express "𝐴 and 𝐵 are completely disjoint subspaces." (1) => (2) in Lemma 5 of [Holland] p. 1520. (Contributed by NM, 21-May-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    ⇒   ⊢ (∃𝑤 ∈ ℝ (0 < 𝑤 ∧ ∀𝑦 ∈ 𝐴 ∀𝑣 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑣)) ≤ (𝑤 · (normℎ‘(𝑦 +ℎ 𝑣)))) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) = 1 → 𝑥 ≤ (normℎ‘(𝑦 −ℎ 𝑧)))))
Theorem	cdj3lem1 32505*	A property of "𝐴 and 𝐵 are completely disjoint subspaces." Part of Lemma 5 of [Holland] p. 1520. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    ⇒   ⊢ (∃𝑥 ∈ ℝ (0 < 𝑥 ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ((normℎ‘𝑦) + (normℎ‘𝑧)) ≤ (𝑥 · (normℎ‘(𝑦 +ℎ 𝑧)))) → (𝐴 ∩ 𝐵) = 0ℋ)
Theorem	cdj3lem2 32506*	Lemma for cdj3i 32512. Value of the first-component function 𝑆. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    &   ⊢ 𝑆 = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑥 = (𝑧 +ℎ 𝑤)))    ⇒   ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑆‘(𝐶 +ℎ 𝐷)) = 𝐶)
Theorem	cdj3lem2a 32507*	Lemma for cdj3i 32512. Closure of the first-component function 𝑆. (Contributed by NM, 25-May-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    &   ⊢ 𝑆 = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑥 = (𝑧 +ℎ 𝑤)))    ⇒   ⊢ ((𝐶 ∈ (𝐴 +ℋ 𝐵) ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑆‘𝐶) ∈ 𝐴)
Theorem	cdj3lem2b 32508*	Lemma for cdj3i 32512. The first-component function 𝑆 is bounded if the subspaces are completely disjoint. (Contributed by NM, 26-May-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    &   ⊢ 𝑆 = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑥 = (𝑧 +ℎ 𝑤)))    ⇒   ⊢ (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢))))
Theorem	cdj3lem3 32509*	Lemma for cdj3i 32512. Value of the second-component function 𝑇. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    &   ⊢ 𝑇 = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑤)))    ⇒   ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑇‘(𝐶 +ℎ 𝐷)) = 𝐷)
Theorem	cdj3lem3a 32510*	Lemma for cdj3i 32512. Closure of the second-component function 𝑇. (Contributed by NM, 26-May-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    &   ⊢ 𝑇 = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑤)))    ⇒   ⊢ ((𝐶 ∈ (𝐴 +ℋ 𝐵) ∧ (𝐴 ∩ 𝐵) = 0ℋ) → (𝑇‘𝐶) ∈ 𝐵)
Theorem	cdj3lem3b 32511*	Lemma for cdj3i 32512. The second-component function 𝑇 is bounded if the subspaces are completely disjoint. (Contributed by NM, 31-May-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    &   ⊢ 𝑇 = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑤)))    ⇒   ⊢ (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦)))) → ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢))))
Theorem	cdj3i 32512*	Two ways to express "𝐴 and 𝐵 are completely disjoint subspaces." (1) <=> (3) in Lemma 5 of [Holland] p. 1520. (Contributed by NM, 1-Jun-2005.) (New usage is discouraged.)
⊢ 𝐴 ∈ Sℋ    &   ⊢ 𝐵 ∈ Sℋ    &   ⊢ 𝑆 = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑥 = (𝑧 +ℎ 𝑤)))    &   ⊢ 𝑇 = (𝑥 ∈ (𝐴 +ℋ 𝐵) ↦ (℩𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 𝑥 = (𝑧 +ℎ 𝑤)))    &   ⊢ (𝜑 ↔ ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑆‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢))))    &   ⊢ (𝜓 ↔ ∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑢 ∈ (𝐴 +ℋ 𝐵)(normℎ‘(𝑇‘𝑢)) ≤ (𝑣 · (normℎ‘𝑢))))    ⇒   ⊢ (∃𝑣 ∈ ℝ (0 < 𝑣 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((normℎ‘𝑥) + (normℎ‘𝑦)) ≤ (𝑣 · (normℎ‘(𝑥 +ℎ 𝑦)))) ↔ ((𝐴 ∩ 𝐵) = 0ℋ ∧ 𝜑 ∧ 𝜓))
PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
21.1  Mathboxes for user contributions
21.1.1  Mathbox guidelines
Theorem	mathbox 32513	(This theorem is a dummy placeholder for these guidelines. The label of this theorem, "mathbox", is hard-coded into the Metamath program to identify the start of the mathbox section for web page generation.) A "mathbox" is a user-contributed section that is maintained by its contributor independently from the main part of set.mm. For contributors: By making a contribution, you agree to release it into the public domain, according to the statement at the beginning of set.mm. Mathboxes are provided to help keep your work synchronized with changes in set.mm while allowing you to work independently without affecting other contributors. Even though in a sense your mathbox belongs to you, it is still part of the shared body of knowledge contained in set.mm, and occasionally other people may make maintenance edits to your mathbox for things like keeping it synchronized with the rest of set.mm, reducing proof lengths, moving your theorems to the main part of set.mm when needed, and fixing typos or other errors. If you want to preserve it the way you left it, you can keep a local copy or keep track of the GitHub commit number. Guidelines: 1. See conventions 30470 for our general style guidelines. For contributing via GitHub, see https://github.com/metamath/set.mm/blob/develop/CONTRIBUTING.md 30470. The Metamath program command "verify markup *" will check that you have followed many of the conventions we use. 2. If at all possible, please use only nullary class constants for new definitions, for example as in df-div 11808. 3. Each $p and $a statement must be immediately preceded with the comment that will be shown on its web page description. The Metamath program "MM> WRITE SOURCE set.mm / REWRAP" command will take care of indentation conventions and line wrapping. 4. All mathbox content will be on public display and should hopefully reflect the overall quality of the website. 5. Mathboxes must be independent from one another (checked by "verify markup *"). If you need a theorem from another mathbox, typically it is moved to the main part of set.mm. New users should consult with more experienced users before doing this. 6. If a contributor is no longer active, we will continue the usual maintenance edits. As time goes on, often theorems will be moved to main or removed in favor of similar replacements. But we are also willing to maintain mathboxes in place, as work by others from years ago may form the foundation of future work; you could even argue that all of mathematics is like that. 7. For theorems of importance (for example, a Metamath 100 theorem or a dependency of one), we prefer to eventually move them out of mathboxes (although a mathbox is perfectly appropriate as proofs are being developed and refined). (Contributed by NM, 20-Feb-2007.) (Revised by the Metamath team, 9-Sep-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    ⇒   ⊢ 𝜑
21.2  Mathbox for Stefan Allan
Theorem	sa-abvi 32514	A theorem about the universal class. Inference associated with bj-abv 37213 (which is proved from fewer axioms). (Contributed by Stefan Allan, 9-Dec-2008.)
⊢ 𝜑    ⇒   ⊢ V = {𝑥 ∣ 𝜑}
Theorem	xfree 32515	A partial converse to 19.9t 2212. (Contributed by Stefan Allan, 21-Dec-2008.) (Revised by Mario Carneiro, 11-Dec-2016.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(∃𝑥𝜑 → 𝜑))
Theorem	xfree2 32516	A partial converse to 19.9t 2212. (Contributed by Stefan Allan, 21-Dec-2008.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑))
Theorem	addltmulALT 32517	A proof readability experiment for addltmul 12413. (Contributed by Stefan Allan, 30-Oct-2010.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (2 < 𝐴 ∧ 2 < 𝐵)) → (𝐴 + 𝐵) < (𝐴 · 𝐵))
21.3  Mathbox for Thierry Arnoux
21.3.1  Propositional Calculus - misc additions
Theorem	ad11antr 32518	Deduction adding 11 conjuncts to antecedent. (Contributed by Thierry Arnoux, 27-Sep-2025.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((((((((((((𝜑 ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) ∧ 𝜅) ∧ 𝜈) → 𝜓)
Theorem	simp-12l 32519	Simplification of a conjunction. (Contributed by Thierry Arnoux, 5-Oct-2025.)
⊢ (((((((((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) ∧ 𝜅) ∧ 𝜈) → 𝜑)
Theorem	simp-12r 32520	Simplification of a conjunction. (Contributed by Thierry Arnoux, 5-Oct-2025.)
⊢ (((((((((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) ∧ 𝜅) ∧ 𝜈) → 𝜓)
Theorem	an52ds 32521	Inference exchanging the last antecedent with the second. (Contributed by Thierry Arnoux, 3-Jun-2025.)
⊢ (((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜂)    ⇒   ⊢ (((((𝜑 ∧ 𝜏) ∧ 𝜒) ∧ 𝜃) ∧ 𝜓) → 𝜂)
Theorem	an62ds 32522	Inference exchanging the last antecedent with the second one. (Contributed by Thierry Arnoux, 3-Jun-2025.)
⊢ ((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜁)    ⇒   ⊢ ((((((𝜑 ∧ 𝜂) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜓) → 𝜁)
Theorem	an72ds 32523	Inference exchanging the last antecedent with the second one. (Contributed by Thierry Arnoux, 3-Jun-2025.)
⊢ (((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) → 𝜎)    ⇒   ⊢ (((((((𝜑 ∧ 𝜁) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜓) → 𝜎)
Theorem	an82ds 32524	Inference exchanging the last antecedent with the second one. (Contributed by Thierry Arnoux, 3-Jun-2025.)
⊢ ((((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) → 𝜌)    ⇒   ⊢ ((((((((𝜑 ∧ 𝜎) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜓) → 𝜌)
Theorem	syl22anbrc 32525	Syllogism inference. (Contributed by Thierry Arnoux, 19-Oct-2025.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (𝜑 → 𝜃)    &   ⊢ (𝜑 → 𝜏)    &   ⊢ (𝜂 ↔ ((𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏)))    ⇒   ⊢ (𝜑 → 𝜂)
Theorem	bian1dOLD 32526	Obsolete version of bian1d 580 as of 29-Jun-2025. (Contributed by Thierry Arnoux, 26-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))    ⇒   ⊢ (𝜑 → ((𝜒 ∧ 𝜓) ↔ (𝜒 ∧ 𝜃)))
Theorem	orim12da 32527	Deduce a disjunction from another one. Variation on orim12d 967. (Contributed by Thierry Arnoux, 18-May-2025.)
⊢ ((𝜑 ∧ 𝜓) → 𝜃)    &   ⊢ ((𝜑 ∧ 𝜒) → 𝜏)    &   ⊢ (𝜑 → (𝜓 ∨ 𝜒))    ⇒   ⊢ (𝜑 → (𝜃 ∨ 𝜏))
Theorem	or3di 32528	Distributive law for disjunction. (Contributed by Thierry Arnoux, 3-Jul-2017.)
⊢ ((𝜑 ∨ (𝜓 ∧ 𝜒 ∧ 𝜏)) ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜏)))
Theorem	or3dir 32529	Distributive law for disjunction. (Contributed by Thierry Arnoux, 3-Jul-2017.)
⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∨ 𝜏) ↔ ((𝜑 ∨ 𝜏) ∧ (𝜓 ∨ 𝜏) ∧ (𝜒 ∨ 𝜏)))
Theorem	3o1cs 32530	Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) → 𝜃)    ⇒   ⊢ (𝜑 → 𝜃)
Theorem	3o2cs 32531	Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) → 𝜃)    ⇒   ⊢ (𝜓 → 𝜃)
Theorem	3o3cs 32532	Deduction eliminating disjunct. (Contributed by Thierry Arnoux, 19-Dec-2016.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) → 𝜃)    ⇒   ⊢ (𝜒 → 𝜃)
Theorem	13an22anass 32533	Associative law for four conjunctions with a triple conjunction. (Contributed by Thierry Arnoux, 21-Jan-2025.)
⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)))
21.3.2  Predicate Calculus
21.3.2.1  Predicate Calculus - misc additions
Theorem	sbc2iedf 32534*	Conversion of implicit substitution to explicit class substitution. (Contributed by Thierry Arnoux, 4-Jul-2023.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜒    &   ⊢ Ⅎ𝑦𝜒    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓 ↔ 𝜒))
Theorem	rspc2daf 32535*	Double restricted specialization, using implicit substitution. (Contributed by Thierry Arnoux, 4-Jul-2023.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜒    &   ⊢ Ⅎ𝑦𝜒    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑊 𝜓)    ⇒   ⊢ (𝜑 → 𝜒)
21.3.2.2  Restricted quantification - misc additions
Theorem	ralcom4f 32536*	Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by Thierry Arnoux, 8-Mar-2017.)
⊢ Ⅎ𝑦𝐴    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦𝜑 ↔ ∀𝑦∀𝑥 ∈ 𝐴 𝜑)
Theorem	rexcom4f 32537*	Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by Thierry Arnoux, 8-Mar-2017.)
⊢ Ⅎ𝑦𝐴    ⇒   ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦𝜑 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝜑)
Theorem	19.9d2rf 32538	A deduction version of one direction of 19.9 2213 with two variables. (Contributed by Thierry Arnoux, 20-Mar-2017.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	19.9d2r 32539*	A deduction version of one direction of 19.9 2213 with two variables. (Contributed by Thierry Arnoux, 30-Jan-2017.)
⊢ (𝜑 → Ⅎ𝑥𝜓)    &   ⊢ (𝜑 → Ⅎ𝑦𝜓)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	r19.29ffa 32540*	A commonly used pattern based on r19.29 3101, version with two restricted quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017.)
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓) → 𝜒)
Theorem	n0limd 32541*	Deduction rule for nonempty classes. (Contributed by Thierry Arnoux, 3-Aug-2025.)
⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	reu6dv 32542*	A condition which implies existential uniqueness. (Contributed by Thierry Arnoux, 13-Oct-2025.)
⊢ (𝜑 → 𝐵 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝑥 = 𝐵))    ⇒   ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 𝜓)
21.3.2.3  Equality
Theorem	eqtrb 32543	A transposition of equality. (Contributed by Thierry Arnoux, 20-Aug-2023.)
⊢ ((𝐴 = 𝐵 ∧ 𝐴 = 𝐶) ↔ (𝐴 = 𝐵 ∧ 𝐵 = 𝐶))
Theorem	eqelbid 32544*	A variable elimination law for equality within a given set 𝐴. See equvel 2461. (Contributed by Thierry Arnoux, 20-Feb-2025.)
⊢ (𝜑 → 𝐵 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝑥 = 𝐵 ↔ 𝑥 = 𝐶) ↔ 𝐵 = 𝐶))
21.3.2.4  Double restricted existential uniqueness quantification
Theorem	opsbc2ie 32545*	Conversion of implicit substitution to explicit class substitution for ordered pairs. (Contributed by Thierry Arnoux, 4-Jul-2023.)
⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑 ↔ 𝜒))    ⇒   ⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ [𝑦 / 𝑏][𝑥 / 𝑎]𝜒))
Theorem	opreu2reuALT 32546*	Correspondence between uniqueness of ordered pairs and double restricted existential uniqueness quantification. Alternate proof of one direction only, use opreu2reurex 6259 instead. (Contributed by Thierry Arnoux, 4-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑 ↔ 𝜒))    ⇒   ⊢ ((∃!𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝜒 ∧ ∃!𝑏 ∈ 𝐵 ∃𝑎 ∈ 𝐴 𝜒) → ∃!𝑝 ∈ (𝐴 × 𝐵)𝜑)
21.3.2.5  Double restricted existential uniqueness quantification syntax
Syntax	w2reu 32547	Syntax for double restricted existential uniqueness quantification.
wff ∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑
Definition	df-2reu 32548	Define the double restricted existential uniqueness quantifier. (Contributed by Thierry Arnoux, 4-Jul-2023.)
⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 ↔ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑))
Theorem	2reucom 32549	Double restricted existential uniqueness commutes. (Contributed by Thierry Arnoux, 4-Jul-2023.)
⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 ↔ ∃!𝑦 ∈ 𝐵 , 𝑥 ∈ 𝐴𝜑)
Theorem	2reu2rex1 32550	Double restricted existential uniqueness implies double restricted existence. (Contributed by Thierry Arnoux, 4-Jul-2023.)
⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)
Theorem	2reureurex 32551	Double restricted existential uniqueness implies restricted existential uniqueness with restricted existence. (Contributed by AV, 5-Jul-2023.)
⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 → ∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)
Theorem	2reu2reu2 32552*	Double restricted existential uniqueness implies two nested restricted existential uniqueness. (Contributed by AV, 5-Jul-2023.)
⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 → ∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑)
Theorem	opreu2reu1 32553*	Equivalent definition of the double restricted existential uniqueness quantifier, using uniqueness of ordered pairs. (Contributed by Thierry Arnoux, 4-Jul-2023.)
⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝜒 ↔ 𝜑))    ⇒   ⊢ (∃!𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵𝜑 ↔ ∃!𝑝 ∈ (𝐴 × 𝐵)𝜒)
Theorem	sq2reunnltb 32554*	There exists a unique decomposition of a prime as a sum of squares of two different positive integers iff the prime is of the form 4𝑘 + 1. Double restricted existential uniqueness variant of 2sqreunnltb 27424. (Contributed by AV, 5-Jul-2023.)
⊢ (𝑃 ∈ ℙ → ((𝑃 mod 4) = 1 ↔ ∃!𝑎 ∈ ℕ , 𝑏 ∈ ℕ(𝑎 < 𝑏 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)))
Theorem	addsqnot2reu 32555*	For each complex number 𝐶, there does not uniquely exist two complex numbers 𝑎 and 𝑏, with 𝑏 squared and added to 𝑎 resulting in the given complex number 𝐶. Double restricted existential uniqueness variant of addsqn2reurex2 27408. (Contributed by AV, 5-Jul-2023.)
⊢ (𝐶 ∈ ℂ → ¬ ∃!𝑎 ∈ ℂ , 𝑏 ∈ ℂ(𝑎 + (𝑏↑2)) = 𝐶)
21.3.2.6  Substitution (without distinct variables) - misc additions
Theorem	sbceqbidf 32556	Equality theorem for class substitution. (Contributed by Thierry Arnoux, 4-Sep-2018.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜒))
Theorem	sbcies 32557*	A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 14-Mar-2019.)
⊢ 𝐴 = (𝐸‘𝑊)    &   ⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝑤 = 𝑊 → ([(𝐸‘𝑤) / 𝑎]𝜓 ↔ 𝜑))
21.3.2.7  Existential "at most one" - misc additions
Theorem	mo5f 32558*	Alternate definition of "at most one." (Contributed by Thierry Arnoux, 1-Mar-2017.)
⊢ Ⅎ𝑖𝜑    &   ⊢ Ⅎ𝑗𝜑    ⇒   ⊢ (∃*𝑥𝜑 ↔ ∀𝑖∀𝑗(([𝑖 / 𝑥]𝜑 ∧ [𝑗 / 𝑥]𝜑) → 𝑖 = 𝑗))
Theorem	nmo 32559*	Negation of "at most one". (Contributed by Thierry Arnoux, 26-Feb-2017.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (¬ ∃*𝑥𝜑 ↔ ∀𝑦∃𝑥(𝜑 ∧ 𝑥 ≠ 𝑦))
21.3.2.8  Existential uniqueness - misc additions
Theorem	reuxfrdf 32560*	Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Cf. reuxfrd 3695 (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.) (Revised by Thierry Arnoux, 30-Mar-2018.)
⊢ Ⅎ𝑦𝐵    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃*𝑦 ∈ 𝐶 𝑥 = 𝐴)    ⇒   ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓) ↔ ∃!𝑦 ∈ 𝐶 𝜓))
Theorem	rexunirn 32561*	Restricted existential quantification over the union of the range of a function. Cf. rexrn 7040 and eluni2 4855. (Contributed by Thierry Arnoux, 19-Sep-2017.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝑉)    ⇒   ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑦 ∈ ∪ ran 𝐹𝜑)
21.3.2.9  Restricted "at most one" - misc additions
Theorem	rmoxfrd 32562*	Transfer "at most one" restricted quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃*𝑥 ∈ 𝐵 𝜓 ↔ ∃*𝑦 ∈ 𝐶 𝜒))
Theorem	rmoun 32563	"At most one" restricted existential quantifier for a union implies the same quantifier on both sets. (Contributed by Thierry Arnoux, 27-Nov-2023.)
⊢ (∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 → (∃*𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐵 𝜑))
Theorem	rmounid 32564*	A case where an "at most one" restricted existential quantifier for a union is equivalent to such a quantifier for one of the sets. (Contributed by Thierry Arnoux, 27-Nov-2023.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ¬ 𝜓)    ⇒   ⊢ (𝜑 → (∃*𝑥 ∈ (𝐴 ∪ 𝐵)𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜓))
21.3.2.10  Restricted iota (description binder)
Theorem	riotaeqbidva 32565*	Equivalent wff's yield equal restricted definition binders (deduction form). (raleqbidva 3302 analog.) (Contributed by Thierry Arnoux, 29-Jan-2025.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐵 𝜒))
21.3.3  General Set Theory
21.3.3.1  Class abstractions (a.k.a. class builders)
Theorem	dmrab 32566*	Domain of a restricted class abstraction over a cartesian product. (Contributed by Thierry Arnoux, 3-Jul-2023.)
⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ dom {𝑧 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {𝑥 ∈ 𝐴 ∣ ∃𝑦 ∈ 𝐵 𝜓}
Theorem	difrab2 32567	Difference of two restricted class abstractions. Compare with difrab 4259. (Contributed by Thierry Arnoux, 3-Jan-2022.)
⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∖ {𝑥 ∈ 𝐵 ∣ 𝜑}) = {𝑥 ∈ (𝐴 ∖ 𝐵) ∣ 𝜑}
Theorem	elrabrd 32568*	Deduction version of elrab 3635, just like elrabd 3637, but backwards direction. (Contributed by Thierry Arnoux, 15-Jan-2026.)
⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → 𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜓})    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	rabexgfGS 32569	Separation Scheme in terms of a restricted class abstraction. To be removed in profit of Glauco's equivalent version. (Contributed by Thierry Arnoux, 11-May-2017.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)
Theorem	rabsnel 32570*	Truth implied by equality of a restricted class abstraction and a singleton. (Contributed by Thierry Arnoux, 15-Sep-2018.)
⊢ 𝐵 ∈ V    ⇒   ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝐵} → 𝐵 ∈ 𝐴)
Theorem	rabsspr 32571*	Conditions for a restricted class abstraction to be a subset of an unordered pair. (Contributed by Thierry Arnoux, 6-Jul-2025.)
⊢ ({𝑥 ∈ 𝑉 ∣ 𝜑} ⊆ {𝑋, 𝑌} ↔ ∀𝑥 ∈ 𝑉 (𝜑 → (𝑥 = 𝑋 ∨ 𝑥 = 𝑌)))
Theorem	rabsstp 32572*	Conditions for a restricted class abstraction to be a subset of an unordered triple. (Contributed by Thierry Arnoux, 6-Jul-2025.)
⊢ ({𝑥 ∈ 𝑉 ∣ 𝜑} ⊆ {𝑋, 𝑌, 𝑍} ↔ ∀𝑥 ∈ 𝑉 (𝜑 → (𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ∨ 𝑥 = 𝑍)))
Theorem	3unrab 32573	Union of three restricted class abstractions. (Contributed by Thierry Arnoux, 6-Jul-2025.)
⊢ (({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ 𝜓}) ∪ {𝑥 ∈ 𝐴 ∣ 𝜒}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ 𝜓 ∨ 𝜒)}
Theorem	foresf1o 32574*	From a surjective function, *choose* a subset of the domain, such that the restricted function is bijective. (Contributed by Thierry Arnoux, 27-Jan-2020.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–onto→𝐵) → ∃𝑥 ∈ 𝒫 𝐴(𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐵)
Theorem	rabfodom 32575*	Domination relation for restricted abstract class builders, based on a surjective function. (Contributed by Thierry Arnoux, 27-Jan-2020.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹‘𝑥)) → (𝜒 ↔ 𝜓))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴–onto→𝐵)    ⇒   ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ 𝜒} ≼ {𝑥 ∈ 𝐴 ∣ 𝜓})
Theorem	rabrexfi 32576*	Conditions for a class abstraction with a restricted existential quantification to be finite. (Contributed by Thierry Arnoux, 6-Jul-2025.)
⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ Fin)    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ ∃𝑦 ∈ 𝐵 𝜓} ∈ Fin)
21.3.3.2  Image Sets
Theorem	abrexdomjm 32577*	An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝑦 ∈ 𝐴 → ∃*𝑥𝜑)    ⇒   ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝜑} ≼ 𝐴)
Theorem	abrexdom2jm 32578*	An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} ≼ 𝐴)
Theorem	abrexexd 32579*	Existence of a class abstraction of existentially restricted sets. (Contributed by Thierry Arnoux, 10-May-2017.)
⊢ Ⅎ𝑥𝐴    &   ⊢ (𝜑 → 𝐴 ∈ V)    ⇒   ⊢ (𝜑 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)
Theorem	elabreximd 32580*	Class substitution in an image set. (Contributed by Thierry Arnoux, 30-Dec-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜒    &   ⊢ (𝐴 = 𝐵 → (𝜒 ↔ 𝜓))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝜓)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵}) → 𝜒)
Theorem	elabreximdv 32581*	Class substitution in an image set. (Contributed by Thierry Arnoux, 30-Dec-2016.)
⊢ (𝐴 = 𝐵 → (𝜒 ↔ 𝜓))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝜓)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑦 = 𝐵}) → 𝜒)
Theorem	abrexss 32582*	A necessary condition for an image set to be a subset. (Contributed by Thierry Arnoux, 6-Feb-2017.)
⊢ Ⅎ𝑥𝐶    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ⊆ 𝐶)
21.3.3.3  Set relations and operations - misc additions
Theorem	nelun 32583	Negated membership for a union. (Contributed by Thierry Arnoux, 13-Dec-2023.)
⊢ (𝐴 = (𝐵 ∪ 𝐶) → (¬ 𝑋 ∈ 𝐴 ↔ (¬ 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ 𝐶)))
Theorem	snsssng 32584	If a singleton is a subset of another, their members are equal. (Contributed by NM, 28-May-2006.) (Revised by Thierry Arnoux, 11-Apr-2024.)
⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ⊆ {𝐵}) → 𝐴 = 𝐵)
Theorem	n0nsnel 32585*	If a class with one element is not a singleton, there is at least another element in this class. (Contributed by AV, 6-Mar-2025.) (Revised by Thierry Arnoux, 28-May-2025.)
⊢ ((𝐶 ∈ 𝐵 ∧ 𝐵 ≠ {𝐴}) → ∃𝑥 ∈ 𝐵 𝑥 ≠ 𝐴)
Theorem	inin 32586	Intersection with an intersection. (Contributed by Thierry Arnoux, 27-Dec-2016.)
⊢ (𝐴 ∩ (𝐴 ∩ 𝐵)) = (𝐴 ∩ 𝐵)
Theorem	difininv 32587	Condition for the intersections of two sets with a given set to be equal. (Contributed by Thierry Arnoux, 28-Dec-2021.)
⊢ ((((𝐴 ∖ 𝐶) ∩ 𝐵) = ∅ ∧ ((𝐶 ∖ 𝐴) ∩ 𝐵) = ∅) → (𝐴 ∩ 𝐵) = (𝐶 ∩ 𝐵))
Theorem	difeq 32588	Rewriting an equation with class difference, without using quantifiers. (Contributed by Thierry Arnoux, 24-Sep-2017.)
⊢ ((𝐴 ∖ 𝐵) = 𝐶 ↔ ((𝐶 ∩ 𝐵) = ∅ ∧ (𝐶 ∪ 𝐵) = (𝐴 ∪ 𝐵)))
Theorem	eqdif 32589	If both set differences of two sets are empty, those sets are equal. (Contributed by Thierry Arnoux, 16-Nov-2023.)
⊢ (((𝐴 ∖ 𝐵) = ∅ ∧ (𝐵 ∖ 𝐴) = ∅) → 𝐴 = 𝐵)
Theorem	indifbi 32590	Two ways to express equality relative to a class 𝐴. (Contributed by Thierry Arnoux, 23-Jun-2024.)
⊢ ((𝐴 ∩ 𝐵) = (𝐴 ∩ 𝐶) ↔ (𝐴 ∖ 𝐵) = (𝐴 ∖ 𝐶))
Theorem	diffib 32591	Case where diffi 9109 is a biconditional. (Contributed by Thierry Arnoux, 27-Jun-2024.)
⊢ (𝐵 ∈ Fin → (𝐴 ∈ Fin ↔ (𝐴 ∖ 𝐵) ∈ Fin))
Theorem	difxp1ss 32592	Difference law for Cartesian products. (Contributed by Thierry Arnoux, 24-Jul-2023.)
⊢ ((𝐴 ∖ 𝐶) × 𝐵) ⊆ (𝐴 × 𝐵)
Theorem	difxp2ss 32593	Difference law for Cartesian products. (Contributed by Thierry Arnoux, 24-Jul-2023.)
⊢ (𝐴 × (𝐵 ∖ 𝐶)) ⊆ (𝐴 × 𝐵)
Theorem	indifundif 32594	A remarkable equation with sets. (Contributed by Thierry Arnoux, 18-May-2020.)
⊢ (((𝐴 ∩ 𝐵) ∖ 𝐶) ∪ (𝐴 ∖ 𝐵)) = (𝐴 ∖ (𝐵 ∩ 𝐶))
Theorem	elpwincl1 32595	Closure of intersection with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐶)    ⇒   ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝒫 𝐶)
Theorem	elpwdifcl 32596	Closure of class difference with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝒫 𝐶)    ⇒   ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ 𝒫 𝐶)
Theorem	elpwiuncl 32597*	Closure of indexed union with regard to elementhood to a power set. (Contributed by Thierry Arnoux, 27-May-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝒫 𝐶)    ⇒   ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ 𝒫 𝐶)
21.3.3.4  Unordered pairs
Theorem	elpreq 32598	Equality wihin a pair. (Contributed by Thierry Arnoux, 23-Aug-2017.)
⊢ (𝜑 → 𝑋 ∈ {𝐴, 𝐵})    &   ⊢ (𝜑 → 𝑌 ∈ {𝐴, 𝐵})    &   ⊢ (𝜑 → (𝑋 = 𝐴 ↔ 𝑌 = 𝐴))    ⇒   ⊢ (𝜑 → 𝑋 = 𝑌)
Theorem	prssad 32599	If a pair is a subset of a class, the first element of the pair is an element of that class. (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → {𝐴, 𝐵} ⊆ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐶)
Theorem	prssbd 32600	If a pair is a subset of a class, the second element of the pair is an element of that class. (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → {𝐴, 𝐵} ⊆ 𝐶)    ⇒   ⊢ (𝜑 → 𝐵 ∈ 𝐶)