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Theorem List for Metamath Proof Explorer - 4101-4200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremiuncom 4101* Commutation of indexed unions. (Contributed by NM, 18-Dec-2008.)

Theoremiuncom4 4102 Commutation of union with indexed union. (Contributed by Mario Carneiro, 18-Jan-2014.)

Theoremiunconst 4103* Indexed union of a constant class, i.e. where does not depend on . (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremiinconst 4104* Indexed intersection of a constant class, i.e. where does not depend on . (Contributed by Mario Carneiro, 6-Feb-2015.)

Theoremiuniin 4105* Law combining indexed union with indexed intersection. Eq. 14 in [KuratowskiMostowski] p. 109. This theorem also appears as the last example at http://en.wikipedia.org/wiki/Union%5F%28set%5Ftheory%29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremiunss1 4106* Subclass theorem for indexed union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremiinss1 4107* Subclass theorem for indexed union. (Contributed by NM, 24-Jan-2012.)

Theoremiuneq1 4108* Equality theorem for indexed union. (Contributed by NM, 27-Jun-1998.)

Theoremiineq1 4109* Equality theorem for restricted existential quantifier. (Contributed by NM, 27-Jun-1998.)

Theoremss2iun 4110 Subclass theorem for indexed union. (Contributed by NM, 26-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremiuneq2 4111 Equality theorem for indexed union. (Contributed by NM, 22-Oct-2003.)

Theoremiineq2 4112 Equality theorem for indexed intersection. (Contributed by NM, 22-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremiuneq2i 4113 Equality inference for indexed union. (Contributed by NM, 22-Oct-2003.)

Theoremiineq2i 4114 Equality inference for indexed intersection. (Contributed by NM, 22-Oct-2003.)

Theoremiineq2d 4115 Equality deduction for indexed intersection. (Contributed by NM, 7-Dec-2011.)

Theoremiuneq2dv 4116* Equality deduction for indexed union. (Contributed by NM, 3-Aug-2004.)

Theoremiineq2dv 4117* Equality deduction for indexed intersection. (Contributed by NM, 3-Aug-2004.)

Theoremiuneq1d 4118* Equality theorem for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)

Theoremiuneq12d 4119* Equality deduction for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)

Theoremiuneq2d 4120* Equality deduction for indexed union. (Contributed by Drahflow, 22-Oct-2015.)

Theoremnfiun 4121 Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.)

Theoremnfiin 4122 Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.)

Theoremnfiu1 4123 Bound-variable hypothesis builder for indexed union. (Contributed by NM, 12-Oct-2003.)

Theoremnfii1 4124 Bound-variable hypothesis builder for indexed intersection. (Contributed by NM, 15-Oct-2003.)

Theoremdfiun2g 4125* Alternate definition of indexed union when is a set. Definition 15(a) of [Suppes] p. 44. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremdfiin2g 4126* Alternate definition of indexed intersection when is a set. (Contributed by Jeff Hankins, 27-Aug-2009.)

Theoremdfiun2 4127* Alternate definition of indexed union when is a set. Definition 15(a) of [Suppes] p. 44. (Contributed by NM, 27-Jun-1998.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremdfiin2 4128* Alternate definition of indexed intersection when is a set. Definition 15(b) of [Suppes] p. 44. (Contributed by NM, 28-Jun-1998.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremdfiunv2 4129* Define double indexed union. (Contributed by FL, 6-Nov-2013.)

Theoremcbviun 4130* Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.)

Theoremcbviin 4131* Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) (Revised by Mario Carneiro, 14-Oct-2016.)

Theoremcbviunv 4132* Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 15-Sep-2003.)

Theoremcbviinv 4133* Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.)

Theoremiunss 4134* Subset theorem for an indexed union. (Contributed by NM, 13-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremssiun 4135* Subset implication for an indexed union. (Contributed by NM, 3-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremssiun2 4136 Identity law for subset of an indexed union. (Contributed by NM, 12-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremssiun2s 4137* Subset relationship for an indexed union. (Contributed by NM, 26-Oct-2003.)

Theoremiunss2 4138* A subclass condition on the members of two indexed classes and that implies a subclass relation on their indexed unions. Generalization of Proposition 8.6 of [TakeutiZaring] p. 59. Compare uniss2 4048. (Contributed by NM, 9-Dec-2004.)

Theoremiunab 4139* The indexed union of a class abstraction. (Contributed by NM, 27-Dec-2004.)

Theoremiunrab 4140* The indexed union of a restricted class abstraction. (Contributed by NM, 3-Jan-2004.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)

Theoremiunxdif2 4141* Indexed union with a class difference as its index. (Contributed by NM, 10-Dec-2004.)

Theoremssiinf 4142 Subset theorem for an indexed intersection. (Contributed by FL, 15-Oct-2012.) (Proof shortened by Mario Carneiro, 14-Oct-2016.)

Theoremssiin 4143* Subset theorem for an indexed intersection. (Contributed by NM, 15-Oct-2003.)

Theoremiinss 4144* Subset implication for an indexed intersection. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremiinss2 4145 An indexed intersection is included in any of its members. (Contributed by FL, 15-Oct-2012.)

Theoremuniiun 4146* Class union in terms of indexed union. Definition in [Stoll] p. 43. (Contributed by NM, 28-Jun-1998.)

Theoremintiin 4147* Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.)

Theoremiunid 4148* An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.)

Theoremiun0 4149 An indexed union of the empty set is empty. (Contributed by NM, 26-Mar-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theorem0iun 4150 An empty indexed union is empty. (Contributed by NM, 4-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theorem0iin 4151 An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.)

Theoremviin 4152* Indexed intersection with a universal index class. When doesn't depend on , this evaluates to by 19.3 1792 and abid2 2555. When , this evaluates to by intiin 4147 and intv 4378. (Contributed by NM, 11-Sep-2008.)

Theoremiunn0 4153* There is a non-empty class in an indexed collection iff the indexed union of them is non-empty. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremiinab 4154* Indexed intersection of a class builder. (Contributed by NM, 6-Dec-2011.)

Theoremiinrab 4155* Indexed intersection of a restricted class builder. (Contributed by NM, 6-Dec-2011.)

Theoremiinrab2 4156* Indexed intersection of a restricted class builder. (Contributed by NM, 6-Dec-2011.)

Theoremiunin2 4157* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4146 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.)

Theoremiunin1 4158* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4146 to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.)

Theoremiinun2 4159* Indexed intersection of union. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4147 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)

Theoremiundif2 4160* Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 4147 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)

Theorem2iunin 4161* Rearrange indexed unions over intersection. (Contributed by NM, 18-Dec-2008.)

Theoremiindif2 4162* Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 4146 to recover Enderton's theorem. (Contributed by NM, 5-Oct-2006.)

Theoremiinin2 4163* Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4147 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.)

Theoremiinin1 4164* Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4147 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.)

Theoremiinvdif 4165* The indexed intersection of a complement. (Contributed by Gérard Lang, 5-Aug-2018.)

Theoremelriin 4166* Elementhood in a relative intersection. (Contributed by Mario Carneiro, 30-Dec-2016.)

Theoremriin0 4167* Relative intersection of an empty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Theoremriinn0 4168* Relative intersection of a nonempty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Theoremriinrab 4169* Relative intersection of a relative abstraction. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Theoremiinxsng 4170* 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.)

Theoremiinxprg 4171* Indexed intersection with an unordered pair index. (Contributed by NM, 25-Jan-2012.)

Theoremiunxsng 4172* A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.)

Theoremiunxsn 4173* 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.)

Theoremiunun 4174 Separate a union in an indexed union. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Theoremiunxun 4175 Separate a union in the index of an indexed union. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Theoremiunxiun 4176* Separate an indexed union in the index of an indexed union. (Contributed by Mario Carneiro, 5-Dec-2016.)

Theoremiinuni 4177* 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.)

Theoremiununi 4178* 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.)

Theoremsspwuni 4179 Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.)

Theorempwssb 4180* Two ways to express a collection of subclasses. (Contributed by NM, 19-Jul-2006.)

Theoremelpwuni 4181 Relationship for power class and union. (Contributed by NM, 18-Jul-2006.)

Theoremiinpw 4182* The power class of an intersection in terms of indexed intersection. Exercise 24(a) of [Enderton] p. 33. (Contributed by NM, 29-Nov-2003.)

Theoremiunpwss 4183* 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.)

Theoremrintn0 4184 Relative intersection of a nonempty set. (Contributed by Stefan O'Rear, 3-Apr-2015.) (Revised by Mario Carneiro, 5-Jun-2015.)

2.1.21  Disjointness

Syntaxwdisj 4185 Extend wff notation to include the statement that a family of classes , for , is a disjoint family.
Disj

Definitiondf-disj 4186* A collection of classes is disjoint when for each element , it is in for at most one . (Contributed by Mario Carneiro, 14-Nov-2016.) (Revised by NM, 16-Jun-2017.)
Disj

Theoremdfdisj2 4187* Alternate definition for disjoint classes. (Contributed by NM, 17-Jun-2017.)
Disj

Theoremdisjss2 4188 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.)
Disj Disj

Theoremdisjeq2 4189 Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremdisjeq2dv 4190* Equality deduction for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremdisjss1 4191* A subset of a disjoint collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremdisjeq1 4192* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremdisjeq1d 4193* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremdisjeq12d 4194* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremcbvdisj 4195* Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremcbvdisjv 4196* Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 11-Dec-2016.)
Disj Disj

Theoremnfdisj 4197 Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoremnfdisj1 4198 Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoremdisjor 4199* Two ways to say that a collection for is disjoint. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 14-Nov-2016.)
Disj

TheoremdisjmoOLD 4200* Two ways to say that a collection for is disjoint. (Contributed by Mario Carneiro, 26-Mar-2015.) (New usage is discouraged.) (Proof modification is discouraged.)

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