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

Theoremssiin 4101* Subset theorem for an indexed intersection. (Contributed by NM, 15-Oct-2003.)

Theoremiinss 4102* Subset implication for an indexed intersection. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremiinss2 4103 An indexed intersection is included in any of its members. (Contributed by FL, 15-Oct-2012.)

Theoremuniiun 4104* Class union in terms of indexed union. Definition in [Stoll] p. 43. (Contributed by NM, 28-Jun-1998.)

Theoremintiin 4105* Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.)

Theoremiunid 4106* An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.)

Theoremiun0 4107 An indexed union of the empty set is empty. (Contributed by NM, 26-Mar-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theorem0iun 4108 An empty indexed union is empty. (Contributed by NM, 4-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theorem0iin 4109 An empty indexed intersection is the universal class. (Contributed by NM, 20-Oct-2005.)

Theoremviin 4110* Indexed intersection with a universal index class. When doesn't depend on , this evaluates to by 19.3 1787 and abid2 2521. When , this evaluates to by intiin 4105 and intv 4335. (Contributed by NM, 11-Sep-2008.)

Theoremiunn0 4111* 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 4112* Indexed intersection of a class builder. (Contributed by NM, 6-Dec-2011.)

Theoremiinrab 4113* Indexed intersection of a restricted class builder. (Contributed by NM, 6-Dec-2011.)

Theoremiinrab2 4114* Indexed intersection of a restricted class builder. (Contributed by NM, 6-Dec-2011.)

Theoremiunin2 4115* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4104 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.)

Theoremiunin1 4116* Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4104 to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.)

Theoremiinun2 4117* Indexed intersection of union. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4105 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)

Theoremiundif2 4118* Indexed union of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use intiin 4105 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)

Theorem2iunin 4119* Rearrange indexed unions over intersection. (Contributed by NM, 18-Dec-2008.)

Theoremiindif2 4120* Indexed intersection of class difference. Generalization of half of theorem "De Morgan's laws" in [Enderton] p. 31. Use uniiun 4104 to recover Enderton's theorem. (Contributed by NM, 5-Oct-2006.)

Theoremiinin2 4121* Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4105 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.)

Theoremiinin1 4122* Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4105 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.)

Theoremelriin 4123* Elementhood in a relative intersection. (Contributed by Mario Carneiro, 30-Dec-2016.)

Theoremriin0 4124* Relative intersection of an empty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Theoremriinn0 4125* Relative intersection of a nonempty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Theoremriinrab 4126* Relative intersection of a relative abstraction. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Theoremiinxsng 4127* 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 4128* Indexed intersection with an unordered pair index. (Contributed by NM, 25-Jan-2012.)

Theoremiunxsng 4129* A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.)

Theoremiunxsn 4130* 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 4131 Separate a union in an indexed union. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Theoremiunxun 4132 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 4133* Separate an indexed union in the index of an indexed union. (Contributed by Mario Carneiro, 5-Dec-2016.)

Theoremiinuni 4134* 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 4135* 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 4136 Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.)

Theorempwssb 4137* Two ways to express a collection of subclasses. (Contributed by NM, 19-Jul-2006.)

Theoremelpwuni 4138 Relationship for power class and union. (Contributed by NM, 18-Jul-2006.)

Theoremiinpw 4139* 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 4140* 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 4141 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 4142 Extend wff notation to include the statement that a family of classes , for , is a disjoint family.
Disj

Definitiondf-disj 4143* 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 4144* Alternate definition for disjoint classes. (Contributed by NM, 17-Jun-2017.)
Disj

Theoremdisjss2 4145 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 4146 Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremdisjeq2dv 4147* Equality deduction for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremdisjss1 4148* A subset of a disjoint collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremdisjeq1 4149* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremdisjeq1d 4150* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremdisjeq12d 4151* Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremcbvdisj 4152* Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj

Theoremcbvdisjv 4153* Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 11-Dec-2016.)
Disj Disj

Theoremnfdisj 4154 Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoremnfdisj1 4155 Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoremdisjor 4156* 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 4157* 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.)

Theoremdisjors 4158* Two ways to say that a collection for is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoremdisji2 4159* Property of a disjoint collection: if and , and , then and are disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoremdisji 4160* Property of a disjoint collection: if and have a common element , then . (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoreminvdisj 4161* If there is a function such that for all , then the sets for distinct are disjoint. (Contributed by Mario Carneiro, 10-Dec-2016.)
Disj

Theoremdisjiun 4162* 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

TheoremdisjiunOLD 4163* A disjoint collection yields disjoint indexed unions for disjoint index sets. (Contributed by Mario Carneiro, 26-Mar-2015.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremsndisj 4164 Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theorem0disj 4165 Any collection of empty sets is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoremdisjxsn 4166* A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoremdisjx0 4167 An empty collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoremdisjprg 4168* A pair collection is disjoint iff the two sets in the family have empty intersection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj

Theoremdisjxiun 4169* 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 and may have the displayed free variables. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj Disj Disj

Theoremdisjxun 4170* The union of two disjoint collections. (Contributed by Mario Carneiro, 14-Nov-2016.)
Disj Disj Disj

Theoremdisjss3 4171* Expand a disjoint collection with any number of empty sets. (Contributed by Mario Carneiro, 15-Nov-2016.)
Disj Disj

2.1.22  Binary relations

Syntaxwbr 4172 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 9248.)

Definitiondf-br 4173 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 often denotes a relation such as " " that compares two classes and , which might be numbers such as and (see df-ltxr 9081 for the specific definition of ). As a wff, relations are true or false. For example, (ex-br 21692). Often class meets the criteria to be defined in df-rel 4844, and in particular may be a function (see df-fun 5415). This definition of relations is well-defined, although not very meaningful, when classes and/or are proper classes (i.e. are not sets). On the other hand, we often find uses for this definition when is a proper class. (Contributed by NM, 31-Dec-1993.)

Theorembreq 4174 Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.)

Theorembreq1 4175 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)

Theorembreq2 4176 Equality theorem for a binary relation. (Contributed by NM, 31-Dec-1993.)

Theorembreq12 4177 Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)

Theorembreqi 4178 Equality inference for binary relations. (Contributed by NM, 19-Feb-2005.)

Theorembreq1i 4179 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)

Theorembreq2i 4180 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)

Theorembreq12i 4181 Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)

Theorembreq1d 4182 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)

Theorembreqd 4183 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)

Theorembreq2d 4184 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)

Theorembreq12d 4185 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theorembreq123d 4186 Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)

Theorembreqan12d 4187 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)

Theorembreqan12rd 4188 Equality deduction for a binary relation. (Contributed by NM, 8-Feb-1996.)

Theoremnbrne1 4189 Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)

Theoremnbrne2 4190 Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)

Theoremeqbrtri 4191 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)

Theoremeqbrtrd 4192 Substitution of equal classes into a binary relation. (Contributed by NM, 8-Oct-1999.)

Theoremeqbrtrri 4193 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)

Theoremeqbrtrrd 4194 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)

Theorembreqtri 4195 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)

Theorembreqtrd 4196 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)

Theorembreqtrri 4197 Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)

Theorembreqtrrd 4198 Substitution of equal classes into a binary relation. (Contributed by NM, 24-Oct-1999.)

Theorem3brtr3i 4199 Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)

Theorem3brtr4i 4200 Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)

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