P. 58 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 5701-5800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	opthprc 5701	Justification theorem for an ordered pair definition that works for any classes, including proper classes. This is a possible definition implied by the footnote in [Jech] p. 78, which says, "The sophisticated reader will not object to our use of a pair of classes." (Contributed by NM, 28-Sep-2003.)
⊢ (((𝐴 × {∅}) ∪ (𝐵 × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	brel 5702	Two things in a binary relation belong to the relation's domain. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ 𝑅 ⊆ (𝐶 × 𝐷)    ⇒   ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷))
Theorem	elxp3 5703*	Membership in a Cartesian product. (Contributed by NM, 5-Mar-1995.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)))
Theorem	opeliunxp 5704	Membership in a union of Cartesian products. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 1-Jan-2017.)
⊢ (⟨𝑥, 𝐶⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵))
Theorem	xpundi 5705	Distributive law for Cartesian product over union. Theorem 103 of [Suppes] p. 52. (Contributed by NM, 12-Aug-2004.)
⊢ (𝐴 × (𝐵 ∪ 𝐶)) = ((𝐴 × 𝐵) ∪ (𝐴 × 𝐶))
Theorem	xpundir 5706	Distributive law for Cartesian product over union. Similar to Theorem 103 of [Suppes] p. 52. (Contributed by NM, 30-Sep-2002.)
⊢ ((𝐴 ∪ 𝐵) × 𝐶) = ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶))
Theorem	xpiundi 5707*	Distributive law for Cartesian product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.)
⊢ (𝐶 × ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)
Theorem	xpiundir 5708*	Distributive law for Cartesian product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.)
⊢ (∪ 𝑥 ∈ 𝐴 𝐵 × 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 × 𝐶)
Theorem	iunxpconst 5709*	Membership in a union of Cartesian products when the second factor is constant. (Contributed by Mario Carneiro, 29-Dec-2014.)
⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵)
Theorem	xpun 5710	The Cartesian product of two unions. (Contributed by NM, 12-Aug-2004.)
⊢ ((𝐴 ∪ 𝐵) × (𝐶 ∪ 𝐷)) = (((𝐴 × 𝐶) ∪ (𝐴 × 𝐷)) ∪ ((𝐵 × 𝐶) ∪ (𝐵 × 𝐷)))
Theorem	elvv 5711*	Membership in universal class of ordered pairs. (Contributed by NM, 4-Jul-1994.)
⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
Theorem	elvvv 5712*	Membership in universal class of ordered triples. (Contributed by NM, 17-Dec-2008.)
⊢ (𝐴 ∈ ((V × V) × V) ↔ ∃𝑥∃𝑦∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
Theorem	elvvuni 5713	An ordered pair contains its union. (Contributed by NM, 16-Sep-2006.)
⊢ (𝐴 ∈ (V × V) → ∪ 𝐴 ∈ 𝐴)
Theorem	brinxp2 5714	Intersection of binary relation with Cartesian product. (Contributed by NM, 3-Mar-2007.) (Revised by Mario Carneiro, 26-Apr-2015.) Group conjuncts and avoid df-3an 1089. (Revised by Peter Mazsa, 18-Sep-2022.)
⊢ (𝐶(𝑅 ∩ (𝐴 × 𝐵))𝐷 ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ 𝐶𝑅𝐷))
Theorem	brinxp 5715	Intersection of binary relation with Cartesian product. (Contributed by NM, 9-Mar-1997.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝑅𝐵 ↔ 𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵))
Theorem	opelinxp 5716	Ordered pair element in an intersection with Cartesian product. (Contributed by Peter Mazsa, 21-Jul-2019.)
⊢ (⟨𝐶, 𝐷⟩ ∈ (𝑅 ∩ (𝐴 × 𝐵)) ↔ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ∧ ⟨𝐶, 𝐷⟩ ∈ 𝑅))
Theorem	poinxp 5717	Intersection of partial order with Cartesian product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)
⊢ (𝑅 Po 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴)
Theorem	soinxp 5718	Intersection of total order with Cartesian product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)
⊢ (𝑅 Or 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Or 𝐴)
Theorem	frinxp 5719	Intersection of well-founded relation with Cartesian product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)
⊢ (𝑅 Fr 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Fr 𝐴)
Theorem	seinxp 5720	Intersection of set-like relation with Cartesian product of its field. (Contributed by Mario Carneiro, 22-Jun-2015.)
⊢ (𝑅 Se 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Se 𝐴)
Theorem	weinxp 5721	Intersection of well-ordering with Cartesian product of its field. (Contributed by NM, 9-Mar-1997.) (Revised by Mario Carneiro, 10-Jul-2014.)
⊢ (𝑅 We 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) We 𝐴)
Theorem	posn 5722	Partial ordering of a singleton. (Contributed by NM, 27-Apr-2009.) (Revised by Mario Carneiro, 23-Apr-2015.)
⊢ (Rel 𝑅 → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))
Theorem	sosn 5723	Strict ordering on a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
⊢ (Rel 𝑅 → (𝑅 Or {𝐴} ↔ ¬ 𝐴𝑅𝐴))
Theorem	frsn 5724	Founded relation on a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 23-Apr-2015.)
⊢ (Rel 𝑅 → (𝑅 Fr {𝐴} ↔ ¬ 𝐴𝑅𝐴))
Theorem	wesn 5725	Well-ordering of a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
⊢ (Rel 𝑅 → (𝑅 We {𝐴} ↔ ¬ 𝐴𝑅𝐴))
Theorem	elopaelxp 5726*	Membership in an ordered-pair class abstraction implies membership in a Cartesian product. (Contributed by Alexander van der Vekens, 23-Jun-2018.) Avoid ax-sep 5261, ax-nul 5268, ax-pr 5389. (Revised by SN, 11-Dec-2024.)
⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → 𝐴 ∈ (V × V))
Theorem	elopaelxpOLD 5727*	Obsolete version of elopaelxp 5726 as of 11-Dec-2024. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → 𝐴 ∈ (V × V))
Theorem	bropaex12 5728*	Two classes related by an ordered-pair class abstraction are sets. (Contributed by AV, 21-Jan-2020.)
⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓}    ⇒   ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Theorem	opabssxp 5729*	An abstraction relation is a subset of a related Cartesian product. (Contributed by NM, 16-Jul-1995.)
⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵)
Theorem	brab2a 5730*	The law of concretion for a binary relation. Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 28-Apr-2015.)
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))    &   ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)}    ⇒   ⊢ (𝐴𝑅𝐵 ↔ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ 𝜓))
Theorem	optocl 5731*	Implicit substitution of class for ordered pair. (Contributed by NM, 5-Mar-1995.)
⊢ 𝐷 = (𝐵 × 𝐶)    &   ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝜑)    ⇒   ⊢ (𝐴 ∈ 𝐷 → 𝜓)
Theorem	2optocl 5732*	Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.)
⊢ 𝑅 = (𝐶 × 𝐷)    &   ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (⟨𝑧, 𝑤⟩ = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ (𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷)) → 𝜑)    ⇒   ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅) → 𝜒)
Theorem	3optocl 5733*	Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.)
⊢ 𝑅 = (𝐷 × 𝐹)    &   ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (⟨𝑧, 𝑤⟩ = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (⟨𝑣, 𝑢⟩ = 𝐶 → (𝜒 ↔ 𝜃))    &   ⊢ (((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐹) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐹) ∧ (𝑣 ∈ 𝐷 ∧ 𝑢 ∈ 𝐹)) → 𝜑)    ⇒   ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑅) → 𝜃)
Theorem	opbrop 5734*	Ordered pair membership in a relation. Special case. (Contributed by NM, 5-Aug-1995.)
⊢ (((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) ∧ (𝑣 = 𝐶 ∧ 𝑢 = 𝐷)) → (𝜑 ↔ 𝜓))    &   ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ 𝜑))}    ⇒   ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (⟨𝐴, 𝐵⟩𝑅⟨𝐶, 𝐷⟩ ↔ 𝜓))
Theorem	0xp 5735	The Cartesian product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by NM, 4-Jul-1994.)
⊢ (∅ × 𝐴) = ∅
Theorem	csbxp 5736	Distribute proper substitution through the Cartesian product of two classes. (Contributed by Alan Sare, 10-Nov-2012.) (Revised by NM, 23-Aug-2018.)
⊢ ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶)
Theorem	releq 5737	Equality theorem for the relation predicate. (Contributed by NM, 1-Aug-1994.)
⊢ (𝐴 = 𝐵 → (Rel 𝐴 ↔ Rel 𝐵))
Theorem	releqi 5738	Equality inference for the relation predicate. (Contributed by NM, 8-Dec-2006.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (Rel 𝐴 ↔ Rel 𝐵)
Theorem	releqd 5739	Equality deduction for the relation predicate. (Contributed by NM, 8-Mar-2014.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (Rel 𝐴 ↔ Rel 𝐵))
Theorem	nfrel 5740	Bound-variable hypothesis builder for a relation. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥Rel 𝐴
Theorem	sbcrel 5741	Distribute proper substitution through a relation predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Rel 𝑅 ↔ Rel ⦋𝐴 / 𝑥⦌𝑅))
Theorem	relss 5742	Subclass theorem for relation predicate. Theorem 2 of [Suppes] p. 58. (Contributed by NM, 15-Aug-1994.)
⊢ (𝐴 ⊆ 𝐵 → (Rel 𝐵 → Rel 𝐴))
Theorem	ssrel 5743*	A subclass relationship depends only on a relation's ordered pairs. Theorem 3.2(i) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Remove dependency on ax-sep 5261, ax-nul 5268, ax-pr 5389. (Revised by KP, 25-Oct-2021.) Remove dependency on ax-12 2171. (Revised by SN, 11-Dec-2024.)
⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
Theorem	ssrelOLD 5744*	Obsolete version of ssrel 5743 as of 11-Dec-2024. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Remove dependency on ax-sep 5261, ax-nul 5268, ax-pr 5389. (Revised by KP, 25-Oct-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
Theorem	eqrel 5745*	Extensionality principle for relations. Theorem 3.2(ii) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.)
⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
Theorem	ssrel2 5746*	A subclass relationship depends only on a relation's ordered pairs. This version of ssrel 5743 is restricted to the relation's domain. (Contributed by Thierry Arnoux, 25-Jan-2018.)
⊢ (𝑅 ⊆ (𝐴 × 𝐵) → (𝑅 ⊆ 𝑆 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆)))
Theorem	ssrel3 5747*	Subclass relation in another form when the subclass is a relation. (Contributed by Peter Mazsa, 16-Feb-2019.)
⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(𝑥𝐴𝑦 → 𝑥𝐵𝑦)))
Theorem	relssi 5748*	Inference from subclass principle for relations. (Contributed by NM, 31-Mar-1998.)
⊢ Rel 𝐴    &   ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)    ⇒   ⊢ 𝐴 ⊆ 𝐵
Theorem	relssdv 5749*	Deduction from subclass principle for relations. (Contributed by NM, 11-Sep-2004.)
⊢ (𝜑 → Rel 𝐴)    &   ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
Theorem	eqrelriv 5750*	Inference from extensionality principle for relations. (Contributed by FL, 15-Oct-2012.)
⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)    ⇒   ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → 𝐴 = 𝐵)
Theorem	eqrelriiv 5751*	Inference from extensionality principle for relations. (Contributed by NM, 17-Mar-1995.)
⊢ Rel 𝐴    &   ⊢ Rel 𝐵    &   ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)    ⇒   ⊢ 𝐴 = 𝐵
Theorem	eqbrriv 5752*	Inference from extensionality principle for relations. (Contributed by NM, 12-Dec-2006.)
⊢ Rel 𝐴    &   ⊢ Rel 𝐵    &   ⊢ (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)    ⇒   ⊢ 𝐴 = 𝐵
Theorem	eqrelrdv 5753*	Deduce equality of relations from equivalence of membership. (Contributed by Rodolfo Medina, 10-Oct-2010.)
⊢ Rel 𝐴    &   ⊢ Rel 𝐵    &   ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	eqbrrdv 5754*	Deduction from extensionality principle for relations. (Contributed by Mario Carneiro, 3-Jan-2017.)
⊢ (𝜑 → Rel 𝐴)    &   ⊢ (𝜑 → Rel 𝐵)    &   ⊢ (𝜑 → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	eqbrrdiv 5755*	Deduction from extensionality principle for relations. (Contributed by Rodolfo Medina, 10-Oct-2010.)
⊢ Rel 𝐴    &   ⊢ Rel 𝐵    &   ⊢ (𝜑 → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	eqrelrdv2 5756*	A version of eqrelrdv 5753. (Contributed by Rodolfo Medina, 10-Oct-2010.)
⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))    ⇒   ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)
Theorem	ssrelrel 5757*	A subclass relationship determined by ordered triples. Use relrelss 6230 to express the antecedent in terms of the relation predicate. (Contributed by NM, 17-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ (𝐴 ⊆ ((V × V) × V) → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵)))
Theorem	eqrelrel 5758*	Extensionality principle for ordered triples (used by 2-place operations df-oprab 7366), analogous to eqrel 5745. Use relrelss 6230 to express the antecedent in terms of the relation predicate. (Contributed by NM, 17-Dec-2008.)
⊢ ((𝐴 ∪ 𝐵) ⊆ ((V × V) × V) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦∀𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵)))
Theorem	elrel 5759*	A member of a relation is an ordered pair. (Contributed by NM, 17-Sep-2006.)
⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
Theorem	rel0 5760	The empty set is a relation. (Contributed by NM, 26-Apr-1998.)
⊢ Rel ∅
Theorem	nrelv 5761	The universal class is not a relation. (Contributed by Thierry Arnoux, 23-Jan-2022.)
⊢ ¬ Rel V
Theorem	relsng 5762	A singleton is a relation iff it is a singleton on an ordered pair. (Contributed by NM, 24-Sep-2013.) (Revised by BJ, 12-Feb-2022.)
⊢ (𝐴 ∈ 𝑉 → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
Theorem	relsnb 5763	An at-most-singleton is a relation iff it is empty (because it is a "singleton on a proper class") or it is a singleton of an ordered pair. (Contributed by BJ, 26-Feb-2023.)
⊢ (Rel {𝐴} ↔ (¬ 𝐴 ∈ V ∨ 𝐴 ∈ (V × V)))
Theorem	relsnopg 5764	A singleton of an ordered pair is a relation. (Contributed by NM, 17-May-1998.) (Revised by BJ, 12-Feb-2022.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Rel {⟨𝐴, 𝐵⟩})
Theorem	relsn 5765	A singleton is a relation iff it is an ordered pair. (Contributed by NM, 24-Sep-2013.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (Rel {𝐴} ↔ 𝐴 ∈ (V × V))
Theorem	relsnop 5766	A singleton of an ordered pair is a relation. (Contributed by NM, 17-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ Rel {⟨𝐴, 𝐵⟩}
Theorem	copsex2gb 5767*	Implicit substitution inference for ordered pairs. Compare copsex2ga 5768. (Contributed by NM, 12-Mar-2014.)
⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ (𝐴 ∈ (V × V) ∧ 𝜑))
Theorem	copsex2ga 5768*	Implicit substitution inference for ordered pairs. Compare copsex2g 5455. (Contributed by NM, 26-Feb-2014.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
Theorem	elopaba 5769*	Membership in an ordered-pair class abstraction. (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ (𝐴 ∈ (V × V) ∧ 𝜑))
Theorem	xpsspw 5770	A Cartesian product is included in the power of the power of the union of its arguments. (Contributed by NM, 13-Sep-2006.)
⊢ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵)
Theorem	unixpss 5771	The double class union of a Cartesian product is included in the union of its arguments. (Contributed by NM, 16-Sep-2006.)
⊢ ∪ ∪ (𝐴 × 𝐵) ⊆ (𝐴 ∪ 𝐵)
Theorem	relun 5772	The union of two relations is a relation. Compare Exercise 5 of [TakeutiZaring] p. 25. (Contributed by NM, 12-Aug-1994.)
⊢ (Rel (𝐴 ∪ 𝐵) ↔ (Rel 𝐴 ∧ Rel 𝐵))
Theorem	relin1 5773	The intersection with a relation is a relation. (Contributed by NM, 16-Aug-1994.)
⊢ (Rel 𝐴 → Rel (𝐴 ∩ 𝐵))
Theorem	relin2 5774	The intersection with a relation is a relation. (Contributed by NM, 17-Jan-2006.)
⊢ (Rel 𝐵 → Rel (𝐴 ∩ 𝐵))
Theorem	relinxp 5775	Intersection with a Cartesian product is a relation. (Contributed by Peter Mazsa, 4-Mar-2019.)
⊢ Rel (𝑅 ∩ (𝐴 × 𝐵))
Theorem	reldif 5776	A difference cutting down a relation is a relation. (Contributed by NM, 31-Mar-1998.)
⊢ (Rel 𝐴 → Rel (𝐴 ∖ 𝐵))
Theorem	reliun 5777	An indexed union is a relation iff each member of its indexed family is a relation. (Contributed by NM, 19-Dec-2008.)
⊢ (Rel ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 Rel 𝐵)
Theorem	reliin 5778	An indexed intersection is a relation if at least one of the member of the indexed family is a relation. (Contributed by NM, 8-Mar-2014.)
⊢ (∃𝑥 ∈ 𝐴 Rel 𝐵 → Rel ∩ 𝑥 ∈ 𝐴 𝐵)
Theorem	reluni 5779*	The union of a class is a relation iff any member is a relation. Exercise 6 of [TakeutiZaring] p. 25 and its converse. (Contributed by NM, 13-Aug-2004.)
⊢ (Rel ∪ 𝐴 ↔ ∀𝑥 ∈ 𝐴 Rel 𝑥)
Theorem	relint 5780*	The intersection of a class is a relation if at least one member is a relation. (Contributed by NM, 8-Mar-2014.)
⊢ (∃𝑥 ∈ 𝐴 Rel 𝑥 → Rel ∩ 𝐴)
Theorem	relopabiv 5781*	A class of ordered pairs is a relation. For a version without a disjoint variable condition, but a longer proof using ax-11 2154 and ax-12 2171, see relopabi 5783. (Contributed by BJ, 22-Jul-2023.)
⊢ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    ⇒   ⊢ Rel 𝐴
Theorem	relopabv 5782*	A class of ordered pairs is a relation. For a version without a disjoint variable condition, but using ax-11 2154 and ax-12 2171, see relopab 5785. (Contributed by SN, 8-Sep-2024.)
⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Theorem	relopabi 5783	A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.) Remove dependency on ax-sep 5261, ax-nul 5268, ax-pr 5389. (Revised by KP, 25-Oct-2021.)
⊢ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    ⇒   ⊢ Rel 𝐴
Theorem	relopabiALT 5784	Alternate proof of relopabi 5783 (shorter but uses more axioms). (Contributed by Mario Carneiro, 21-Dec-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    ⇒   ⊢ Rel 𝐴
Theorem	relopab 5785	A class of ordered pairs is a relation. (Contributed by NM, 8-Mar-1995.) Remove disjoint variable conditions. (Revised by Alan Sare, 9-Jul-2013.) (Proof shortened by Mario Carneiro, 21-Dec-2013.)
⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Theorem	mptrel 5786	The maps-to notation always describes a binary relation. (Contributed by Scott Fenton, 16-Apr-2012.)
⊢ Rel (𝑥 ∈ 𝐴 ↦ 𝐵)
Theorem	reli 5787	The identity relation is a relation. Part of Exercise 4.12(p) of [Mendelson] p. 235. (Contributed by NM, 26-Apr-1998.) (Revised by Mario Carneiro, 21-Dec-2013.)
⊢ Rel I
Theorem	rele 5788	The membership relation is a relation. (Contributed by NM, 26-Apr-1998.) (Revised by Mario Carneiro, 21-Dec-2013.)
⊢ Rel E
Theorem	opabid2 5789*	A relation expressed as an ordered pair abstraction. (Contributed by NM, 11-Dec-2006.)
⊢ (Rel 𝐴 → {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = 𝐴)
Theorem	inopab 5790*	Intersection of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)
⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)}
Theorem	difopab 5791*	Difference of two ordered-pair class abstractions. (Contributed by Stefan O'Rear, 17-Jan-2015.) (Proof shortened by SN, 19-Dec-2024.)
⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ ¬ 𝜓)}
Theorem	difopabOLD 5792*	Obsolete version of difopab 5791 as of 19-Dec-2024. (Contributed by Stefan O'Rear, 17-Jan-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ ¬ 𝜓)}
Theorem	inxp 5793	Intersection of two Cartesian products. Exercise 9 of [TakeutiZaring] p. 25. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ 𝐷))
Theorem	xpindi 5794	Distributive law for Cartesian product over intersection. Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)
⊢ (𝐴 × (𝐵 ∩ 𝐶)) = ((𝐴 × 𝐵) ∩ (𝐴 × 𝐶))
Theorem	xpindir 5795	Distributive law for Cartesian product over intersection. Similar to Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)
⊢ ((𝐴 ∩ 𝐵) × 𝐶) = ((𝐴 × 𝐶) ∩ (𝐵 × 𝐶))
Theorem	xpiindi 5796*	Distributive law for Cartesian product over indexed intersection. (Contributed by Mario Carneiro, 21-Mar-2015.)
⊢ (𝐴 ≠ ∅ → (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) = ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))
Theorem	xpriindi 5797*	Distributive law for Cartesian product over relativized indexed intersection. (Contributed by Mario Carneiro, 21-Mar-2015.)
⊢ (𝐶 × (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))
Theorem	eliunxp 5798*	Membership in a union of Cartesian products. Analogue of elxp 5661 for nonconstant 𝐵(𝑥). (Contributed by Mario Carneiro, 29-Dec-2014.)
⊢ (𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
Theorem	opeliunxp2 5799*	Membership in a union of Cartesian products. (Contributed by Mario Carneiro, 14-Feb-2015.)
⊢ (𝑥 = 𝐶 → 𝐵 = 𝐸)    ⇒   ⊢ (⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸))
Theorem	raliunxp 5800*	Write a double restricted quantification as one universal quantifier. In this version of ralxp 5802, 𝐵(𝑦) is not assumed to be constant. (Contributed by Mario Carneiro, 29-Dec-2014.)
⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜓)