P. 48 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4701-4800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	brrelex12 4701	A true binary relation on a relation implies the arguments are sets. (This is a property of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)
⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Theorem	brrelex1 4702	A true binary relation on a relation implies the first argument is a set. (This is a property of our ordered pair definition.) (Contributed by NM, 18-May-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V)
Theorem	brrelex 4703	A true binary relation on a relation implies the first argument is a set. (This is a property of our ordered pair definition.) (Contributed by NM, 18-May-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V)
Theorem	brrelex2 4704	A true binary relation on a relation implies the second argument is a set. (This is a property of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)
⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ V)
Theorem	brrelex12i 4705	Two classes that are related by a binary relation are sets. (An artifact of our ordered pair definition.) (Contributed by BJ, 3-Oct-2022.)
⊢ Rel 𝑅    ⇒   ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Theorem	brrelex1i 4706	The first argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by NM, 4-Jun-1998.)
⊢ Rel 𝑅    ⇒   ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ V)
Theorem	brrelex2i 4707	The second argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)
⊢ Rel 𝑅    ⇒   ⊢ (𝐴𝑅𝐵 → 𝐵 ∈ V)
Theorem	nprrel 4708	No proper class is related to anything via any relation. (Contributed by Roy F. Longton, 30-Jul-2005.)
⊢ Rel 𝑅    &   ⊢ ¬ 𝐴 ∈ V    ⇒   ⊢ ¬ 𝐴𝑅𝐵
Theorem	0nelrel 4709	A binary relation does not contain the empty set. (Contributed by AV, 15-Nov-2021.)
⊢ (Rel 𝑅 → ∅ ∉ 𝑅)
Theorem	fconstmpt 4710*	Representation of a constant function using the mapping operation. (Note that 𝑥 cannot appear free in 𝐵.) (Contributed by NM, 12-Oct-1999.) (Revised by Mario Carneiro, 16-Nov-2013.)
⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵)
Theorem	vtoclr 4711*	Variable to class conversion of transitive relation. (Contributed by NM, 9-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ Rel 𝑅    &   ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)    ⇒   ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)
Theorem	opelvvg 4712	Ordered pair membership in the universal class of ordered pairs. (Contributed by Mario Carneiro, 3-May-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ ∈ (V × V))
Theorem	opelvv 4713	Ordered pair membership in the universal class of ordered pairs. (Contributed by NM, 22-Aug-2013.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ⟨𝐴, 𝐵⟩ ∈ (V × V)
Theorem	opthprc 4714	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 4715	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	brab2a 4716*	Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 9-Nov-2015.)
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))    &   ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)}    ⇒   ⊢ (𝐴𝑅𝐵 ↔ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ 𝜓))
Theorem	elxp3 4717*	Membership in a cross product. (Contributed by NM, 5-Mar-1995.)
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)))
Theorem	opeliunxp 4718	Membership in a union of cross 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 4719	Distributive law for cross product over union. Theorem 103 of [Suppes] p. 52. (Contributed by NM, 12-Aug-2004.)
⊢ (𝐴 × (𝐵 ∪ 𝐶)) = ((𝐴 × 𝐵) ∪ (𝐴 × 𝐶))
Theorem	xpundir 4720	Distributive law for cross 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 4721*	Distributive law for cross product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.)
⊢ (𝐶 × ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)
Theorem	xpiundir 4722*	Distributive law for cross product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.)
⊢ (∪ 𝑥 ∈ 𝐴 𝐵 × 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 × 𝐶)
Theorem	iunxpconst 4723*	Membership in a union of cross products when the second factor is constant. (Contributed by Mario Carneiro, 29-Dec-2014.)
⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵)
Theorem	xpun 4724	The cross product of two unions. (Contributed by NM, 12-Aug-2004.)
⊢ ((𝐴 ∪ 𝐵) × (𝐶 ∪ 𝐷)) = (((𝐴 × 𝐶) ∪ (𝐴 × 𝐷)) ∪ ((𝐵 × 𝐶) ∪ (𝐵 × 𝐷)))
Theorem	elvv 4725*	Membership in universal class of ordered pairs. (Contributed by NM, 4-Jul-1994.)
⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
Theorem	elvvv 4726*	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 4727	An ordered pair contains its union. (Contributed by NM, 16-Sep-2006.)
⊢ (𝐴 ∈ (V × V) → ∪ 𝐴 ∈ 𝐴)
Theorem	mosubopt 4728*	"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 4729*	"At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-May-1995.)
⊢ ∃*𝑥𝜑    ⇒   ⊢ ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
Theorem	brinxp2 4730	Intersection of binary relation with Cartesian product. (Contributed by NM, 3-Mar-2007.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ (𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵))
Theorem	brinxp 4731	Intersection of binary relation with Cartesian product. (Contributed by NM, 9-Mar-1997.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝑅𝐵 ↔ 𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵))
Theorem	poinxp 4732	Intersection of partial order with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)
⊢ (𝑅 Po 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴)
Theorem	soinxp 4733	Intersection of linear order with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)
⊢ (𝑅 Or 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Or 𝐴)
Theorem	seinxp 4734	Intersection of set-like relation with cross product of its field. (Contributed by Mario Carneiro, 22-Jun-2015.)
⊢ (𝑅 Se 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Se 𝐴)
Theorem	posng 4735	Partial ordering of a singleton. (Contributed by Jim Kingdon, 5-Dec-2018.)
⊢ ((Rel 𝑅 ∧ 𝐴 ∈ V) → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))
Theorem	sosng 4736	Strict linear ordering on a singleton. (Contributed by Jim Kingdon, 5-Dec-2018.)
⊢ ((Rel 𝑅 ∧ 𝐴 ∈ V) → (𝑅 Or {𝐴} ↔ ¬ 𝐴𝑅𝐴))
Theorem	opabssxp 4737*	An abstraction relation is a subset of a related cross product. (Contributed by NM, 16-Jul-1995.)
⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵)
Theorem	brab2ga 4738*	The law of concretion for a binary relation. See brab2a 4716 for alternate proof. TODO: should one of them be deleted? (Contributed by Mario Carneiro, 28-Apr-2015.) (Proof modification is discouraged.)
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))    &   ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)}    ⇒   ⊢ (𝐴𝑅𝐵 ↔ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ 𝜓))
Theorem	optocl 4739*	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 4740*	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 4741*	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 4742*	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 4743	The cross 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	csbxpg 4744	Distribute proper substitution through the cross product of two classes. (Contributed by Alan Sare, 10-Nov-2012.)
⊢ (𝐴 ∈ 𝐷 → ⦋𝐴 / 𝑥⦌(𝐵 × 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 × ⦋𝐴 / 𝑥⦌𝐶))
Theorem	releq 4745	Equality theorem for the relation predicate. (Contributed by NM, 1-Aug-1994.)
⊢ (𝐴 = 𝐵 → (Rel 𝐴 ↔ Rel 𝐵))
Theorem	releqi 4746	Equality inference for the relation predicate. (Contributed by NM, 8-Dec-2006.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (Rel 𝐴 ↔ Rel 𝐵)
Theorem	releqd 4747	Equality deduction for the relation predicate. (Contributed by NM, 8-Mar-2014.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (Rel 𝐴 ↔ Rel 𝐵))
Theorem	nfrel 4748	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 4749	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 4750	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 4751*	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.)
⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
Theorem	eqrel 4752*	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 4753*	A subclass relationship depends only on a relation's ordered pairs. This version of ssrel 4751 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	relssi 4754*	Inference from subclass principle for relations. (Contributed by NM, 31-Mar-1998.)
⊢ Rel 𝐴    &   ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)    ⇒   ⊢ 𝐴 ⊆ 𝐵
Theorem	relssdv 4755*	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 4756*	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 4757*	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 4758*	Inference from extensionality principle for relations. (Contributed by NM, 12-Dec-2006.)
⊢ Rel 𝐴    &   ⊢ Rel 𝐵    &   ⊢ (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)    ⇒   ⊢ 𝐴 = 𝐵
Theorem	eqrelrdv 4759*	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 4760*	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 4761*	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 4762*	A version of eqrelrdv 4759. (Contributed by Rodolfo Medina, 10-Oct-2010.)
⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))    ⇒   ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)
Theorem	ssrelrel 4763*	A subclass relationship determined by ordered triples. Use relrelss 5196 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 4764*	Extensionality principle for ordered triples, analogous to eqrel 4752. Use relrelss 5196 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 4765*	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	relsng 4766	A singleton is a relation iff it is an ordered pair. (Contributed by NM, 24-Sep-2013.) (Revised by BJ, 12-Feb-2022.)
⊢ (𝐴 ∈ 𝑉 → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
Theorem	relsnopg 4767	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 4768	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 4769	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	xpss12 4770	Subset theorem for cross product. Generalization of Theorem 101 of [Suppes] p. 52. (Contributed by NM, 26-Aug-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐷))
Theorem	xpss 4771	A cross product is included in the ordered pair universe. Exercise 3 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)
⊢ (𝐴 × 𝐵) ⊆ (V × V)
Theorem	relxp 4772	A cross product is a relation. Theorem 3.13(i) of [Monk1] p. 37. (Contributed by NM, 2-Aug-1994.)
⊢ Rel (𝐴 × 𝐵)
Theorem	xpss1 4773	Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)
⊢ (𝐴 ⊆ 𝐵 → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐶))
Theorem	xpss2 4774	Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)
⊢ (𝐴 ⊆ 𝐵 → (𝐶 × 𝐴) ⊆ (𝐶 × 𝐵))
Theorem	xpsspw 4775	A cross 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 4776	The double class union of a cross product is included in the union of its arguments. (Contributed by NM, 16-Sep-2006.)
⊢ ∪ ∪ (𝐴 × 𝐵) ⊆ (𝐴 ∪ 𝐵)
Theorem	xpexg 4777	The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. (Contributed by NM, 14-Aug-1994.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ∈ V)
Theorem	xpex 4778	The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. (Contributed by NM, 14-Aug-1994.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 × 𝐵) ∈ V
Theorem	sqxpexg 4779	The Cartesian square of a set is a set. (Contributed by AV, 13-Jan-2020.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V)
Theorem	relun 4780	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 4781	The intersection with a relation is a relation. (Contributed by NM, 16-Aug-1994.)
⊢ (Rel 𝐴 → Rel (𝐴 ∩ 𝐵))
Theorem	relin2 4782	The intersection with a relation is a relation. (Contributed by NM, 17-Jan-2006.)
⊢ (Rel 𝐵 → Rel (𝐴 ∩ 𝐵))
Theorem	reldif 4783	A difference cutting down a relation is a relation. (Contributed by NM, 31-Mar-1998.)
⊢ (Rel 𝐴 → Rel (𝐴 ∖ 𝐵))
Theorem	reliun 4784	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 4785	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 4786*	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 4787*	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	rel0 4788	The empty set is a relation. (Contributed by NM, 26-Apr-1998.)
⊢ Rel ∅
Theorem	relopabiv 4789*	A class of ordered pairs is a relation. For a version without a disjoint variable condition, see relopabi 4791. (Contributed by BJ, 22-Jul-2023.)
⊢ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    ⇒   ⊢ Rel 𝐴
Theorem	relopabv 4790*	A class of ordered pairs is a relation. For a version without a disjoint variable condition, see relopab 4792. (Contributed by SN, 8-Sep-2024.)
⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Theorem	relopabi 4791	A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.)
⊢ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    ⇒   ⊢ Rel 𝐴
Theorem	relopab 4792	A class of ordered pairs is a relation. (Contributed by NM, 8-Mar-1995.) (Unnecessary distinct variable restrictions were removed by Alan Sare, 9-Jul-2013.) (Proof shortened by Mario Carneiro, 21-Dec-2013.)
⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Theorem	brabv 4793	If two classes are in a relationship given by an ordered-pair class abstraction, the classes are sets. (Contributed by Alexander van der Vekens, 5-Nov-2017.)
⊢ (𝑋{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V))
Theorem	mptrel 4794	The maps-to notation always describes a relationship. (Contributed by Scott Fenton, 16-Apr-2012.)
⊢ Rel (𝑥 ∈ 𝐴 ↦ 𝐵)
Theorem	reli 4795	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 4796	The membership relation is a relation. (Contributed by NM, 26-Apr-1998.) (Revised by Mario Carneiro, 21-Dec-2013.)
⊢ Rel E
Theorem	opabid2 4797*	A relation expressed as an ordered pair abstraction. (Contributed by NM, 11-Dec-2006.)
⊢ (Rel 𝐴 → {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = 𝐴)
Theorem	inopab 4798*	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 4799*	The difference of two ordered-pair abstractions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∖ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ ¬ 𝜓)}
Theorem	inxp 4800	The intersection of two cross 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))