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Theorem List for Metamath Proof Explorer - 4901-5000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsoinxp 4901 Intersection of total order with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)

Theoremfrinxp 4902 Intersection of well-founded relation with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)

Theoremseinxp 4903 Intersection of set-like relation with cross product of its field. (Contributed by Mario Carneiro, 22-Jun-2015.)
Se Se

Theoremweinxp 4904 Intersection of well-ordering with cross product of its field. (Contributed by NM, 9-Mar-1997.) (Revised by Mario Carneiro, 10-Jul-2014.)

Theoremposn 4905 Partial ordering of a singleton. (Contributed by NM, 27-Apr-2009.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremsosn 4906 Strict ordering on a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)

Theoremfrsn 4907 Founded relation on a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremwesn 4908 Well-ordering of a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)

Theoremopabssxp 4909* An abstraction relation is a subset of a related cross product. (Contributed by NM, 16-Jul-1995.)

Theorembrab2ga 4910* The law of concretion for a binary relation. See brab2a 4886 for alternate proof. TODO: should one of them be deleted? (Contributed by Mario Carneiro, 28-Apr-2015.) (Proof modification is discouraged.)

Theoremoptocl 4911* Implicit substitution of class for ordered pair. (Contributed by NM, 5-Mar-1995.)

Theorem2optocl 4912* Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.)

Theorem3optocl 4913* Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.)

Theoremopbrop 4914* Ordered pair membership in a relation. Special case. (Contributed by NM, 5-Aug-1995.)

Theoremxp0r 4915 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.)

Theoremonxpdisj 4916 Ordinal numbers and ordered pairs are disjoint collections. This theorem can be used if we want to extend a set of ordinal numbers or ordered pairs with disjoint elements. See also snsn0non 4659. (Contributed by NM, 1-Jun-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremonnev 4917 The class of ordinal numbers is not equal to the universe. (Contributed by NM, 16-Jun-2007.) (Proof shortened by Mario Carneiro, 10-Jan-2013.)

Theoremreleq 4918 Equality theorem for the relation predicate. (Contributed by NM, 1-Aug-1994.)

Theoremreleqi 4919 Equality inference for the relation predicate. (Contributed by NM, 8-Dec-2006.)

Theoremreleqd 4920 Equality deduction for the relation predicate. (Contributed by NM, 8-Mar-2014.)

Theoremnfrel 4921 Bound-variable hypothesis builder for a relation. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremrelss 4922 Subclass theorem for relation predicate. Theorem 2 of [Suppes] p. 58. (Contributed by NM, 15-Aug-1994.)

Theoremssrel 4923* 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.)

Theoremeqrel 4924* Extensionality principle for relations. Theorem 3.2(ii) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.)

Theoremssrel2 4925* A subclass relationship depends only on a relation's ordered pairs. This version of ssrel 4923 is restricted to the relation's domain. (Contributed by Thierry Arnoux, 25-Jan-2018.)

Theoremrelssi 4926* Inference from subclass principle for relations. (Contributed by NM, 31-Mar-1998.)

Theoremrelssdv 4927* Deduction from subclass principle for relations. (Contributed by NM, 11-Sep-2004.)

Theoremeqrelriv 4928* Inference from extensionality principle for relations. (Contributed by FL, 15-Oct-2012.)

Theoremeqrelriiv 4929* Inference from extensionality principle for relations. (Contributed by NM, 17-Mar-1995.)

Theoremeqbrriv 4930* Inference from extensionality principle for relations. (Contributed by NM, 12-Dec-2006.)

Theoremeqrelrdv 4931* Deduce equality of relations from equivalence of membership. (Contributed by Rodolfo Medina, 10-Oct-2010.)

Theoremeqbrrdv 4932* Deduction from extensionality principle for relations. (Contributed by Mario Carneiro, 3-Jan-2017.)

Theoremeqbrrdiv 4933* Deduction from extensionality principle for relations. (Contributed by Rodolfo Medina, 10-Oct-2010.)

Theoremeqrelrdv2 4934* A version of eqrelrdv 4931. (Contributed by Rodolfo Medina, 10-Oct-2010.)

Theoremssrelrel 4935* A subclass relationship determined by ordered triples. Use relrelss 5352 to express the antecedent in terms of the relation predicate. (Contributed by NM, 17-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremeqrelrel 4936* Extensionality principle for ordered triples (used by 2-place operations df-oprab 6044), analogous to eqrel 4924. Use relrelss 5352 to express the antecedent in terms of the relation predicate. (Contributed by NM, 17-Dec-2008.)

Theoremelrel 4937* A member of a relation is an ordered pair. (Contributed by NM, 17-Sep-2006.)

Theoremrelsn 4938 A singleton is a relation iff it is an ordered pair. (Contributed by NM, 24-Sep-2013.)

Theoremrelsnop 4939 A singleton of an ordered pair is a relation. (Contributed by NM, 17-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremxpss12 4940 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.)

Theoremxpss 4941 A cross product is included in the ordered pair universe. Exercise 3 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)

Theoremrelxp 4942 A cross product is a relation. Theorem 3.13(i) of [Monk1] p. 37. (Contributed by NM, 2-Aug-1994.)

Theoremxpss1 4943 Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)

Theoremxpss2 4944 Subset relation for cross product. (Contributed by Jeff Hankins, 30-Aug-2009.)

Theoremxpsspw 4945 A cross product is included in the power of the power of the union of its arguments. (Contributed by NM, 13-Sep-2006.)

TheoremxpsspwOLD 4946 A cross product is included in the power of the power of the union of its arguments. (Contributed by NM, 13-Sep-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremunixpss 4947 The double class union of a cross product is included in the union of its arguments. (Contributed by NM, 16-Sep-2006.)

Theoremxpexg 4948 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. See also xpexgALT 6256. (Contributed by NM, 14-Aug-1994.)

Theoremxpex 4949 The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. (Contributed by NM, 14-Aug-1994.)

Theoremrelun 4950 The union of two relations is a relation. Compare Exercise 5 of [TakeutiZaring] p. 25. (Contributed by NM, 12-Aug-1994.)

Theoremrelin1 4951 The intersection with a relation is a relation. (Contributed by NM, 16-Aug-1994.)

Theoremrelin2 4952 The intersection with a relation is a relation. (Contributed by NM, 17-Jan-2006.)

Theoremreldif 4953 A difference cutting down a relation is a relation. (Contributed by NM, 31-Mar-1998.)

Theoremreliun 4954 An indexed union is a relation iff each member of its indexed family is a relation. (Contributed by NM, 19-Dec-2008.)

Theoremreliin 4955 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.)

Theoremreluni 4956* 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.)

Theoremrelint 4957* The intersection of a class is a relation if at least one member is a relation. (Contributed by NM, 8-Mar-2014.)

Theoremrel0 4958 The empty set is a relation. (Contributed by NM, 26-Apr-1998.)

Theoremrelopabi 4959 A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.)

Theoremrelopab 4960 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.)

Theoremreli 4961 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.)

Theoremrele 4962 The membership relation is a relation. (Contributed by NM, 26-Apr-1998.) (Revised by Mario Carneiro, 21-Dec-2013.)

Theoremopabid2 4963* A relation expressed as an ordered pair abstraction. (Contributed by NM, 11-Dec-2006.)

Theoreminopab 4964* Intersection of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)

Theoremdifopab 4965* The difference of two ordered-pair abstractions. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoreminxp 4966 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.)

Theoremxpindi 4967 Distributive law for cross product over intersection. Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)

Theoremxpindir 4968 Distributive law for cross product over intersection. Similar to Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)

Theoremxpiindi 4969* Distributive law for cross product over indexed intersection. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theoremxpriindi 4970* Distributive law for cross product over relativized indexed intersection. (Contributed by Mario Carneiro, 21-Mar-2015.)

Theoremeliunxp 4971* Membership in a union of cross products. Analogue of elxp 4854 for nonconstant . (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremopeliunxp2 4972* Membership in a union of cross products. (Contributed by Mario Carneiro, 14-Feb-2015.)

Theoremraliunxp 4973* Write a double restricted quantification as one universal quantifier. In this version of ralxp 4975, is not assumed to be constant. (Contributed by Mario Carneiro, 29-Dec-2014.)

Theoremrexiunxp 4974* Write a double restricted quantification as one universal quantifier. In this version of rexxp 4976, is not assumed to be constant. (Contributed by Mario Carneiro, 14-Feb-2015.)

Theoremralxp 4975* Universal quantification restricted to a cross product is equivalent to a double restricted quantification. The hypothesis specifies an implicit substitution. (Contributed by NM, 7-Feb-2004.) (Revised by Mario Carneiro, 29-Dec-2014.)

Theoremrexxp 4976* Existential quantification restricted to a cross product is equivalent to a double restricted quantification. (Contributed by NM, 11-Nov-1995.) (Revised by Mario Carneiro, 14-Feb-2015.)

Theoremdjussxp 4977* Disjoint union is a subset of a cross product. (Contributed by Stefan O'Rear, 21-Nov-2014.)

Theoremralxpf 4978* Version of ralxp 4975 with bound-variable hypotheses. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremrexxpf 4979* Version of rexxp 4976 with bound-variable hypotheses. (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremiunxpf 4980* Indexed union on a cross product is equals a double indexed union. The hypothesis specifies an implicit substitution. (Contributed by NM, 19-Dec-2008.)

Theoremopabbi2dv 4981* Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2519. (Contributed by NM, 24-Feb-2014.)

Theoremrelop 4982* A necessary and sufficient condition for a Kuratowski ordered pair to be a relation. (Contributed by NM, 3-Jun-2008.)

Theoremideqg 4983 For sets, the identity relation is the same as equality. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremideq 4984 For sets, the identity relation is the same as equality. (Contributed by NM, 13-Aug-1995.)

Theoremididg 4985 A set is identical to itself. (Contributed by NM, 28-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Theoremissetid 4986 Two ways of expressing set existence. (Contributed by NM, 16-Feb-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremcoss1 4987 Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.)

Theoremcoss2 4988 Subclass theorem for composition. (Contributed by NM, 5-Apr-2013.)

Theoremcoeq1 4989 Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)

Theoremcoeq2 4990 Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)

Theoremcoeq1i 4991 Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)

Theoremcoeq2i 4992 Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)

Theoremcoeq1d 4993 Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)

Theoremcoeq2d 4994 Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)

Theoremcoeq12i 4995 Equality inference for composition of two classes. (Contributed by FL, 7-Jun-2012.)

Theoremcoeq12d 4996 Equality deduction for composition of two classes. (Contributed by FL, 7-Jun-2012.)

Theoremnfco 4997 Bound-variable hypothesis builder for function value. (Contributed by NM, 1-Sep-1999.)

Theorembrcog 4998* Ordered pair membership in a composition. (Contributed by NM, 24-Feb-2015.)

Theoremopelco2g 4999* Ordered pair membership in a composition. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 24-Feb-2015.)

Theorembrcogw 5000 Ordered pair membership in a composition. (Contributed by Thierry Arnoux, 14-Jan-2018.)

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