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Theorem List for Metamath Proof Explorer - 4401-4500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremiunrab 4401* The indexed union of a restricted class abstraction. (Contributed by NM, 3-Jan-2004.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
𝑥𝐴 {𝑦𝐵𝜑} = {𝑦𝐵 ∣ ∃𝑥𝐴 𝜑}
 
Theoremiunxdif2 4402* Indexed union with a class difference as its index. (Contributed by NM, 10-Dec-2004.)
(𝑥 = 𝑦𝐶 = 𝐷)       (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝐶𝐷 𝑦 ∈ (𝐴𝐵)𝐷 = 𝑥𝐴 𝐶)
 
Theoremssiinf 4403 Subset theorem for an indexed intersection. (Contributed by FL, 15-Oct-2012.) (Proof shortened by Mario Carneiro, 14-Oct-2016.)
𝑥𝐶       (𝐶 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝐶𝐵)
 
Theoremssiin 4404* Subset theorem for an indexed intersection. (Contributed by NM, 15-Oct-2003.)
(𝐶 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝐶𝐵)
 
Theoremiinss 4405* Subset implication for an indexed intersection. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(∃𝑥𝐴 𝐵𝐶 𝑥𝐴 𝐵𝐶)
 
Theoremiinss2 4406 An indexed intersection is included in any of its members. (Contributed by FL, 15-Oct-2012.)
(𝑥𝐴 𝑥𝐴 𝐵𝐵)
 
Theoremuniiun 4407* Class union in terms of indexed union. Definition in [Stoll] p. 43. (Contributed by NM, 28-Jun-1998.)
𝐴 = 𝑥𝐴 𝑥
 
Theoremintiin 4408* Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.)
𝐴 = 𝑥𝐴 𝑥
 
Theoremiunid 4409* An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.)
𝑥𝐴 {𝑥} = 𝐴
 
Theoremiun0 4410 An indexed union of the empty set is empty. (Contributed by NM, 26-Mar-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
𝑥𝐴 ∅ = ∅
 
Theorem0iun 4411 An empty indexed union is empty. (Contributed by NM, 4-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
𝑥 ∈ ∅ 𝐴 = ∅
 
Theorem0iin 4412 An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.)
𝑥 ∈ ∅ 𝐴 = V
 
Theoremviin 4413* Indexed intersection with a universal index class. When 𝐴 doesn't depend on 𝑥, this evaluates to 𝐴 by 19.3 2018 and abid2 2636. When 𝐴 = 𝑥, this evaluates to by intiin 4408 and intv 4666. (Contributed by NM, 11-Sep-2008.)
𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 𝑦𝐴}
 
Theoremiunn0 4414* There is a nonempty class in an indexed collection 𝐵(𝑥) iff the indexed union of them is nonempty. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(∃𝑥𝐴 𝐵 ≠ ∅ ↔ 𝑥𝐴 𝐵 ≠ ∅)
 
Theoremiinab 4415* Indexed intersection of a class builder. (Contributed by NM, 6-Dec-2011.)
𝑥𝐴 {𝑦𝜑} = {𝑦 ∣ ∀𝑥𝐴 𝜑}
 
Theoremiinrab 4416* Indexed intersection of a restricted class builder. (Contributed by NM, 6-Dec-2011.)
(𝐴 ≠ ∅ → 𝑥𝐴 {𝑦𝐵𝜑} = {𝑦𝐵 ∣ ∀𝑥𝐴 𝜑})
 
Theoremiinrab2 4417* Indexed intersection of a restricted class builder. (Contributed by NM, 6-Dec-2011.)
( 𝑥𝐴 {𝑦𝐵𝜑} ∩ 𝐵) = {𝑦𝐵 ∣ ∀𝑥𝐴 𝜑}
 
Theoremiunin2 4418* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4407 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.)
𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶)
 
Theoremiunin1 4419* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4407 to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.)
𝑥𝐴 (𝐶𝐵) = ( 𝑥𝐴 𝐶𝐵)
 
Theoremiinun2 4420* Indexed intersection of union. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4408 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)
𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶)
 
Theoremiundif2 4421* Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 4408 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)
𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶)
 
Theorem2iunin 4422* Rearrange indexed unions over intersection. (Contributed by NM, 18-Dec-2008.)
𝑥𝐴 𝑦𝐵 (𝐶𝐷) = ( 𝑥𝐴 𝐶 𝑦𝐵 𝐷)
 
Theoremiindif2 4423* Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 4407 to recover Enderton's theorem. (Contributed by NM, 5-Oct-2006.)
(𝐴 ≠ ∅ → 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶))
 
Theoremiinin2 4424* Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4408 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.)
(𝐴 ≠ ∅ → 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶))
 
Theoremiinin1 4425* Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4408 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.)
(𝐴 ≠ ∅ → 𝑥𝐴 (𝐶𝐵) = ( 𝑥𝐴 𝐶𝐵))
 
Theoremiinvdif 4426* The indexed intersection of a complement. (Contributed by Gérard Lang, 5-Aug-2018.)
𝑥𝐴 (V ∖ 𝐵) = (V ∖ 𝑥𝐴 𝐵)
 
Theoremelriin 4427* Elementhood in a relative intersection. (Contributed by Mario Carneiro, 30-Dec-2016.)
(𝐵 ∈ (𝐴 𝑥𝑋 𝑆) ↔ (𝐵𝐴 ∧ ∀𝑥𝑋 𝐵𝑆))
 
Theoremriin0 4428* Relative intersection of an empty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝑋 = ∅ → (𝐴 𝑥𝑋 𝑆) = 𝐴)
 
Theoremriinn0 4429* Relative intersection of a nonempty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)
((∀𝑥𝑋 𝑆𝐴𝑋 ≠ ∅) → (𝐴 𝑥𝑋 𝑆) = 𝑥𝑋 𝑆)
 
Theoremriinrab 4430* Relative intersection of a relative abstraction. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝐴 𝑥𝑋 {𝑦𝐴𝜑}) = {𝑦𝐴 ∣ ∀𝑥𝑋 𝜑}
 
Theoremsymdif0 4431 Symmetric difference with the empty class. (Contributed by Scott Fenton, 24-Apr-2012.)
(𝐴 △ ∅) = 𝐴
 
Theoremsymdifv 4432 Symmetric difference with the universal class. (Contributed by Scott Fenton, 24-Apr-2012.)
(𝐴 △ V) = (V ∖ 𝐴)
 
Theoremsymdifid 4433 Symmetric difference with self yields the empty class. (Contributed by Scott Fenton, 25-Apr-2012.)
(𝐴𝐴) = ∅
 
Theoremiinxsng 4434* A singleton index picks out an instance of an indexed intersection's argument. (Contributed by NM, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
(𝑥 = 𝐴𝐵 = 𝐶)       (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)
 
Theoremiinxprg 4435* Indexed intersection with an unordered pair index. (Contributed by NM, 25-Jan-2012.)
(𝑥 = 𝐴𝐶 = 𝐷)    &   (𝑥 = 𝐵𝐶 = 𝐸)       ((𝐴𝑉𝐵𝑊) → 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷𝐸))
 
Theoremiunxsng 4436* A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.)
(𝑥 = 𝐴𝐵 = 𝐶)       (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)
 
Theoremiunxsn 4437* A singleton index picks out an instance of an indexed union's argument. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 25-Jun-2016.)
𝐴 ∈ V    &   (𝑥 = 𝐴𝐵 = 𝐶)        𝑥 ∈ {𝐴}𝐵 = 𝐶
 
Theoremiunun 4438 Separate a union in an indexed union. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
𝑥𝐴 (𝐵𝐶) = ( 𝑥𝐴 𝐵 𝑥𝐴 𝐶)
 
Theoremiunxun 4439 Separate a union in the index of an indexed union. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
𝑥 ∈ (𝐴𝐵)𝐶 = ( 𝑥𝐴 𝐶 𝑥𝐵 𝐶)
 
Theoremiunxdif3 4440* An indexed union where some terms are the empty set. See iunxdif2 4402. (Contributed by Thierry Arnoux, 4-May-2020.)
𝑥𝐸       (∀𝑥𝐸 𝐵 = ∅ → 𝑥 ∈ (𝐴𝐸)𝐵 = 𝑥𝐴 𝐵)
 
Theoremiunxprg 4441* A pair index picks out two instances of an indexed union's argument. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
(𝑥 = 𝐴𝐶 = 𝐷)    &   (𝑥 = 𝐵𝐶 = 𝐸)       ((𝐴𝑉𝐵𝑊) → 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷𝐸))
 
Theoremiunxiun 4442* Separate an indexed union in the index of an indexed union. (Contributed by Mario Carneiro, 5-Dec-2016.)
𝑥 𝑦𝐴 𝐵𝐶 = 𝑦𝐴 𝑥𝐵 𝐶
 
Theoremiinuni 4443* A relationship involving union and indexed intersection. Exercise 23 of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
(𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥)
 
Theoremiununi 4444* A relationship involving union and indexed union. Exercise 25 of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
((𝐵 = ∅ → 𝐴 = ∅) ↔ (𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥))
 
Theoremsspwuni 4445 Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
(𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
 
Theorempwssb 4446* Two ways to express a collection of subclasses. (Contributed by NM, 19-Jul-2006.)
(𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
 
Theoremelpwuni 4447 Relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
(𝐵𝐴 → (𝐴 ⊆ 𝒫 𝐵 𝐴 = 𝐵))
 
Theoremiinpw 4448* The power class of an intersection in terms of indexed intersection. Exercise 24(a) of [Enderton] p. 33. (Contributed by NM, 29-Nov-2003.)
𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥
 
Theoremiunpwss 4449* Inclusion of an indexed union of a power class in the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.)
𝑥𝐴 𝒫 𝑥 ⊆ 𝒫 𝐴
 
Theoremrintn0 4450 Relative intersection of a nonempty set. (Contributed by Stefan O'Rear, 3-Apr-2015.) (Revised by Mario Carneiro, 5-Jun-2015.)
((𝑋 ⊆ 𝒫 𝐴𝑋 ≠ ∅) → (𝐴 𝑋) = 𝑋)
 
2.1.21  Disjointness
 
Syntaxwdisj 4451 Extend wff notation to include the statement that a family of classes 𝐵(𝑥), for 𝑥𝐴, is a disjoint family.
wff Disj 𝑥𝐴 𝐵
 
Definitiondf-disj 4452* A collection of classes 𝐵(𝑥) is disjoint when for each element 𝑦, it is in 𝐵(𝑥) for at most one 𝑥. (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by NM, 16-Jun-2017.)
(Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥𝐴 𝑦𝐵)
 
Theoremdfdisj2 4453* Alternate definition for disjoint classes. (Contributed by NM, 17-Jun-2017.)
(Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝐵))
 
Theoremdisjss2 4454 If each element of a collection is contained in a disjoint collection, the original collection is also disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
(∀𝑥𝐴 𝐵𝐶 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐴 𝐵))
 
Theoremdisjeq2 4455 Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
(∀𝑥𝐴 𝐵 = 𝐶 → (Disj 𝑥𝐴 𝐵Disj 𝑥𝐴 𝐶))
 
Theoremdisjeq2dv 4456* Equality deduction for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
((𝜑𝑥𝐴) → 𝐵 = 𝐶)       (𝜑 → (Disj 𝑥𝐴 𝐵Disj 𝑥𝐴 𝐶))
 
Theoremdisjss1 4457* A subset of a disjoint collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
(𝐴𝐵 → (Disj 𝑥𝐵 𝐶Disj 𝑥𝐴 𝐶))
 
Theoremdisjeq1 4458* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
(𝐴 = 𝐵 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
 
Theoremdisjeq1d 4459* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
(𝜑𝐴 = 𝐵)       (𝜑 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
 
Theoremdisjeq12d 4460* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐷))
 
Theoremcbvdisj 4461* Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)       (Disj 𝑥𝐴 𝐵Disj 𝑦𝐴 𝐶)
 
Theoremcbvdisjv 4462* Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 11-Dec-2016.)
(𝑥 = 𝑦𝐵 = 𝐶)       (Disj 𝑥𝐴 𝐵Disj 𝑦𝐴 𝐶)
 
Theoremnfdisj 4463 Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
𝑦𝐴    &   𝑦𝐵       𝑦Disj 𝑥𝐴 𝐵
 
Theoremnfdisj1 4464 Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
𝑥Disj 𝑥𝐴 𝐵
 
Theoremdisjor 4465* Two ways to say that a collection 𝐵(𝑖) for 𝑖𝐴 is disjoint. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 14-Nov-2016.)
(𝑖 = 𝑗𝐵 = 𝐶)       (Disj 𝑖𝐴 𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝐵𝐶) = ∅))
 
Theoremdisjors 4466* Two ways to say that a collection 𝐵(𝑖) for 𝑖𝐴 is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
(Disj 𝑥𝐴 𝐵 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗 ∨ (𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵) = ∅))
 
Theoremdisji2 4467* Property of a disjoint collection: if 𝐵(𝑋) = 𝐶 and 𝐵(𝑌) = 𝐷, and 𝑋𝑌, then 𝐶 and 𝐷 are disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
(𝑥 = 𝑋𝐵 = 𝐶)    &   (𝑥 = 𝑌𝐵 = 𝐷)       ((Disj 𝑥𝐴 𝐵 ∧ (𝑋𝐴𝑌𝐴) ∧ 𝑋𝑌) → (𝐶𝐷) = ∅)
 
Theoremdisji 4468* Property of a disjoint collection: if 𝐵(𝑋) = 𝐶 and 𝐵(𝑌) = 𝐷 have a common element 𝑍, then 𝑋 = 𝑌. (Contributed by Mario Carneiro, 14-Nov-2016.)
(𝑥 = 𝑋𝐵 = 𝐶)    &   (𝑥 = 𝑌𝐵 = 𝐷)       ((Disj 𝑥𝐴 𝐵 ∧ (𝑋𝐴𝑌𝐴) ∧ (𝑍𝐶𝑍𝐷)) → 𝑋 = 𝑌)
 
Theoreminvdisj 4469* If there is a function 𝐶(𝑦) such that 𝐶(𝑦) = 𝑥 for all 𝑦𝐵(𝑥), then the sets 𝐵(𝑥) for distinct 𝑥𝐴 are disjoint. (Contributed by Mario Carneiro, 10-Dec-2016.)
(∀𝑥𝐴𝑦𝐵 𝐶 = 𝑥Disj 𝑥𝐴 𝐵)
 
Theoreminvdisjrab 4470* The restricted class abstractions {𝑥𝐵𝐶 = 𝑦} for distinct 𝑦𝐴 are disjoint. (Contributed by AV, 6-May-2020.)
Disj 𝑦𝐴 {𝑥𝐵𝐶 = 𝑦}
 
Theoremdisjiun 4471* A disjoint collection yields disjoint indexed unions for disjoint index sets. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 14-Nov-2016.)
((Disj 𝑥𝐴 𝐵 ∧ (𝐶𝐴𝐷𝐴 ∧ (𝐶𝐷) = ∅)) → ( 𝑥𝐶 𝐵 𝑥𝐷 𝐵) = ∅)
 
Theoremsndisj 4472 Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj 𝑥𝐴 {𝑥}
 
Theorem0disj 4473 Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj 𝑥𝐴
 
Theoremdisjxsn 4474* A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj 𝑥 ∈ {𝐴}𝐵
 
Theoremdisjx0 4475 An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj 𝑥 ∈ ∅ 𝐵
 
Theoremdisjprg 4476* A pair collection is disjoint iff the two sets in the family have empty intersection. (Contributed by Mario Carneiro, 14-Nov-2016.)
(𝑥 = 𝐴𝐶 = 𝐷)    &   (𝑥 = 𝐵𝐶 = 𝐸)       ((𝐴𝑉𝐵𝑉𝐴𝐵) → (Disj 𝑥 ∈ {𝐴, 𝐵}𝐶 ↔ (𝐷𝐸) = ∅))
 
Theoremdisjxiun 4477* An indexed union of a disjoint collection of disjoint collections is disjoint if each component is disjoint, and the disjoint unions in the collection are also disjoint. Note that 𝐵(𝑦) and 𝐶(𝑥) may have the displayed free variables. (Contributed by Mario Carneiro, 14-Nov-2016.) (Proof shortened by JJ, 27-May-2021.)
(Disj 𝑦𝐴 𝐵 → (Disj 𝑥 𝑦𝐴 𝐵𝐶 ↔ (∀𝑦𝐴 Disj 𝑥𝐵 𝐶Disj 𝑦𝐴 𝑥𝐵 𝐶)))
 
TheoremdisjxiunOLD 4478* Obsolete proof of disjxiun 4477 as of 27-May-2021. An indexed union of a disjoint collection of disjoint collections is disjoint if each component is disjoint, and the disjoint unions in the collection are also disjoint. Note that 𝐵(𝑦) and 𝐶(𝑥) may have the displayed free variables. (Contributed by Mario Carneiro, 14-Nov-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
(Disj 𝑦𝐴 𝐵 → (Disj 𝑥 𝑦𝐴 𝐵𝐶 ↔ (∀𝑦𝐴 Disj 𝑥𝐵 𝐶Disj 𝑦𝐴 𝑥𝐵 𝐶)))
 
Theoremdisjxun 4479* The union of two disjoint collections. (Contributed by Mario Carneiro, 14-Nov-2016.)
(𝑥 = 𝑦𝐶 = 𝐷)       ((𝐴𝐵) = ∅ → (Disj 𝑥 ∈ (𝐴𝐵)𝐶 ↔ (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶 ∧ ∀𝑥𝐴𝑦𝐵 (𝐶𝐷) = ∅)))
 
Theoremdisjss3 4480* Expand a disjoint collection with any number of empty sets. (Contributed by Mario Carneiro, 15-Nov-2016.)
((𝐴𝐵 ∧ ∀𝑥 ∈ (𝐵𝐴)𝐶 = ∅) → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶))
 
2.1.22  Binary relations
 
Syntaxwbr 4481 Extend wff notation to include the general binary relation predicate. Note that the syntax is simply three class symbols in a row. Since binary relations are the only possible wff expressions consisting of three class expressions in a row, the syntax is unambiguous. (For an example of how syntax could become ambiguous if we are not careful, see the comment in cneg 10018.)
wff 𝐴𝑅𝐵
 
Definitiondf-br 4482 Define a general binary relation. Note that the syntax is simply three class symbols in a row. Definition 6.18 of [TakeutiZaring] p. 29 generalized to arbitrary classes. Class 𝑅 often denotes a relation such as "< " that compares two classes 𝐴 and 𝐵, which might be numbers such as 1 and 2 (see df-ltxr 9834 for the specific definition of <). As a wff, relations are true or false. For example, (𝑅 = {⟨2, 6⟩, ⟨3, 9⟩} → 3𝑅9) (ex-br 26418). Often class 𝑅 meets the Rel criteria to be defined in df-rel 4939, and in particular 𝑅 may be a function (see df-fun 5691). This definition of relations is well-defined, although not very meaningful, when classes 𝐴 and/or 𝐵 are proper classes (i.e. are not sets). On the other hand, we often find uses for this definition when 𝑅 is a proper class (see for example iprc 6869). (Contributed by NM, 31-Dec-1993.)
(𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
 
Theorembreq 4483 Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.)
(𝑅 = 𝑆 → (𝐴𝑅𝐵𝐴𝑆𝐵))
 
Theorembreq1 4484 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
(𝐴 = 𝐵 → (𝐴𝑅𝐶𝐵𝑅𝐶))
 
Theorembreq2 4485 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)
(𝐴 = 𝐵 → (𝐶𝑅𝐴𝐶𝑅𝐵))
 
Theorembreq12 4486 Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝑅𝐶𝐵𝑅𝐷))
 
Theorembreqi 4487 Equality inference for binary relations. (Contributed by NM, 19-Feb-2005.)
𝑅 = 𝑆       (𝐴𝑅𝐵𝐴𝑆𝐵)
 
Theorembreq1i 4488 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
𝐴 = 𝐵       (𝐴𝑅𝐶𝐵𝑅𝐶)
 
Theorembreq2i 4489 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
𝐴 = 𝐵       (𝐶𝑅𝐴𝐶𝑅𝐵)
 
Theorembreq12i 4490 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝑅𝐶𝐵𝑅𝐷)
 
Theorembreq1d 4491 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝑅𝐶𝐵𝑅𝐶))
 
Theorembreqd 4492 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴𝐷𝐶𝐵𝐷))
 
Theorembreq2d 4493 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝑅𝐴𝐶𝑅𝐵))
 
Theorembreq12d 4494 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝑅𝐶𝐵𝑅𝐷))
 
Theorembreq123d 4495 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝑅 = 𝑆)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝑅𝐶𝐵𝑆𝐷))
 
Theorembreqdi 4496 Equality deduction for a binary relation. (Contributed by Thierry Arnoux, 5-Oct-2020.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶𝐴𝐷)       (𝜑𝐶𝐵𝐷)
 
Theorembreqan12d 4497 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜑𝜓) → (𝐴𝑅𝐶𝐵𝑅𝐷))
 
Theorembreqan12rd 4498 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜓𝜑) → (𝐴𝑅𝐶𝐵𝑅𝐷))
 
Theoremeqnbrtrd 4499 Substitution of equal classes into the negation of a binary relation. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → ¬ 𝐵𝑅𝐶)       (𝜑 → ¬ 𝐴𝑅𝐶)
 
Theoremnbrne1 4500 Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)
((𝐴𝑅𝐵 ∧ ¬ 𝐴𝑅𝐶) → 𝐵𝐶)
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