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Theorem List for Intuitionistic Logic Explorer - 4401-4500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremopelxpi 4401 Ordered pair membership in a cross product (implication). (Contributed by NM, 28-May-1995.)
((𝐴𝐶𝐵𝐷) → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷))

Theoremopelxp1 4402 The first member of an ordered pair of classes in a cross product belongs to first cross product argument. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
(⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → 𝐴𝐶)

Theoremopelxp2 4403 The second member of an ordered pair of classes in a cross product belongs to second cross product argument. (Contributed by Mario Carneiro, 26-Apr-2015.)
(⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → 𝐵𝐷)

Theoremotelxp1 4404 The first member of an ordered triple of classes in a cross product belongs to first cross product argument. (Contributed by NM, 28-May-2008.)
(⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝑅 × 𝑆) × 𝑇) → 𝐴𝑅)

Theoremrabxp 4405* Membership in a class builder restricted to a cross product. (Contributed by NM, 20-Feb-2014.)
(𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))       {𝑥 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵𝜓)}

Theorembrrelex12 4406 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 𝑅𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Theorembrrelex 4407 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 𝑅𝐴𝑅𝐵) → 𝐴 ∈ V)

Theorembrrelex2 4408 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 𝑅𝐴𝑅𝐵) → 𝐵 ∈ V)

Theorembrrelexi 4409 The first argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by NM, 4-Jun-1998.)
Rel 𝑅       (𝐴𝑅𝐵𝐴 ∈ V)

Theorembrrelex2i 4410 The second argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)
Rel 𝑅       (𝐴𝑅𝐵𝐵 ∈ V)

Theoremnprrel 4411 No proper class is related to anything via any relation. (Contributed by Roy F. Longton, 30-Jul-2005.)
Rel 𝑅    &    ¬ 𝐴 ∈ V        ¬ 𝐴𝑅𝐵

Theoremfconstmpt 4412* 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}) = (𝑥𝐴𝐵)

Theoremvtoclr 4413* Variable to class conversion of transitive relation. (Contributed by NM, 9-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
Rel 𝑅    &   ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)       ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)

Theoremopelvvg 4414 Ordered pair membership in the universal class of ordered pairs. (Contributed by Mario Carneiro, 3-May-2015.)
((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ ∈ (V × V))

Theoremopelvv 4415 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)

Theoremopthprc 4416 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 = 𝐶𝐵 = 𝐷))

Theorembrel 4417 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 → (𝐴𝐶𝐵𝐷))

Theorembrab2a 4418* Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 9-Nov-2015.)
((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))    &   𝑅 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)}       (𝐴𝑅𝐵 ↔ ((𝐴𝐶𝐵𝐷) ∧ 𝜓))

Theoremelxp3 4419* Membership in a cross product. (Contributed by NM, 5-Mar-1995.)
(𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝑦(⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)))

Theoremopeliunxp 4420 Membership in a union of cross products. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 1-Jan-2017.)
(⟨𝑥, 𝐶⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝐶𝐵))

Theoremxpundi 4421 Distributive law for cross product over union. Theorem 103 of [Suppes] p. 52. (Contributed by NM, 12-Aug-2004.)
(𝐴 × (𝐵𝐶)) = ((𝐴 × 𝐵) ∪ (𝐴 × 𝐶))

Theoremxpundir 4422 Distributive law for cross product over union. Similar to Theorem 103 of [Suppes] p. 52. (Contributed by NM, 30-Sep-2002.)
((𝐴𝐵) × 𝐶) = ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶))

Theoremxpiundi 4423* Distributive law for cross product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.)
(𝐶 × 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶 × 𝐵)

Theoremxpiundir 4424* Distributive law for cross product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.)
( 𝑥𝐴 𝐵 × 𝐶) = 𝑥𝐴 (𝐵 × 𝐶)

Theoremiunxpconst 4425* Membership in a union of cross products when the second factor is constant. (Contributed by Mario Carneiro, 29-Dec-2014.)
𝑥𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵)

Theoremxpun 4426 The cross product of two unions. (Contributed by NM, 12-Aug-2004.)
((𝐴𝐵) × (𝐶𝐷)) = (((𝐴 × 𝐶) ∪ (𝐴 × 𝐷)) ∪ ((𝐵 × 𝐶) ∪ (𝐵 × 𝐷)))

Theoremelvv 4427* Membership in universal class of ordered pairs. (Contributed by NM, 4-Jul-1994.)
(𝐴 ∈ (V × V) ↔ ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)

Theoremelvvv 4428* Membership in universal class of ordered triples. (Contributed by NM, 17-Dec-2008.)
(𝐴 ∈ ((V × V) × V) ↔ ∃𝑥𝑦𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)

Theoremelvvuni 4429 An ordered pair contains its union. (Contributed by NM, 16-Sep-2006.)
(𝐴 ∈ (V × V) → 𝐴𝐴)

Theoremmosubopt 4430* "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-Aug-2007.)
(∀𝑦𝑧∃*𝑥𝜑 → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑))

Theoremmosubop 4431* "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-May-1995.)
∃*𝑥𝜑       ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)

Theorembrinxp2 4432 Intersection of binary relation with cross product. (Contributed by NM, 3-Mar-2007.) (Revised by Mario Carneiro, 26-Apr-2015.)
(𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵 ↔ (𝐴𝐶𝐵𝐷𝐴𝑅𝐵))

Theorembrinxp 4433 Intersection of binary relation with cross product. (Contributed by NM, 9-Mar-1997.)
((𝐴𝐶𝐵𝐷) → (𝐴𝑅𝐵𝐴(𝑅 ∩ (𝐶 × 𝐷))𝐵))

Theorempoinxp 4434 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)

Theoremsoinxp 4435 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)

Theoremseinxp 4436 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)

Theoremposng 4437 Partial ordering of a singleton. (Contributed by Jim Kingdon, 5-Dec-2018.)
((Rel 𝑅𝐴 ∈ V) → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))

Theoremsosng 4438 Strict linear ordering on a singleton. (Contributed by Jim Kingdon, 5-Dec-2018.)
((Rel 𝑅𝐴 ∈ V) → (𝑅 Or {𝐴} ↔ ¬ 𝐴𝑅𝐴))

Theoremopabssxp 4439* An abstraction relation is a subset of a related cross product. (Contributed by NM, 16-Jul-1995.)
{⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵)

Theorembrab2ga 4440* The law of concretion for a binary relation. See brab2a 4418 for alternate proof. TODO: should one of them be deleted? (Contributed by Mario Carneiro, 28-Apr-2015.) (Proof modification is discouraged.)
((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))    &   𝑅 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)}       (𝐴𝑅𝐵 ↔ ((𝐴𝐶𝐵𝐷) ∧ 𝜓))

Theoremoptocl 4441* Implicit substitution of class for ordered pair. (Contributed by NM, 5-Mar-1995.)
𝐷 = (𝐵 × 𝐶)    &   (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑𝜓))    &   ((𝑥𝐵𝑦𝐶) → 𝜑)       (𝐴𝐷𝜓)

Theorem2optocl 4442* Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.)
𝑅 = (𝐶 × 𝐷)    &   (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑𝜓))    &   (⟨𝑧, 𝑤⟩ = 𝐵 → (𝜓𝜒))    &   (((𝑥𝐶𝑦𝐷) ∧ (𝑧𝐶𝑤𝐷)) → 𝜑)       ((𝐴𝑅𝐵𝑅) → 𝜒)

Theorem3optocl 4443* Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.)
𝑅 = (𝐷 × 𝐹)    &   (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑𝜓))    &   (⟨𝑧, 𝑤⟩ = 𝐵 → (𝜓𝜒))    &   (⟨𝑣, 𝑢⟩ = 𝐶 → (𝜒𝜃))    &   (((𝑥𝐷𝑦𝐹) ∧ (𝑧𝐷𝑤𝐹) ∧ (𝑣𝐷𝑢𝐹)) → 𝜑)       ((𝐴𝑅𝐵𝑅𝐶𝑅) → 𝜃)

Theoremopbrop 4444* Ordered pair membership in a relation. Special case. (Contributed by NM, 5-Aug-1995.)
(((𝑧 = 𝐴𝑤 = 𝐵) ∧ (𝑣 = 𝐶𝑢 = 𝐷)) → (𝜑𝜓))    &   𝑅 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ 𝜑))}       (((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑆𝐷𝑆)) → (⟨𝐴, 𝐵𝑅𝐶, 𝐷⟩ ↔ 𝜓))

Theorem0xp 4445 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) = ∅

Theoremcsbxpg 4446 Distribute proper substitution through the cross product of two classes. (Contributed by Alan Sare, 10-Nov-2012.)
(𝐴𝐷𝐴 / 𝑥(𝐵 × 𝐶) = (𝐴 / 𝑥𝐵 × 𝐴 / 𝑥𝐶))

Theoremreleq 4447 Equality theorem for the relation predicate. (Contributed by NM, 1-Aug-1994.)
(𝐴 = 𝐵 → (Rel 𝐴 ↔ Rel 𝐵))

Theoremreleqi 4448 Equality inference for the relation predicate. (Contributed by NM, 8-Dec-2006.)
𝐴 = 𝐵       (Rel 𝐴 ↔ Rel 𝐵)

Theoremreleqd 4449 Equality deduction for the relation predicate. (Contributed by NM, 8-Mar-2014.)
(𝜑𝐴 = 𝐵)       (𝜑 → (Rel 𝐴 ↔ Rel 𝐵))

Theoremnfrel 4450 Bound-variable hypothesis builder for a relation. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐴       𝑥Rel 𝐴

Theoremsbcrel 4451 Distribute proper substitution through a relation predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
(𝐴𝑉 → ([𝐴 / 𝑥]Rel 𝑅 ↔ Rel 𝐴 / 𝑥𝑅))

Theoremrelss 4452 Subclass theorem for relation predicate. Theorem 2 of [Suppes] p. 58. (Contributed by NM, 15-Aug-1994.)
(𝐴𝐵 → (Rel 𝐵 → Rel 𝐴))

Theoremssrel 4453* 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 → (𝐴𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))

Theoremeqrel 4454* Extensionality principle for relations. Theorem 3.2(ii) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.)
((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))

Theoremssrel2 4455* A subclass relationship depends only on a relation's ordered pairs. This version of ssrel 4453 is restricted to the relation's domain. (Contributed by Thierry Arnoux, 25-Jan-2018.)
(𝑅 ⊆ (𝐴 × 𝐵) → (𝑅𝑆 ↔ ∀𝑥𝐴𝑦𝐵 (⟨𝑥, 𝑦⟩ ∈ 𝑅 → ⟨𝑥, 𝑦⟩ ∈ 𝑆)))

Theoremrelssi 4456* Inference from subclass principle for relations. (Contributed by NM, 31-Mar-1998.)
Rel 𝐴    &   (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)       𝐴𝐵

Theoremrelssdv 4457* Deduction from subclass principle for relations. (Contributed by NM, 11-Sep-2004.)
(𝜑 → Rel 𝐴)    &   (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))       (𝜑𝐴𝐵)

Theoremeqrelriv 4458* Inference from extensionality principle for relations. (Contributed by FL, 15-Oct-2012.)
(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)       ((Rel 𝐴 ∧ Rel 𝐵) → 𝐴 = 𝐵)

Theoremeqrelriiv 4459* Inference from extensionality principle for relations. (Contributed by NM, 17-Mar-1995.)
Rel 𝐴    &   Rel 𝐵    &   (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)       𝐴 = 𝐵

Theoremeqbrriv 4460* Inference from extensionality principle for relations. (Contributed by NM, 12-Dec-2006.)
Rel 𝐴    &   Rel 𝐵    &   (𝑥𝐴𝑦𝑥𝐵𝑦)       𝐴 = 𝐵

Theoremeqrelrdv 4461* 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))       (𝜑𝐴 = 𝐵)

Theoremeqbrrdv 4462* Deduction from extensionality principle for relations. (Contributed by Mario Carneiro, 3-Jan-2017.)
(𝜑 → Rel 𝐴)    &   (𝜑 → Rel 𝐵)    &   (𝜑 → (𝑥𝐴𝑦𝑥𝐵𝑦))       (𝜑𝐴 = 𝐵)

Theoremeqbrrdiv 4463* Deduction from extensionality principle for relations. (Contributed by Rodolfo Medina, 10-Oct-2010.)
Rel 𝐴    &   Rel 𝐵    &   (𝜑 → (𝑥𝐴𝑦𝑥𝐵𝑦))       (𝜑𝐴 = 𝐵)

Theoremeqrelrdv2 4464* A version of eqrelrdv 4461. (Contributed by Rodolfo Medina, 10-Oct-2010.)
(𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))       (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)

Theoremssrelrel 4465* A subclass relationship determined by ordered triples. Use relrelss 4869 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) → (𝐴𝐵 ↔ ∀𝑥𝑦𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵)))

Theoremeqrelrel 4466* Extensionality principle for ordered triples, analogous to eqrel 4454. Use relrelss 4869 to express the antecedent in terms of the relation predicate. (Contributed by NM, 17-Dec-2008.)
((𝐴𝐵) ⊆ ((V × V) × V) → (𝐴 = 𝐵 ↔ ∀𝑥𝑦𝑧(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐴 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ 𝐵)))

Theoremelrel 4467* A member of a relation is an ordered pair. (Contributed by NM, 17-Sep-2006.)
((Rel 𝑅𝐴𝑅) → ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)

Theoremrelsn 4468 A singleton is a relation iff it is an ordered pair. (Contributed by NM, 24-Sep-2013.)
𝐴 ∈ V       (Rel {𝐴} ↔ 𝐴 ∈ (V × V))

Theoremrelsnop 4469 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⟩}

Theoremxpss12 4470 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.)
((𝐴𝐵𝐶𝐷) → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐷))

Theoremxpss 4471 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)

Theoremrelxp 4472 A cross product is a relation. Theorem 3.13(i) of [Monk1] p. 37. (Contributed by NM, 2-Aug-1994.)
Rel (𝐴 × 𝐵)

Theoremxpss1 4473 Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)
(𝐴𝐵 → (𝐴 × 𝐶) ⊆ (𝐵 × 𝐶))

Theoremxpss2 4474 Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)
(𝐴𝐵 → (𝐶 × 𝐴) ⊆ (𝐶 × 𝐵))

Theoremxpsspw 4475 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 (𝐴𝐵)

Theoremunixpss 4476 The double class union of a cross product is included in the union of its arguments. (Contributed by NM, 16-Sep-2006.)
(𝐴 × 𝐵) ⊆ (𝐴𝐵)

Theoremxpexg 4477 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. (Contributed by NM, 14-Aug-1994.)
((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)

Theoremxpex 4478 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

Theoremrelun 4479 The union of two relations is a relation. Compare Exercise 5 of [TakeutiZaring] p. 25. (Contributed by NM, 12-Aug-1994.)
(Rel (𝐴𝐵) ↔ (Rel 𝐴 ∧ Rel 𝐵))

Theoremrelin1 4480 The intersection with a relation is a relation. (Contributed by NM, 16-Aug-1994.)
(Rel 𝐴 → Rel (𝐴𝐵))

Theoremrelin2 4481 The intersection with a relation is a relation. (Contributed by NM, 17-Jan-2006.)
(Rel 𝐵 → Rel (𝐴𝐵))

Theoremreldif 4482 A difference cutting down a relation is a relation. (Contributed by NM, 31-Mar-1998.)
(Rel 𝐴 → Rel (𝐴𝐵))

Theoremreliun 4483 An indexed union is a relation iff each member of its indexed family is a relation. (Contributed by NM, 19-Dec-2008.)
(Rel 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 Rel 𝐵)

Theoremreliin 4484 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.)
(∃𝑥𝐴 Rel 𝐵 → Rel 𝑥𝐴 𝐵)

Theoremreluni 4485* 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 ↔ ∀𝑥𝐴 Rel 𝑥)

Theoremrelint 4486* The intersection of a class is a relation if at least one member is a relation. (Contributed by NM, 8-Mar-2014.)
(∃𝑥𝐴 Rel 𝑥 → Rel 𝐴)

Theoremrel0 4487 The empty set is a relation. (Contributed by NM, 26-Apr-1998.)
Rel ∅

Theoremrelopabi 4488 A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.)
𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}       Rel 𝐴

Theoremrelopab 4489 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}

Theoremreli 4490 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

Theoremrele 4491 The membership relation is a relation. (Contributed by NM, 26-Apr-1998.) (Revised by Mario Carneiro, 21-Dec-2013.)
Rel E

Theoremopabid2 4492* A relation expressed as an ordered pair abstraction. (Contributed by NM, 11-Dec-2006.)
(Rel 𝐴 → {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝐴} = 𝐴)

Theoreminopab 4493* Intersection of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑𝜓)}

Theoremdifopab 4494* 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)}

Theoreminxp 4495 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)) = ((𝐴𝐶) × (𝐵𝐷))

Theoremxpindi 4496 Distributive law for cross product over intersection. Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)
(𝐴 × (𝐵𝐶)) = ((𝐴 × 𝐵) ∩ (𝐴 × 𝐶))

Theoremxpindir 4497 Distributive law for cross product over intersection. Similar to Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)
((𝐴𝐵) × 𝐶) = ((𝐴 × 𝐶) ∩ (𝐵 × 𝐶))

Theoremxpiindim 4498* Distributive law for cross product over indexed intersection. (Contributed by Jim Kingdon, 7-Dec-2018.)
(∃𝑦 𝑦𝐴 → (𝐶 × 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶 × 𝐵))

Theoremxpriindim 4499* Distributive law for cross product over relativized indexed intersection. (Contributed by Jim Kingdon, 7-Dec-2018.)
(∃𝑦 𝑦𝐴 → (𝐶 × (𝐷 𝑥𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ 𝑥𝐴 (𝐶 × 𝐵)))

Theoremeliunxp 4500* Membership in a union of cross products. Analogue of elxp 4387 for nonconstant 𝐵(𝑥). (Contributed by Mario Carneiro, 29-Dec-2014.)
(𝐶 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))

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