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Statement List for Metamath Proof Explorer - 4101-4200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremonnev 4101 The class of ordinal numbers is not equal to the universe. (The proof was shortened by Mario Carneiro, 10-Jan-2013.)
 
TheoremonnevOLD 4102 The class of ordinal numbers is not equal to the universe.
 
Theoremreleq 4103 Equality theorem for the relation predicate.
 
Theoremreleqi 4104 Equality inference for the relation predicate.
   =>   
 
Theoremreleqd 4105 Equality deduction for the relation predicate.
   =>   
 
Theoremhbrel 4106* Bound-variable hypothesis builder for a relation.
   =>   
 
Theoremrelss 4107 Subclass theorem for relation predicate. Theorem 2 of [Suppes] p. 58.
 
Theoremssrel 4108* A subclass relationship depends only on a relation's ordered pairs. Theorem 3.2(i) of [Monk1] p. 33. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
 
Theoremeqrel 4109* Extensionality principle for relations. Theorem 3.2(ii) of [Monk1] p. 33.
 
Theoremrelssi 4110* Inference from subclass principle for relations.
   &       =>   
 
Theoremrelssdv 4111* Deduction from subclass principle for relations.
   &       =>   
 
Theoremeqrelriv 4112* Inference from extensionality principle for relations. (Contributed by FL, 15-Oct-2012.)
   =>   
 
Theoremeqrelriiv 4113* Inference from extensionality principle for relations.
   &       &       =>   
 
Theoremeqbrriv 4114* Inference from extensionality principle for relations.
   &       &       =>   
 
Theoremeqrelrdv 4115* Deduce equality of relations from equivalence of membership. (Contributed by Rodolfo Medina, 10-Oct-2010.)
   &       &       =>   
 
Theoremeqrelrdv2 4116* A version of eqrelrdv 4115. (Contributed by Rodolfo Medina, 10-Oct-2010.)
   =>   
 
Theoremssrelrel 4117* A subclass relationship determined by ordered triples. Use relrelss 4465 to express the antecedent in terms of the relation predicate. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
 
Theoremeqrelrel 4118* Extensionality principle for ordered triples (used by 2-place operations df-oprab 4982), analogous to eqrel 4109. Use relrelss 4465 to express the antecedent in terms of the relation predicate.
 
Theoremelrel 4119* A member of a relation is an ordered pair.
 
Theoremrelsn 4120 A singleton is a relation iff it is an ordered pair.
   =>   
 
Theoremrelsnop 4121 A singleton of an ordered pair is a relation.
   =>   
 
Theoremxpss12 4122 Subset theorem for cross product. Generalization of Theorem 101 of [Suppes] p. 52. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
 
Theoremxpss 4123 A cross product is included in the ordered pair universe. Exercise 3 of [TakeutiZaring] p. 25.
 
Theoremrelxp 4124 A cross product is a relation. Theorem 3.13(i) of [Monk1] p. 37.
 
Theoremxpss1 4125 Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)
 
Theoremxpss2 4126 Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)
 
Theoremxpsspw 4127 A cross product is included in the power of the power of the union of its arguments.
 
TheoremxpsspwOLD 4128 A cross product is included in the power of the power of the union of its arguments.
 
Theoremunixpss 4129 The double class union of a cross product is included in the union of its arguments.
 
Theoremxpexg 4130 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23.
 
Theoremxpex 4131 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23.
   &       =>   
 
Theoremrelun 4132 The union of two relations is a relation. Compare Exercise 5 of [TakeutiZaring] p. 25.
 
Theoremrelin1 4133 The intersection with a relation is a relation.
 
Theoremrelin2 4134 The intersection with a relation is a relation.
 
Theoremreldif 4135 A difference cutting down a relation is a relation.
 
Theoremreliun 4136 An indexed union is a relation iff each member of its indexed family is a relation.
 
Theoremreliin 4137 An indexed intersection is a relation if if at least one of the member of the indexed family is a relation.
 
Theoremreluni 4138* The union of a class is a relation iff any member is a relation. Exercise 6 of [TakeutiZaring] p. 25 and its converse.
 
Theoremrelint 4139* The intersection of a class is a relation if at least one member is a relation.
 
Theoremrel0 4140 The empty set is a relation.
 
Theoremrelopabi 4141 A class of ordered pairs is a relation. (The proof was shortened by Mario Carneiro, 21-Dec-2013.)
   =>   
 
Theoremrelopab 4142 A class of ordered pairs is a relation. (Unnecessary distinct variable restrictions were removed by Alan Sare, 9-Jul-2013.) (The proof was shortened by Mario Carneiro, 21-Dec-2013.)
 
Theoremreli 4143 The identity relation is a relation. Part of Exercise 4.12(p) of [Mendelson] p. 235.
 
Theoremrele 4144 The membership relation is a relation.
 
Theoremopabid2 4145* A relation expressed as an ordered pair abstraction.
 
Theoreminopab 4146* Intersection of two ordered pair class abstractions.
 
Theoremdifopab 4147* The difference of two ordered-pair abstractions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
 
Theoreminxp 4148 The intersection of two cross products. Exercise 9 of [TakeutiZaring] p. 25. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
 
Theoremxpindi 4149 Distributive law for cross product over intersection. Theorem 102 of [Suppes] p. 52.
 
Theoremxpindir 4150 Distributive law for cross product over intersection. Similar to Theorem 102 of [Suppes] p. 52.
 
Theoremeliunxp 4151* Membership in a union of cross products. Analogue of elxp 4040 for nonconstant .
 
Theoremopeliunxp2 4152* Membership in a union of cross products.
   =>   
 
Theoremraliunxp 4153* Write a double restricted quantification as one universal quantifier. In this version of ralxp 4155, is not assumed to be constant.
   =>   
 
Theoremrexiunxp 4154* Write a double restricted quantification as one universal quantifier. In this version of rexxp 4156, is not assumed to be constant. (Contributed by Mario Carneiro, 14-Feb-2015.)
   =>   
 
Theoremralxp 4155* Universal quantification restricted to a cross product is equivalent to a double restricted quantification. The hypothesis specifies an implicit substitution.
   =>   
 
Theoremrexxp 4156* Existential quantification restricted to a cross product is equivalent to a double restricted quantification.
   =>   
 
Theoremdjussxp 4157* Disjoint union is a subset of a cross product. (Contributed by Stefan O'Rear, 21-Nov-2014.)
 
Theoremralxpf 4158* Version of ralxp 4155 with bound-variable hypotheses.
   &       &       &       =>   
 
Theoremrexxpf 4159* Version of rexxp 4156 with bound-variable hypotheses.
   &       &       &       =>   
 
Theoremiunxpf 4160* Indexed union on a cross product is equals a double indexed union. The hypothesis specifies an implicit substitution.
   &       &       &       =>   
 
Theoremopabbi2dv 4161* Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2027.
   &       =>   
 
Theoremrelop 4162* A necessary and sufficient condition for a Kuratowski ordered pair to be a relation.
   &       =>   
 
Theoremideqg 4163 For sets, the identity relation is the same as equality. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
 
Theoremideq 4164 For sets, the identity relation is the same as equality.
   =>   
 
Theoremididg 4165 A set is identical to itself. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
 
Theoremopelxpex2 4166 The second member of an ordered pair of classes in a cross product exists when the order pair doesn't belong to . (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
 
Theoremissetid 4167 Two ways of expressing set existence. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
 
Theoremcoss1 4168 Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.)
 
Theoremcoss2 4169 Subclass theorem for composition.
 
Theoremcoeq1 4170 Equality theorem for composition of two classes.
 
Theoremcoeq2 4171 Equality theorem for composition of two classes.
 
Theoremcoeq1i 4172 Equality inference for composition of two classes.
   =>   
 
Theoremcoeq2i 4173 Equality inference for composition of two classes.
   =>   
 
Theoremcoeq1d 4174 Equality deduction for composition of two classes.
   =>   
 
Theoremcoeq2d 4175 Equality deduction for composition of two classes.
   =>   
 
Theoremcoeq12i 4176 Equality inference for composition of two classes. (Contributed by FL, 7-Jun-2012.)
   &       =>   
 
Theoremcoeq12d 4177 Equality deduction for composition of two classes. (Contributed by FL, 7-Jun-2012.)
   &       =>   
 
Theoremhbco 4178* Bound-variable hypothesis builder for function value.
   &       =>   
 
Theoremopelco 4179* Ordered pair membership in a composition. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
   &       =>   
 
Theorembrco 4180* Binary relation on a composition.
   &       =>   
 
Theoremopelco2g 4181* Ordered pair membership in a composition.
 
Theoremcnvss 4182 Subset theorem for converse.
 
Theoremcnveq 4183 Equality theorem for converse.
 
Theoremcnveqi 4184 Equality inference for converse.
   =>   
 
Theoremcnveqd 4185 Equality deduction for converse.
   =>   
 
Theoremelcnv 4186* Membership in a converse. Equation 5 of [Suppes] p. 62.
 
Theoremelcnv2 4187* Membership in a converse. Equation 5 of [Suppes] p. 62.
 
Theoremhbcnv 4188* Bound-variable hypothesis builder for converse.
   =>   
 
Theoremopelcnvg 4189 Ordered-pair membership in converse. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
 
Theorembrcnvg 4190 The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61.
 
Theoremopelcnv 4191 Ordered-pair membership in converse.
   &       =>   
 
Theorembrcnv 4192 The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61.
   &       =>   
 
Theoremcnvco 4193 Distributive law of converse over class composition. Theorem 26 of [Suppes] p. 64. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)
 
Theoremcnvuni 4194* The converse of a class union is the (indexed) union of the converses of its members.
 
Theoremdfdm3 4195* Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24.
 
Theoremdfrn2 4196* Alternate definition of range. Definition 4 of [Suppes] p. 60.
 
Theoremdfrn3 4197* Alternate definition of range. Definition 6.5(2) of [TakeutiZaring] p. 24.
 
Theoremdfdm4 4198 Alternate definition of domain.
 
Theoremdfdmf 4199* Definition of domain, using bound-variable hypotheses instead of distinct variable conditions.
   &       =>   
 
Theoremeldmg 4200* Domain membership. Theorem 4 of [Suppes] p. 59.
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