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	axprlem3 5401*	Lemma for axpr 5403. Eliminate the antecedent of the relevant replacement instance. (Contributed by Rohan Ridenour, 10-Aug-2023.) (Proof shortened by Matthew House, 18-Sep-2025.)
⊢ ∃𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ ∃𝑠(𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))
Theorem	axprlem4 5402*	Lemma for axpr 5403. If an existing set of empty sets corresponds to one element of the pair, then the element 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.) (Revised by Matthew House, 18-Sep-2025.)
⊢ ∃𝑠∀𝑛𝜑    &   ⊢ (𝜑 → (𝑛 ∈ 𝑠 → ∀𝑡 ¬ 𝑡 ∈ 𝑛))    &   ⊢ (∀𝑛𝜑 → (if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) ↔ 𝑤 = 𝑣))    ⇒   ⊢ (∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) → (𝑤 = 𝑣 → ∃𝑠(𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))))
Theorem	axpr 5403*	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 5409 below so that the uses of the Axiom of Pairing can be more easily identified. For a shorter proof using ax-ext 2738, see axprALT 5398. (Contributed by NM, 14-Nov-2006.) Remove dependency on ax-ext 2738. (Revised by Rohan Ridenour, 10-Aug-2023.) (Proof shortened by BJ, 13-Aug-2023.) (Proof shortened by Matthew House, 18-Sep-2025.) Use ax-pr 5409 instead. (New usage is discouraged.)
⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)
Theorem	axprlem1OLD 5404*	Obsolete version of axprlem1 5399 as of 6-Apr-2026. (Contributed by Rohan Ridenour, 10-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑥∀𝑦(∀𝑧 ¬ 𝑧 ∈ 𝑦 → 𝑦 ∈ 𝑥)
Theorem	axprlem3OLD 5405*	Obsolete version of axprlem3 5401 as of 18-Sep-2025. (Contributed by Rohan Ridenour, 10-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑧∀𝑤(𝑤 ∈ 𝑧 ↔ ∃𝑠(𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))
Theorem	axprlem4OLD 5406*	Obsolete version of axprlem4 5402 as of 18-Sep-2025. (Contributed by Rohan Ridenour, 10-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑥) → ∃𝑠(𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))
Theorem	axprlem5OLD 5407*	Obsolete version of axprlem4 5402 as of 18-Sep-2025. (Contributed by Rohan Ridenour, 10-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((∀𝑠(∀𝑛 ∈ 𝑠 ∀𝑡 ¬ 𝑡 ∈ 𝑛 → 𝑠 ∈ 𝑝) ∧ 𝑤 = 𝑦) → ∃𝑠(𝑠 ∈ 𝑝 ∧ if-(∃𝑛 𝑛 ∈ 𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))
Theorem	axprOLD 5408*	Obsolete version of axpr 5403 as of 18-Sep-2025. (Contributed by NM, 14-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)
Axiom	ax-pr 5409*	The Axiom of Pairing of ZF set theory. It was derived as Theorem axpr 5403 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 5410	Derive the abbreviated version of the Axiom of Pairing from ax-pr 5409. See zfpair 5397 for its derivation from the other axioms. (Contributed by NM, 14-Nov-2006.)
⊢ {𝑥, 𝑦} ∈ V
Theorem	vsnex 5411	A singleton built on a setvar is a set. (Contributed by BJ, 15-Jan-2025.)
⊢ {𝑥} ∈ V
Theorem	axprglem 5412*	Lemma for axprg 5413. (Contributed by GG, 11-Mar-2026.)
⊢ (𝑥 = 𝐴 → ∃𝑧∀𝑤((𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧))
Theorem	axprg 5413*	Derive The Axiom of Pairing with class variables. (Contributed by GG, 6-Mar-2026.)
⊢ ∃𝑧∀𝑤((𝑤 = 𝐴 ∨ 𝑤 = 𝐵) → 𝑤 ∈ 𝑧)
Theorem	prex 5414	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 4738), so we can dispense with hypotheses requiring them to be sets. (Contributed by NM, 15-Jul-1993.) Avoid ax-nul 5274 and shorten proof. (Revised by GG, 6-Mar-2026.)
⊢ {𝐴, 𝐵} ∈ V
Theorem	snex 5415	A singleton is a set. Theorem 7.12 of [Quine] p. 51, proved using Extensionality, Separation and Pairing. See also snexALT 5359. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 19-May-2013.) Avoid ax-nul 5274 and shorten proof. (Revised by GG, 6-Mar-2026.)
⊢ {𝐴} ∈ V
Theorem	snexg 5416	A singleton built on a set is a set. Special case of snex 5415 which is intuitionistically valid. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 19-May-2013.) Extract from snex 5415 and shorten proof. (Revised by BJ, 15-Jan-2025.) (Proof shortened by GG, 6-Mar-2026.)
⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)
Theorem	snexgALT 5417	Alternate proof of snexg 5416 based on vsnex 5411, which uses an instance of ax-sep 5262. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 19-May-2013.) Extract from snex 5415 and shorten proof. (Revised by BJ, 15-Jan-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)
Theorem	snexOLD 5418	Obsolete version of snex 5415 as of 6-Mar-2026. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 19-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ {𝐴} ∈ V
Theorem	prexOLD 5419	Obsolete version of prex 5414 as of 6-Mar-2026. (Contributed by NM, 15-Jul-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ {𝐴, 𝐵} ∈ V
Theorem	exel 5420*	There exist two sets, one a member of the other. This theorem looks similar to el 5424, but its meaning is different. It only depends on the axioms ax-mp 5 to ax-4 1842, ax-6 2000, and ax-pr 5409. 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 5421. (Contributed by SN, 23-Dec-2024.)
⊢ ∃𝑦∃𝑥 𝑥 ∈ 𝑦
Theorem	exexneq 5421*	There exist two different sets. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2407. (Revised by BJ, 31-May-2019.) Avoid ax-8 2148. (Revised by SN, 21-Sep-2023.) Avoid ax-12 2216. (Revised by Rohan Ridenour, 9-Oct-2024.) Use ax-pr 5409 instead of ax-pow 5341. (Revised by BTernaryTau, 3-Dec-2024.) Extract this result from the proof of dtru 5423. (Revised by BJ, 2-Jan-2025.)
⊢ ∃𝑥∃𝑦 ¬ 𝑥 = 𝑦
Theorem	exneq 5422*	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 2738, ax-sep 5262, or ax-pow 5341 nor auxiliary logical axiom schemes ax-10 2179 to ax-13 2407. See dtruALT 5364 for a shorter proof using more axioms, and dtruALT2 5346 for a proof using ax-pow 5341 instead of ax-pr 5409. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2407. (Revised by BJ, 31-May-2019.) Avoid ax-8 2148. (Revised by SN, 21-Sep-2023.) Avoid ax-12 2216. (Revised by Rohan Ridenour, 9-Oct-2024.) Use ax-pr 5409 instead of ax-pow 5341. (Revised by BTernaryTau, 3-Dec-2024.) Extract this result from the proof of dtru 5423. (Revised by BJ, 2-Jan-2025.)
⊢ ∃𝑥 ¬ 𝑥 = 𝑦
Theorem	dtru 5423*	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 2061. The same comments and revision history concerning axiom usage as in exneq 5422 apply. See dtruALT 5364 and dtruALT2 5346 for alternate proofs avoiding ax-pr 5409. (Contributed by NM, 7-Nov-2006.) Extract exneq 5422 as an intermediate result. (Revised by BJ, 2-Jan-2025.)
⊢ ¬ ∀𝑥 𝑥 = 𝑦
Theorem	el 5424*	Any set is an element of some other set. See elALT 5428 for a shorter proof using more axioms, and see elALT2 5345 for a proof that uses ax-9 2156 and ax-pow 5341 instead of ax-pr 5409. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) Use ax-pr 5409 instead of ax-9 2156 and ax-pow 5341. (Revised by BTernaryTau, 2-Dec-2024.) (Proof shortened by Matthew House, 6-Apr-2026.)
⊢ ∃𝑦 𝑥 ∈ 𝑦
Theorem	el.OLD 5425*	Obsolete version of el 5424 as of 6-Apr-2026. (Contributed by NM, 4-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑦 𝑥 ∈ 𝑦
Theorem	sels 5426*	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 5428. (Revised by BJ, 3-Apr-2019.) Avoid ax-sep 5262, ax-nul 5274, ax-pow 5341. (Revised by BTernaryTau, 15-Jan-2025.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥)
Theorem	selsALT 5427*	Alternate proof of sels 5426, requiring ax-sep 5262 but not using el 5424 (which is proved from it as elALT 5428). (especially when the proof of el 5424 is inlined in sels 5426). (Contributed by NM, 4-Jan-2002.) Generalize from the proof of elALT 5428. (Revised by BJ, 3-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥)
Theorem	elALT 5428*	Alternate proof of el 5424, shorter but requiring ax-sep 5262. (Contributed by NM, 4-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃𝑦 𝑥 ∈ 𝑦
Theorem	snelpwg 5429	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 5274. (Revised by BJ, 17-Jan-2025.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ 𝒫 𝐵))
Theorem	snelpwi 5430	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	snelpw 5431	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 5432	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 5433	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 5434*	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 5435	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 5436	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 5437	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 5438	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 5439*	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 5440*	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 5441*	Negation of subclass relationship. Compare nss 4004. (Contributed by NM, 30-Jun-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ (¬ 𝐴 ⊆ 𝐵 ↔ ∃𝑥(𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵))
Theorem	pweqb 5442	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 5443*	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 5274. (Revised by BJ, 17-Jan-2025.)
⊢ (𝐴 ∈ 𝑉 → ∩ {𝑥 ∣ 𝐴 ∈ 𝑥} = {𝐴})
Theorem	moabex 5444	"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)
Theorem	moabexOLD 5445	Obsolete version of moabex 5444 as of 2-Feb-2026. (Contributed by NM, 30-Dec-1996.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∃*𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V)
Theorem	rmorabex 5446	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 5447	The abstraction of a wff with existential uniqueness exists. (Contributed by NM, 25-Nov-1994.)
⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V)
Theorem	nnullss 5448*	A nonempty class (even if proper) has a nonempty subset. (Contributed by NM, 23-Aug-2003.)
⊢ (𝐴 ≠ ∅ → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅))
Theorem	exss 5449*	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 5450	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 5274. (Revised by GG, 6-Mar-2026.)
⊢ ⟨𝐴, 𝐵⟩ ∈ V
Theorem	opexOLD 5451	Obsolete version of opex 5450 as of 6-Mar-2026. (Contributed by NM, 18-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ⟨𝐴, 𝐵⟩ ∈ V
Theorem	otex 5452	An ordered triple of classes is a set. (Contributed by NM, 3-Apr-2015.)
⊢ ⟨𝐴, 𝐵, 𝐶⟩ ∈ V
Theorem	elopg 5453	Characterization of the elements of an ordered pair. Closed form of elop 5454. (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 5454	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 5455	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 5456	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 5457	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 5458	Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (See op2ndb 6233 to extract the second member, op1sta 6231 for an alternate version, and op1st 8003 for the preferred version.) (Contributed by NM, 25-Nov-2003.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ∩ ∩ ⟨𝐴, 𝐵⟩ = 𝐴
Theorem	brv 5459	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 5791. (Contributed by Scott Fenton, 11-Apr-2012.)
⊢ 𝐴V𝐵
2.3.3  Ordered pair theorem
Theorem	opnz 5460	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 5461	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 5462	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 5463	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 5464	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 5465	Equality of the first members of equal ordered pairs. Closed form of opth1 5462. (Contributed by AV, 14-Oct-2018.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐴 = 𝐶))
Theorem	opthg2 5466	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 5467	Ordered pair theorem. (Contributed by NM, 21-Sep-2014.)
⊢ 𝐶 ∈ V    &   ⊢ 𝐷 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	opthneg 5468	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 5469	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 5470	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 5471	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 5472	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 5473	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 5474*	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 5475*	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 5476*	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 5477*	Version of copsexg 5479 with a disjoint variable condition, which does not require ax-13 2407. (Contributed by GG, 26-Jan-2024.) Shorten proof and remove dependency on ax-10 2179. (Revised by Eric Schmidt, 2-May-2026.)
⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
Theorem	copsexgwOLD 5478*	Obsolete version of copsexgw 5477 as of 2-May-2026. (Contributed by GG, 26-Jan-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
Theorem	copsexg 5479*	Substitution of class 𝐴 for ordered pair ⟨𝑥, 𝑦⟩. Usage of this theorem is discouraged because it depends on ax-13 2407. Use the weaker copsexgw 5477 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 5480*	Closed theorem form of copsex2g 5481. (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 5481*	Implicit substitution inference for ordered pairs. (Contributed by NM, 28-May-1995.) Use a similar proof to copsex4g 5483 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	copsex2dv 5482*	Implicit substitution deduction for ordered pairs. (Contributed by Thierry Arnoux, 4-May-2025.)
⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒))
Theorem	copsex4g 5483*	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 5484	A property of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)
⊢ ¬ ∅ ∈ ⟨𝐴, 𝐵⟩
Theorem	opwo0id 5485	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 5486	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 5487	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 5488	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 5489	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)
Theorem	moop2 5490*	"At most one" property of an ordered pair. (Contributed by NM, 11-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ 𝐵 ∈ V    ⇒   ⊢ ∃*𝑥 𝐴 = ⟨𝐵, 𝑥⟩
Theorem	opeqsng 5491	Equivalence for an ordered pair equal to a singleton. (Contributed by NM, 3-Jun-2008.) (Revised by AV, 15-Jul-2022.) (Avoid depending on this detail.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴})))
Theorem	opeqsn 5492	Equivalence for an ordered pair equal to a singleton. (Contributed by NM, 3-Jun-2008.) (Revised by AV, 15-Jul-2022.) (Avoid depending on this detail.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩ = {𝐶} ↔ (𝐴 = 𝐵 ∧ 𝐶 = {𝐴}))
Theorem	opeqpr 5493	Equivalence for an ordered pair equal to an unordered pair. (Contributed by NM, 3-Jun-2008.) (Avoid depending on this detail.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    &   ⊢ 𝐷 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩ = {𝐶, 𝐷} ↔ ((𝐶 = {𝐴} ∧ 𝐷 = {𝐴, 𝐵}) ∨ (𝐶 = {𝐴, 𝐵} ∧ 𝐷 = {𝐴})))
Theorem	snopeqop 5494	Equivalence for an ordered pair equal to a singleton of an ordered pair. (Contributed by AV, 18-Sep-2020.) (Revised by AV, 15-Jul-2022.) (Avoid depending on this detail.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ({⟨𝐴, 𝐵⟩} = ⟨𝐶, 𝐷⟩ ↔ (𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ∧ 𝐶 = {𝐴}))
Theorem	propeqop 5495	Equivalence for an ordered pair equal to a pair of ordered pairs. (Contributed by AV, 18-Sep-2020.) (Proof shortened by AV, 16-Jun-2022.) (Avoid depending on this detail.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    &   ⊢ 𝐷 ∈ V    &   ⊢ 𝐸 ∈ V    &   ⊢ 𝐹 ∈ V    ⇒   ⊢ ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ⟨𝐸, 𝐹⟩ ↔ ((𝐴 = 𝐶 ∧ 𝐸 = {𝐴}) ∧ ((𝐴 = 𝐵 ∧ 𝐹 = {𝐴, 𝐷}) ∨ (𝐴 = 𝐷 ∧ 𝐹 = {𝐴, 𝐵}))))
Theorem	propssopi 5496	If a pair of ordered pairs is a subset of an ordered pair, their first components are equal. (Contributed by AV, 20-Sep-2020.) (Proof shortened by AV, 16-Jun-2022.) (Avoid depending on this detail.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    &   ⊢ 𝐷 ∈ V    &   ⊢ 𝐸 ∈ V    &   ⊢ 𝐹 ∈ V    ⇒   ⊢ ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ⊆ ⟨𝐸, 𝐹⟩ → 𝐴 = 𝐶)
Theorem	snopeqopsnid 5497	Equivalence for an ordered pair of two identical singletons equal to a singleton of an ordered pair. (Contributed by AV, 24-Sep-2020.) (Revised by AV, 15-Jul-2022.) (Avoid depending on this detail.)
⊢ 𝐴 ∈ V    ⇒   ⊢ {⟨𝐴, 𝐴⟩} = ⟨{𝐴}, {𝐴}⟩
Theorem	mosubopt 5498*	"At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-Aug-2007.)
⊢ (∀𝑦∀𝑧∃*𝑥𝜑 → ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑))
Theorem	mosubop 5499*	"At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-May-1995.)
⊢ ∃*𝑥𝜑    ⇒   ⊢ ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
Theorem	euop2 5500*	Transfer existential uniqueness to second member of an ordered pair. (Contributed by NM, 10-Apr-2004.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (∃!𝑥∃𝑦(𝑥 = ⟨𝐴, 𝑦⟩ ∧ 𝜑) ↔ ∃!𝑦𝜑)