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Theorem List for New Foundations Explorer - 5901-6000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-antisym 5901* Define the set of all antisymmetric relationships over a base set. (Contributed by SF, 19-Feb-2015.)
Antisym

Definitiondf-partial 5902 Define the set of all partial orderings over a base set. (Contributed by SF, 19-Feb-2015.)
Po Ref Trans Antisym

Definitiondf-connex 5903* Define the set of all connected relationships over a base set. (Contributed by SF, 19-Feb-2015.)
Connex

Definitiondf-strict 5904 Define the set of all strict orderings over a base set. (Contributed by SF, 19-Feb-2015.)
Or Po Connex

Definitiondf-found 5905* Define the set of all founded relationships over a base set. (Contributed by SF, 19-Feb-2015.)
Fr

Definitiondf-we 5906 Define the set of all well orderings over a base set. (Contributed by SF, 19-Feb-2015.)
We Or Fr

Definitiondf-ext 5907* Define the set of all extensional relationships over a base set. (Contributed by SF, 19-Feb-2015.)
Ext

Definitiondf-sym 5908* Define the set of all symmetric relationships over a base set. (Contributed by SF, 19-Feb-2015.)
Sym

Definitiondf-er 5909 Define the set of all equivalence relationships over a base set. (Contributed by SF, 19-Feb-2015.)
Er Sym Trans

Theoremtransex 5910 The class of all transitive relationships is a set. (Contributed by SF, 19-Feb-2015.)
Trans

Theoremrefex 5911 The class of all reflexive relationships is a set. (Contributed by SF, 11-Mar-2015.)
Ref

Theoremantisymex 5912 The class of all antisymmetric relationships is a set. (Contributed by SF, 11-Mar-2015.)
Antisym

Theoremconnexex 5913 The class of all connected relationships is a set. (Contributed by SF, 11-Mar-2015.)
Connex

Theoremfoundex 5914 The class of all founded relationships is a set. (Contributed by SF, 19-Feb-2015.)
Fr

Theoremextex 5915 The class of all extensional relationships is a set. (Contributed by SF, 19-Feb-2015.)
Ext

Theoremsymex 5916 The class of all symmetric relationships is a set. (Contributed by SF, 20-Feb-2015.)
Sym

Theorempartialex 5917 The class of all partial orderings is a set. (Contributed by SF, 11-Mar-2015.)
Po

Theoremstrictex 5918 The class of all strict orderings is a set. (Contributed by SF, 19-Feb-2015.)
Or

Theoremweex 5919 The class of all well orderings is a set. (Contributed by SF, 19-Feb-2015.)
We

Theoremerex 5920 The class of all equivalence relationships is a set. (Contributed by SF, 20-Feb-2015.)
Er

Theoremtrd 5921 Transitivity law in natural deduction form. (Contributed by SF, 20-Feb-2015.)
Trans

Theoremfrd 5922* Founded relationship in natural deduction form. (Contributed by SF, 12-Mar-2015.)
Fr

Theoremextd 5923* Extensional relationship in natural deduction form. (Contributed by SF, 20-Feb-2015.)
Ext

Theoremsymd 5924 Symmetric relationship in natural deduction form. (Contributed by SF, 20-Feb-2015.)
Sym

Theoremtrrd 5925* Deduce transitivity from its properties. (Contributed by SF, 22-Feb-2015.)
Trans

Theoremrefrd 5926* Deduce reflexitiviy from its properties. (Contributed by SF, 12-Mar-2015.)
Ref

Theoremrefd 5927 Natural deduction form of reflexitivity. (Contributed by SF, 20-Mar-2015.)
Ref

Theoremantird 5928* Deduce antisymmetry from its properties. (Contributed by SF, 12-Mar-2015.)
Antisym

Theoremantid 5929 The antisymmetry property. (Contributed by SF, 18-Mar-2015.)
Antisym

Theoremconnexrd 5930* Deduce connectivity from its properties. (Contributed by SF, 12-Mar-2015.)
Connex

Theoremconnexd 5931 The connectivity property. (Contributed by SF, 18-Mar-2015.)
Connex

Theoremersymtr 5932 Equivalence relationship as symmetric, transitive relationship. (Contributed by SF, 22-Feb-2015.)
Er Sym Trans

Theoremporta 5933 Partial ordering as reflexive, transitive, antisymmetric relationship. (Contributed by SF, 12-Mar-2015.)
Po Ref Trans Antisym

Theoremsopc 5934 Linear ordering as partial, connected relationship. (Contributed by SF, 12-Mar-2015.)
Or Po Connex

Theoremfrds 5935* Substitution schema verson of frd 5922. (Contributed by SF, 19-Mar-2015.)
Fr

Theorempod 5936* A reflexive, transitive, and anti-symmetric ordering is a partial ordering. (Contributed by SF, 22-Feb-2015.)
Po

Theoremsod 5937* A reflexive, transitive, antisymmetric, and connected relationship is a strict ordering. (Contributed by SF, 12-Mar-2015.)
Or

Theoremweds 5938* Any property that holds for some element of a well-ordered set has an minimal element satisfying that property. (Contributed by SF, 20-Mar-2015.)
We

Theorempo0 5939 Anything partially orders the empty set. (Contributed by SF, 12-Mar-2015.)
Po

Theoremconnex0 5940 Anything is connected over the empty set. (Contributed by SF, 12-Mar-2015.)
Connex

Theoremso0 5941 Anything totally orders the empty set. (Contributed by SF, 12-Mar-2015.)
Or

Theoremiserd 5942* A symmetric, transitive relationship is an equivalence relationship. (Contributed by SF, 22-Feb-2015.)
Er

Theoremider 5943 The identity relationship is an equivalence relationship over the universe. (Contributed by SF, 22-Feb-2015.)
Er

Theoremssetpov 5944 The subset relationship partially orders the universe. (Contributed by SF, 12-Mar-2015.)
S Po

2.4.2  Equivalence relations and classes

Syntaxcec 5945 Extend the definition of a class to include equivalence class.

Syntaxcqs 5946 Extend the definition of a class to include quotient set.

Definitiondf-ec 5947 Define the -coset of . Exercise 35 of [Enderton] p. 61. This is called the equivalence class of modulo when is an equivalence relation. In this case, is a representative (member) of the equivalence class , which contains all sets that are equivalent to . Definition of [Enderton] p. 57 uses the notation (subscript) , although we simply follow the brackets by since we don't have subscripted expressions. For an alternate definition, see dfec2 5948. (Contributed by set.mm contributors, 22-Feb-2015.)

Theoremdfec2 5948* Alternate definition of -coset of . Definition 34 of [Suppes] p. 81. (Contributed by set.mm contributors, 22-Feb-2015.)

Theoremecexg 5949 An equivalence class modulo a set is a set. (Contributed by set.mm contributors, 24-Jul-1995.)

Theoremecexr 5950 A nonempty equivalence class implies the representative is a set. (Contributed by set.mm contributors, 9-Jul-2014.)

Definitiondf-qs 5951* Define quotient set. is usually an equivalence relation. Definition of [Enderton] p. 58. (Contributed by set.mm contributors, 22-Feb-2015.)

Theoremersym 5952 An equivalence relation is symmetric. (Contributed by set.mm contributors, 22-Feb-2015.)
Er

Theoremersymb 5953 An equivalence relation is symmetric. (Contributed by set.mm contributors, 30-Jul-1995.) (Revised by set.mm contributors, 9-Jul-2014.)
Er

Theoremertr 5954 An equivalence relation is transitive. (Contributed by set.mm contributors, 4-Jun-1995.) (Revised by set.mm contributors, 9-Jul-2014.)
Er

Theoremertrd 5955 A transitivity relation for equivalences. (Contributed by set.mm contributors, 9-Jul-2014.)
Er

Theoremertr2d 5956 A transitivity relation for equivalences. (Contributed by set.mm contributors, 9-Jul-2014.)
Er

Theoremertr3d 5957 A transitivity relation for equivalences. (Contributed by set.mm contributors, 9-Jul-2014.)
Er

Theoremertr4d 5958 A transitivity relation for equivalences. (Contributed by set.mm contributors, 9-Jul-2014.)
Er

Theoremerref 5959 An equivalence relation is reflexive on its field. Compare Theorem 3M of [Enderton] p. 56. (Contributed by set.mm contributors, 6-May-2013.)
Er

Theoremeqerlem 5960* Lemma for eqer 5961. (Contributed by set.mm contributors, 17-Mar-2008.)

Theoremeqer 5961* Equivalence relation involving equality of dependent classes and . (Contributed by set.mm contributors, 17-Mar-2008.)
Er

Theoremeceq1 5962 Equality theorem for equivalence class. (Contributed by set.mm contributors, 23-Jul-1995.)

Theoremeceq2 5963 Equality theorem for equivalence class. (Contributed by set.mm contributors, 23-Jul-1995.)

Theoremelec 5964 Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by set.mm contributors, 9-Jul-2014.)

Theoremerdmrn 5965 The range and domain of an equivalence relation are equal. (Contributed by Rodolfo Medina, 11-Oct-2010.)
Er

Theoremecss 5966 An equivalence class is a subset of the domain. (Contributed by set.mm contributors, 6-Aug-1995.) (Revised by set.mm contributors, 9-Jul-2014.)
Er

Theoremecdmn0 5967 A representative of a nonempty equivalence class belongs to the domain of the equivalence relation. (Contributed by set.mm contributors, 15-Feb-1996.) (Revised by set.mm contributors, 9-Jul-2014.)

Theoremerth 5968 Basic property of equivalence relations. Theorem 73 of [Suppes] p. 82. (Contributed by set.mm contributors, 23-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
Er

Theoremerth2 5969 Basic property of equivalence relations. Compare Theorem 73 of [Suppes] p. 82. Assumes membership of the second argument in the domain. (Contributed by set.mm contributors, 30-Jul-1995.) (Revised by set.mm contributors, 9-Jul-2014.)
Er

Theoremerthi 5970 Basic property of equivalence relations. Part of Lemma 3N of [Enderton] p. 57. (Contributed by set.mm contributors, 30-Jul-1995.) (Revised by set.mm contributors, 9-Jul-2014.)
Er

Theoremereldm 5971 Equality of equivalence classes implies equivalence of domain membership. (Contributed by set.mm contributors, 28-Jan-1996.) (Revised by set.mm contributors, 9-Jul-2014.)
Er

Theoremerdisj 5972 Equivalence classes do not overlap. In other words, two equivalence classes are either equal or disjoint. Theorem 74 of [Suppes] p. 83. (Contributed by set.mm contributors, 15-Jun-2004.) (Revised by Mario Carneiro, 9-Jul-2014.)
Er

Theoremecidsn 5973 An equivalence class modulo the identity relation is a singleton. (Contributed by set.mm contributors, 24-Oct-2004.)

Theoremqseq1 5974 Equality theorem for quotient set. (Contributed by set.mm contributors, 23-Jul-1995.)

Theoremqseq2 5975 Equality theorem for quotient set. (Contributed by set.mm contributors, 23-Jul-1995.)

Theoremelqsg 5976* Closed form of elqs 5977. (Contributed by Rodolfo Medina, 12-Oct-2010.)

Theoremelqs 5977* Membership in a quotient set. (Contributed by set.mm contributors, 23-Jul-1995.) (Revised by set.mm contributors, 12-Nov-2008.)

Theoremelqsi 5978* Membership in a quotient set. (Contributed by set.mm contributors, 23-Jul-1995.)

Theoremecelqsg 5979 Membership of an equivalence class in a quotient set. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremecelqsi 5980 Membership of an equivalence class in a quotient set. (Contributed by set.mm contributors, 25-Jul-1995.) (Revised by set.mm contributors, 9-Jul-2014.)

Theoremecopqsi 5981 "Closure" law for equivalence class of ordered pairs. (Contributed by set.mm contributors, 25-Mar-1996.)

Theoremqsexg 5982 A quotient set exists. (Contributed by FL, 19-May-2007.)

Theoremqsex 5983 A quotient set exists. (Contributed by set.mm contributors, 14-Aug-1995.)

Theoremuniqs 5984 The union of a quotient set. (Contributed by set.mm contributors, 9-Dec-2008.)

Theoremuniqs2 5985 The union of a quotient set. (Contributed by set.mm contributors, 11-Jul-2014.)
Er

Theoremqsss 5986 A quotient set is a set of subsets of the base set. (Contributed by Mario Carneiro, 9-Jul-2014.)
Er

Theoremsnec 5987 The singleton of an equivalence class. (Contributed by set.mm contributors, 29-Jan-1999.) (Revised by set.mm contributors, 9-Jul-2014.)

Theoremecqs 5988 Equivalence class in terms of quotient set. (Contributed by set.mm contributors, 29-Jan-1999.) (Revised by set.mm contributors, 15-Jan-2009.)

Theoremecid 5989 A set is equal to its converse epsilon coset. (Note: converse epsilon is not an equivalence relation.) (Contributed by set.mm contributors, 13-Aug-1995.) (Revised by set.mm contributors, 9-Jul-2014.)

Theoremqsid 5990 A set is equal to its quotient set mod converse epsilon. (Note: converse epsilon is not an equivalence relation.) (Contributed by set.mm contributors, 13-Aug-1995.) (Revised by set.mm contributors, 9-Jul-2014.)

Theoremectocld 5991* Implicit substitution of class for equivalence class. (Contributed by set.mm contributors, 9-Jul-2014.)

Theoremectocl 5992* Implicit substitution of class for equivalence class. (Contributed by set.mm contributors, 23-Jul-1995.) (Revised by set.mm contributors, 9-Jul-2014.)

Theoremelqsn0 5993 A quotient set doesn't contain the empty set. (Contributed by set.mm contributors, 24-Aug-1995.) (Revised by set.mm contributors, 21-Mar-2007.)

Theoremecelqsdm 5994 Membership of an equivalence class in a quotient set. (Contributed by set.mm contributors, 30-Jul-1995.) (Revised by set.mm contributors, 21-Mar-2007.)

Theoremqsdisj 5995 Members of a quotient set do not overlap. (Contributed by Rodolfo Medina, 12-Oct-2010.) (Revised by Mario Carneiro, 11-Jul-2014.)
Er

Theoremecoptocl 5996* Implicit substitution of class for equivalence class of ordered pair. (Contributed by set.mm contributors, 23-Jul-1995.)

Theorem2ecoptocl 5997* Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by set.mm contributors, 23-Jul-1995.)

Theorem3ecoptocl 5998* Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by set.mm contributors, 9-Aug-1995.)

2.4.3  The mapping operation

Syntaxcmap 5999 Extend the definition of a class to include the mapping operation. (Read for , "the set of all functions that map from to .)

Syntaxcpm 6000 Extend the definition of a class to include the partial mapping operation. (Read for , "the set of all partial functions that map from to .)

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