P. 55 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 5401-5500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	reusv3i 5401*	Two ways of expressing existential uniqueness via an indirect equality. (Contributed by NM, 23-Dec-2012.)
⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝑧 → 𝐶 = 𝐷)    ⇒   ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷))
Theorem	reusv3 5402*	Two ways to express single-valuedness of a class expression 𝐶(𝑦). See reusv1 5394 for the connection to uniqueness. (Contributed by NM, 27-Dec-2012.)
⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝑧 → 𝐶 = 𝐷)    ⇒   ⊢ (∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝐶 ∈ 𝐴) → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))
Theorem	eusv4 5403*	Two ways to express single-valuedness of a class expression 𝐵(𝑦). (Contributed by NM, 27-Oct-2010.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)
Theorem	alxfr 5404*	Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 18-Feb-2007.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((∀𝑦 𝐴 ∈ 𝐵 ∧ ∀𝑥∃𝑦 𝑥 = 𝐴) → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
Theorem	ralxfrd 5405*	Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 15-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.) (Proof shortened by JJ, 7-Aug-2021.)
⊢ ((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	rexxfrd 5406*	Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by FL, 10-Apr-2007.) (Revised by Mario Carneiro, 15-Aug-2014.)
⊢ ((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	ralxfr2d 5407*	Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by Mario Carneiro, 20-Aug-2014.)
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐶 𝑥 = 𝐴))    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑦 ∈ 𝐶 𝜒))
Theorem	rexxfr2d 5408*	Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by Mario Carneiro, 20-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐶 𝑥 = 𝐴))    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑦 ∈ 𝐶 𝜒))
Theorem	ralxfrd2 5409*	Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Variant of ralxfrd 5405. (Contributed by Alexander van der Vekens, 25-Apr-2018.)
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑦 ∈ 𝐶 𝜒))
Theorem	rexxfrd2 5410*	Transfer existence from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Variant of rexxfrd 5406. (Contributed by Alexander van der Vekens, 25-Apr-2018.)
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑦 ∈ 𝐶 𝜒))
Theorem	ralxfr 5411*	Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.)
⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵)    &   ⊢ (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐶 𝜓)
Theorem	ralxfrALT 5412*	Alternate proof of ralxfr 5411 which does not use ralxfrd 5405. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵)    &   ⊢ (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐶 𝜓)
Theorem	rexxfr 5413*	Transfer existence from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.)
⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵)    &   ⊢ (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐶 𝜓)
Theorem	rabxfrd 5414*	Membership in a restricted class abstraction after substituting an expression 𝐴 (containing 𝑦) for 𝑥 in the formula defining the class abstraction. (Contributed by NM, 16-Jan-2012.)
⊢ Ⅎ𝑦𝐵    &   ⊢ Ⅎ𝑦𝐶    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝐴 ∈ 𝐷)    &   ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑦 = 𝐵 → 𝐴 = 𝐶)    ⇒   ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐷) → (𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜓} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜒}))
Theorem	rabxfr 5415*	Membership in a restricted class abstraction after substituting an expression 𝐴 (containing 𝑦) for 𝑥 in the formula defining the class abstraction. (Contributed by NM, 10-Jun-2005.)
⊢ Ⅎ𝑦𝐵    &   ⊢ Ⅎ𝑦𝐶    &   ⊢ (𝑦 ∈ 𝐷 → 𝐴 ∈ 𝐷)    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐵 → 𝐴 = 𝐶)    ⇒   ⊢ (𝐵 ∈ 𝐷 → (𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜑} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜓}))
Theorem	reuhypd 5416*	A theorem useful for eliminating the restricted existential uniqueness hypotheses in riotaxfrd 7402. (Contributed by NM, 16-Jan-2012.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 ↔ 𝑦 = 𝐵))    ⇒   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴)
Theorem	reuhyp 5417*	A theorem useful for eliminating the restricted existential uniqueness hypotheses in reuxfr1 3747. (Contributed by NM, 15-Nov-2004.)
⊢ (𝑥 ∈ 𝐶 → 𝐵 ∈ 𝐶)    &   ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 = 𝐴 ↔ 𝑦 = 𝐵))    ⇒   ⊢ (𝑥 ∈ 𝐶 → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴)
Theorem	zfpair 5418	The Axiom of Pairing of Zermelo-Fraenkel set theory. Axiom 2 of [TakeutiZaring] p. 15. In some textbooks this is stated as a separate axiom; here we show it is redundant since it can be derived from the other axioms. This theorem should not be referenced by any proof other than axprALT 5419. Instead, use zfpair2 5427 below so that the uses of the Axiom of Pairing can be more easily identified. (Contributed by NM, 18-Oct-1995.) (New usage is discouraged.)
⊢ {𝑥, 𝑦} ∈ V
Theorem	axprALT 5419*	Alternate proof of axpr 5425. (Contributed by NM, 14-Nov-2006.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)
2.3.2  Derive the Axiom of Pairing
Theorem	axprlem1 5420*	Lemma for axpr 5425. There exists a set to which all empty sets belong. (Contributed by Rohan Ridenour, 10-Aug-2023.) (Revised by BJ, 13-Aug-2023.)
⊢ ∃𝑥∀𝑦(∀𝑧 ¬ 𝑧 ∈ 𝑦 → 𝑦 ∈ 𝑥)
Theorem	axprlem2 5421*	Lemma for axpr 5425. There exists a set to which all sets whose only members are empty sets belong. (Contributed by Rohan Ridenour, 9-Aug-2023.) (Revised by BJ, 13-Aug-2023.)
⊢ ∃𝑥∀𝑦(∀𝑧 ∈ 𝑦 ∀𝑤 ¬ 𝑤 ∈ 𝑧 → 𝑦 ∈ 𝑥)
Theorem	axprlem3 5422*	Lemma for axpr 5425. Eliminate the antecedent of the relevant replacement instance. (Contributed by Rohan Ridenour, 10-Aug-2023.)
⊢ ∃𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ ∃𝑠(𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))
Theorem	axprlem4 5423*	Lemma for axpr 5425. The first element of the pair is included in any superset of the set whose existence is asserted by the axiom of replacement. (Contributed by Rohan Ridenour, 10-Aug-2023.) (Revised by BJ, 13-Aug-2023.)
⊢ ((∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑥) → ∃𝑠(𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))
Theorem	axprlem5 5424*	Lemma for axpr 5425. The second element of the pair is included in any superset of the set whose existence is asserted by the axiom of replacement. (Contributed by Rohan Ridenour, 10-Aug-2023.) (Revised by BJ, 13-Aug-2023.)
⊢ ((∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦) → ∃𝑠(𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))
Theorem	axpr 5425*	Unabbreviated version of the Axiom of Pairing of ZF set theory, derived as a theorem from the other axioms. This theorem should not be referenced by any proof. Instead, use ax-pr 5426 below so that the uses of the Axiom of Pairing can be more easily identified. For a shorter proof using ax-ext 2701, see axprALT 5419. (Contributed by NM, 14-Nov-2006.) Remove dependency on ax-ext 2701. (Revised by Rohan Ridenour, 10-Aug-2023.) (Proof shortened by BJ, 13-Aug-2023.) Use ax-pr 5426 instead. (New usage is discouraged.)
⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)
Axiom	ax-pr 5426*	The Axiom of Pairing of ZF set theory. It was derived as Theorem axpr 5425 above and is therefore redundant, but we state it as a separate axiom here so that its uses can be identified more easily. (Contributed by NM, 14-Nov-2006.)
⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)
Theorem	zfpair2 5427	Derive the abbreviated version of the Axiom of Pairing from ax-pr 5426. See zfpair 5418 for its derivation from the other axioms. (Contributed by NM, 14-Nov-2006.)
⊢ {𝑥, 𝑦} ∈ V
Theorem	vsnex 5428	A singleton built on a setvar is a set. (Contributed by BJ, 15-Jan-2025.)
⊢ {𝑥} ∈ V
Theorem	snexg 5429	A singleton built on a set is a set. Special case of snex 5430 which does not require ax-nul 5305 and is intuitionistically valid. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 19-May-2013.) Extract from snex 5430 and shorten proof. (Revised by BJ, 15-Jan-2025.)
⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)
Theorem	snex 5430	A singleton is a set. Theorem 7.12 of [Quine] p. 51, proved using Extensionality, Separation, Null Set, and Pairing. See also snexALT 5380. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 19-May-2013.)
⊢ {𝐴} ∈ V
Theorem	prex 5431	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 4770), so we can dispense with hypotheses requiring them to be sets. (Contributed by NM, 15-Jul-1993.)
⊢ {𝐴, 𝐵} ∈ V
Theorem	exel 5432*	There exist two sets, one a member of the other. This theorem looks similar to el 5436, but its meaning is different. It only depends on the axioms ax-mp 5 to ax-4 1809, ax-6 1969, and ax-pr 5426. This theorem does not exclude that these two sets could actually be one single set containing itself. That two different sets exist is proved by exexneq 5433. (Contributed by SN, 23-Dec-2024.)
⊢ ∃𝑦∃𝑥 𝑥 ∈ 𝑦
Theorem	exexneq 5433*	There exist two different sets. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2369. (Revised by BJ, 31-May-2019.) Avoid ax-8 2106. (Revised by SN, 21-Sep-2023.) Avoid ax-12 2169. (Revised by Rohan Ridenour, 9-Oct-2024.) Use ax-pr 5426 instead of ax-pow 5362. (Revised by BTernaryTau, 3-Dec-2024.) Extract this result from the proof of dtru 5435. (Revised by BJ, 2-Jan-2025.)
⊢ ∃𝑥∃𝑦 ¬ 𝑥 = 𝑦
Theorem	exneq 5434*	Given any set (the "𝑦 " in the statement), there exists a set not equal to it. The same statement without disjoint variable condition is false, since we do not have ∃𝑥¬ 𝑥 = 𝑥. This theorem is proved directly from set theory axioms (no class definitions) and does not depend on ax-ext 2701, ax-sep 5298, or ax-pow 5362 nor auxiliary logical axiom schemes ax-10 2135 to ax-13 2369. See dtruALT 5385 for a shorter proof using more axioms, and dtruALT2 5367 for a proof using ax-pow 5362 instead of ax-pr 5426. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2369. (Revised by BJ, 31-May-2019.) Avoid ax-8 2106. (Revised by SN, 21-Sep-2023.) Avoid ax-12 2169. (Revised by Rohan Ridenour, 9-Oct-2024.) Use ax-pr 5426 instead of ax-pow 5362. (Revised by BTernaryTau, 3-Dec-2024.) Extract this result from the proof of dtru 5435. (Revised by BJ, 2-Jan-2025.)
⊢ ∃𝑥 ¬ 𝑥 = 𝑦
Theorem	dtru 5435*	Given any set (the "𝑦 " in the statement), not all sets are equal to it. The same statement without disjoint variable condition is false since it contradicts stdpc6 2029. The same comments and revision history concerning axiom usage as in exneq 5434 apply. (Contributed by NM, 7-Nov-2006.) Extract exneq 5434 as an intermediate result. (Revised by BJ, 2-Jan-2025.)
⊢ ¬ ∀𝑥 𝑥 = 𝑦
Theorem	el 5436*	Any set is an element of some other set. See elALT 5439 for a shorter proof using more axioms, and see elALT2 5366 for a proof that uses ax-9 2114 and ax-pow 5362 instead of ax-pr 5426. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) Use ax-pr 5426 instead of ax-9 2114 and ax-pow 5362. (Revised by BTernaryTau, 2-Dec-2024.)
⊢ ∃𝑦 𝑥 ∈ 𝑦
Theorem	sels 5437*	If a class is a set, then it is a member of a set. (Contributed by NM, 4-Jan-2002.) Generalize from the proof of elALT 5439. (Revised by BJ, 3-Apr-2019.) Avoid ax-sep 5298, ax-nul 5305, ax-pow 5362. (Revised by BTernaryTau, 15-Jan-2025.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥)
Theorem	selsALT 5438*	Alternate proof of sels 5437, requiring ax-sep 5298 but not using el 5436 (which is proved from it as elALT 5439). (especially when the proof of el 5436 is inlined in sels 5437). (Contributed by NM, 4-Jan-2002.) Generalize from the proof of elALT 5439. (Revised by BJ, 3-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥)
Theorem	elALT 5439*	Alternate proof of el 5436, shorter but requiring ax-sep 5298. (Contributed by NM, 4-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑦 𝑥 ∈ 𝑦
Theorem	dtruOLD 5440*	Obsolete proof of dtru 5435 as of 01-Jan-2025. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2369. (Revised by BJ, 31-May-2019.) Avoid ax-12 2169. (Revised by Rohan Ridenour, 9-Oct-2024.) Use ax-pr 5426 instead of ax-pow 5362. (Revised by BTernaryTau, 3-Dec-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ¬ ∀𝑥 𝑥 = 𝑦
Theorem	snelpwg 5441	A singleton of a set is a member of the powerclass of a class if and only if that set is a member of that class. (Contributed by NM, 1-Apr-1998.) Put in closed form and avoid ax-nul 5305. (Revised by BJ, 17-Jan-2025.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ 𝒫 𝐵))
Theorem	snelpwi 5442	If a set is a member of a class, then the singleton of that set is a member of the powerclass of that class. (Contributed by Alan Sare, 25-Aug-2011.)
⊢ (𝐴 ∈ 𝐵 → {𝐴} ∈ 𝒫 𝐵)
Theorem	snelpwiOLD 5443	Obsolete version of snelpwi 5442 as of 17-Jan-2025. (Contributed by NM, 28-May-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝐵 → {𝐴} ∈ 𝒫 𝐵)
Theorem	snelpw 5444	A singleton of a set is a member of the powerclass of a class if and only if that set is a member of that class. (Contributed by NM, 1-Apr-1998.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ 𝒫 𝐵)
Theorem	prelpw 5445	An unordered pair of two sets is a member of the powerclass of a class if and only if the two sets are members of that class. (Contributed by AV, 8-Jan-2020.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ↔ {𝐴, 𝐵} ∈ 𝒫 𝐶))
Theorem	prelpwi 5446	If two sets are members of a class, then the unordered pair of those two sets is a member of the powerclass of that class. (Contributed by Thierry Arnoux, 10-Mar-2017.) (Proof shortened by AV, 23-Oct-2021.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → {𝐴, 𝐵} ∈ 𝒫 𝐶)
Theorem	rext 5447*	A theorem similar to extensionality, requiring the existence of a singleton. Exercise 8 of [TakeutiZaring] p. 16. (Contributed by NM, 10-Aug-1993.)
⊢ (∀𝑧(𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧) → 𝑥 = 𝑦)
Theorem	sspwb 5448	The powerclass construction preserves and reflects inclusion. Classes are subclasses if and only if their power classes are subclasses. Exercise 18 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.)
⊢ (𝐴 ⊆ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
Theorem	unipw 5449	A class equals the union of its power class. Exercise 6(a) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.) (Proof shortened by Alan Sare, 28-Dec-2008.)
⊢ ∪ 𝒫 𝐴 = 𝐴
Theorem	univ 5450	The union of the universe is the universe. Exercise 4.12(c) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.)
⊢ ∪ V = V
Theorem	pwtr 5451	A class is transitive iff its power class is transitive. (Contributed by Alan Sare, 25-Aug-2011.) (Revised by Mario Carneiro, 15-Jun-2014.)
⊢ (Tr 𝐴 ↔ Tr 𝒫 𝐴)
Theorem	ssextss 5452*	An extensionality-like principle defining subclass in terms of subsets. (Contributed by NM, 30-Jun-2004.)
⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵))
Theorem	ssext 5453*	An extensionality-like principle that uses the subset instead of the membership relation: two classes are equal iff they have the same subsets. (Contributed by NM, 30-Jun-2004.)
⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵))
Theorem	nssss 5454*	Negation of subclass relationship. Compare nss 4045. (Contributed by NM, 30-Jun-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ (¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥(𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵))
Theorem	pweqb 5455	Classes are equal if and only if their power classes are equal. Exercise 19 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.)
⊢ (𝐴 = 𝐵 ↔ 𝒫 𝐴 = 𝒫 𝐵)
Theorem	intidg 5456*	The intersection of all sets to which a set belongs is the singleton of that set. (Contributed by NM, 5-Jun-2009.) Put in closed form and avoid ax-nul 5305. (Revised by BJ, 17-Jan-2025.)
⊢ (𝐴 ∈ 𝑉 → ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} = {𝐴})
Theorem	intidOLD 5457*	Obsolete version of intidg 5456 as of 18-Jan-2025. (Contributed by NM, 5-Jun-2009.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} = {𝐴}
Theorem	moabex 5458	"At most one" existence implies a class abstraction exists. (Contributed by NM, 30-Dec-1996.)
⊢ (∃*𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V)
Theorem	rmorabex 5459	Restricted "at most one" existence implies a restricted class abstraction exists. (Contributed by NM, 17-Jun-2017.)
⊢ (∃*𝑥 ∈ 𝐴 𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)
Theorem	euabex 5460	The abstraction of a wff with existential uniqueness exists. (Contributed by NM, 25-Nov-1994.)
⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V)
Theorem	nnullss 5461*	A nonempty class (even if proper) has a nonempty subset. (Contributed by NM, 23-Aug-2003.)
⊢ (𝐴 ≠ ∅ → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅))
Theorem	exss 5462*	Restricted existence in a class (even if proper) implies restricted existence in a subset. (Contributed by NM, 23-Aug-2003.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑦(𝑦 ⊆ 𝐴 ∧ ∃𝑥 ∈ 𝑦 𝜑))
Theorem	opex 5463	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.)
⊢ ⟨𝐴, 𝐵⟩ ∈ V
Theorem	otex 5464	An ordered triple of classes is a set. (Contributed by NM, 3-Apr-2015.)
⊢ ⟨𝐴, 𝐵, 𝐶⟩ ∈ V
Theorem	elopg 5465	Characterization of the elements of an ordered pair. Closed form of elop 5466. (Contributed by BJ, 22-Jun-2019.) (Avoid depending on this detail.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ ⟨𝐴, 𝐵⟩ ↔ (𝐶 = {𝐴} ∨ 𝐶 = {𝐴, 𝐵})))
Theorem	elop 5466	Characterization of the elements of an ordered pair. Exercise 3 of [TakeutiZaring] p. 15. (Contributed by NM, 15-Jul-1993.) (Revised by Mario Carneiro, 26-Apr-2015.) Remove an extraneous hypothesis. (Revised by BJ, 25-Dec-2020.) (Avoid depending on this detail.)
⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (𝐴 ∈ ⟨𝐵, 𝐶⟩ ↔ (𝐴 = {𝐵} ∨ 𝐴 = {𝐵, 𝐶}))
Theorem	opi1 5467	One of the two elements in an ordered pair. (Contributed by NM, 15-Jul-1993.) (Revised by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ {𝐴} ∈ ⟨𝐴, 𝐵⟩
Theorem	opi2 5468	One of the two elements of an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ {𝐴, 𝐵} ∈ ⟨𝐴, 𝐵⟩
Theorem	opeluu 5469	Each member of an ordered pair belongs to the union of the union of a class to which the ordered pair belongs. Lemma 3D of [Enderton] p. 41. (Contributed by NM, 31-Mar-1995.) (Revised by Mario Carneiro, 27-Feb-2016.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → (𝐴 ∈ ∪ ∪ 𝐶 ∧ 𝐵 ∈ ∪ ∪ 𝐶))
Theorem	op1stb 5470	Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (See op2ndb 6225 to extract the second member, op1sta 6223 for an alternate version, and op1st 7985 for the preferred version.) (Contributed by NM, 25-Nov-2003.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ∩ ∩ ⟨𝐴, 𝐵⟩ = 𝐴
Theorem	brv 5471	Two classes are always in relation by V. This is simply equivalent to ⟨𝐴, 𝐵⟩ ∈ V, and does not imply that V is a relation: see nrelv 5799. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ 𝐴V𝐵
2.3.3  Ordered pair theorem
Theorem	opnz 5472	An ordered pair is nonempty iff the arguments are sets. (Contributed by NM, 24-Jan-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ (⟨𝐴, 𝐵⟩ ≠ ∅ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
Theorem	opnzi 5473	An ordered pair is nonempty if the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ⟨𝐴, 𝐵⟩ ≠ ∅
Theorem	opth1 5474	Equality of the first members of equal ordered pairs. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐴 = 𝐶)
Theorem	opth 5475	The ordered pair theorem. If two ordered pairs are equal, their first elements are equal and their second elements are equal. Exercise 6 of [TakeutiZaring] p. 16. Note that 𝐶 and 𝐷 are not required to be sets due our specific ordered pair definition. (Contributed by NM, 28-May-1995.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	opthg 5476	Ordered pair theorem. 𝐶 and 𝐷 are not required to be sets under our specific ordered pair definition. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
Theorem	opth1g 5477	Equality of the first members of equal ordered pairs. Closed form of opth1 5474. (Contributed by AV, 14-Oct-2018.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐴 = 𝐶))
Theorem	opthg2 5478	Ordered pair theorem. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ ((𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
Theorem	opth2 5479	Ordered pair theorem. (Contributed by NM, 21-Sep-2014.)
⊢ 𝐶 ∈ V    &   ⊢ 𝐷 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	opthneg 5480	Two ordered pairs are not equal iff their first components or their second components are not equal. (Contributed by AV, 13-Dec-2018.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷)))
Theorem	opthne 5481	Two ordered pairs are not equal iff their first components or their second components are not equal. (Contributed by AV, 13-Dec-2018.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩ ↔ (𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷))
Theorem	otth2 5482	Ordered triple theorem, with triple expressed with ordered pairs. (Contributed by NM, 1-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝑅 ∈ V    ⇒   ⊢ (⟨⟨𝐴, 𝐵⟩, 𝑅⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑆⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆))
Theorem	otth 5483	Ordered triple theorem. (Contributed by NM, 25-Sep-2014.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝑅 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵, 𝑅⟩ = ⟨𝐶, 𝐷, 𝑆⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆))
Theorem	otthg 5484	Ordered triple theorem, closed form. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝐷, 𝐸, 𝐹⟩ ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹)))
Theorem	otthne 5485	Contrapositive of the ordered triple theorem. (Contributed by Scott Fenton, 31-Jan-2025.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵, 𝐶⟩ ≠ ⟨𝐷, 𝐸, 𝐹⟩ ↔ (𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐸 ∨ 𝐶 ≠ 𝐹))
Theorem	eqvinop 5486*	A variable introduction law for ordered pairs. Analogue of Lemma 15 of [Monk2] p. 109. (Contributed by NM, 28-May-1995.)
⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (𝐴 = ⟨𝐵, 𝐶⟩ ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩))
Theorem	sbcop1 5487*	The proper substitution of an ordered pair for a setvar variable corresponds to a proper substitution of its first component. (Contributed by AV, 8-Apr-2023.)
⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝑎 / 𝑥]𝜓 ↔ [⟨𝑎, 𝑦⟩ / 𝑧]𝜑)
Theorem	sbcop 5488*	The proper substitution of an ordered pair for a setvar variable corresponds to a proper substitution of each of its components. (Contributed by AV, 8-Apr-2023.)
⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝑏 / 𝑦][𝑎 / 𝑥]𝜓 ↔ [⟨𝑎, 𝑏⟩ / 𝑧]𝜑)
Theorem	copsexgw 5489*	Version of copsexg 5490 with a disjoint variable condition, which does not require ax-13 2369. (Contributed by Gino Giotto, 26-Jan-2024.)
⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
Theorem	copsexg 5490*	Substitution of class 𝐴 for ordered pair ⟨𝑥, 𝑦⟩. Usage of this theorem is discouraged because it depends on ax-13 2369. Use the weaker copsexgw 5489 when possible. (Contributed by NM, 27-Dec-1996.) (Revised by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 25-Aug-2019.) (New usage is discouraged.)
⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
Theorem	copsex2t 5491*	Closed theorem form of copsex2g 5492. (Contributed by NM, 17-Feb-2013.)
⊢ ((∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))
Theorem	copsex2g 5492*	Implicit substitution inference for ordered pairs. (Contributed by NM, 28-May-1995.) Use a similar proof to copsex4g 5494 to reduce axiom usage. (Revised by SN, 1-Sep-2024.)
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))
Theorem	copsex2gOLD 5493*	Obsolete version of copsex2g 5492 as of 1-Sep-2024. (Contributed by NM, 28-May-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))
Theorem	copsex4g 5494*	An implicit substitution inference for 2 ordered pairs. (Contributed by NM, 5-Aug-1995.)
⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆)) → (∃𝑥∃𝑦∃𝑧∃𝑤((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑧, 𝑤⟩) ∧ 𝜑) ↔ 𝜓))
Theorem	0nelop 5495	A property of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)
⊢ ¬ ∅ ∈ ⟨𝐴, 𝐵⟩
Theorem	opwo0id 5496	An ordered pair is equal to the ordered pair without the empty set. This is because no ordered pair contains the empty set. (Contributed by AV, 15-Nov-2021.)
⊢ ⟨𝑋, 𝑌⟩ = (⟨𝑋, 𝑌⟩ ∖ {∅})
Theorem	opeqex 5497	Equivalence of existence implied by equality of ordered pairs. (Contributed by NM, 28-May-2008.)
⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐶 ∈ V ∧ 𝐷 ∈ V)))
Theorem	oteqex2 5498	Equivalence of existence implied by equality of ordered triples. (Contributed by NM, 26-Apr-2015.)
⊢ (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩ → (𝐶 ∈ V ↔ 𝑇 ∈ V))
Theorem	oteqex 5499	Equivalence of existence implied by equality of ordered triples. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩ → ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ↔ (𝑅 ∈ V ∧ 𝑆 ∈ V ∧ 𝑇 ∈ V)))
Theorem	opcom 5500	An ordered pair commutes iff its members are equal. (Contributed by NM, 28-May-2009.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐵, 𝐴⟩ ↔ 𝐴 = 𝐵)