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Theorem List for Intuitionistic Logic Explorer - 6101-6200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnnawordex 6101* Equivalence for weak ordering of natural numbers. (Contributed by NM, 8-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremnnm00 6102 The product of two natural numbers is zero iff at least one of them is zero. (Contributed by Jim Kingdon, 11-Nov-2004.)

2.6.24  Equivalence relations and classes

Syntaxwer 6103 Extend the definition of a wff to include the equivalence predicate.

Syntaxcec 6104 Extend the definition of a class to include equivalence class.

Syntaxcqs 6105 Extend the definition of a class to include quotient set.

Definitiondf-er 6106 Define the equivalence relation predicate. Our notation is not standard. A formal notation doesn't seem to exist in the literature; instead only informal English tends to be used. The present definition, although somewhat cryptic, nicely avoids dummy variables. In dfer2 6107 we derive a more typical definition. We show that an equivalence relation is reflexive, symmetric, and transitive in erref 6126, ersymb 6120, and ertr 6121. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 2-Nov-2015.)

Theoremdfer2 6107* Alternate definition of equivalence predicate. (Contributed by NM, 3-Jan-1997.) (Revised by Mario Carneiro, 12-Aug-2015.)

Definitiondf-ec 6108 Define the -coset of . Exercise 35 of [Enderton] p. 61. This is called the equivalence class of modulo when is an equivalence relation (i.e. when ; see dfer2 6107). 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 6109. (Contributed by NM, 23-Jul-1995.)

Theoremdfec2 6109* Alternate definition of -coset of . Definition 34 of [Suppes] p. 81. (Contributed by NM, 3-Jan-1997.) (Proof shortened by Mario Carneiro, 9-Jul-2014.)

Theoremecexg 6110 An equivalence class modulo a set is a set. (Contributed by NM, 24-Jul-1995.)

Theoremecexr 6111 An inhabited equivalence class implies the representative is a set. (Contributed by Mario Carneiro, 9-Jul-2014.)

Definitiondf-qs 6112* Define quotient set. is usually an equivalence relation. Definition of [Enderton] p. 58. (Contributed by NM, 23-Jul-1995.)

Theoremereq1 6113 Equality theorem for equivalence predicate. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremereq2 6114 Equality theorem for equivalence predicate. (Contributed by Mario Carneiro, 12-Aug-2015.)

Theoremerrel 6115 An equivalence relation is a relation. (Contributed by Mario Carneiro, 12-Aug-2015.)

Theoremerdm 6116 The domain of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)

Theoremercl 6117 Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)

Theoremersym 6118 An equivalence relation is symmetric. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremercl2 6119 Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)

Theoremersymb 6120 An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremertr 6121 An equivalence relation is transitive. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremertrd 6122 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremertr2d 6123 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremertr3d 6124 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremertr4d 6125 A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremerref 6126 An equivalence relation is reflexive on its field. Compare Theorem 3M of [Enderton] p. 56. (Contributed by Mario Carneiro, 6-May-2013.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremercnv 6127 The converse of an equivalence relation is itself. (Contributed by Mario Carneiro, 12-Aug-2015.)

Theoremerrn 6128 The range and domain of an equivalence relation are equal. (Contributed by Rodolfo Medina, 11-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremerssxp 6129 An equivalence relation is a subset of the cartesian product of the field. (Contributed by Mario Carneiro, 12-Aug-2015.)

Theoremerex 6130 An equivalence relation is a set if its domain is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Proof shortened by Mario Carneiro, 12-Aug-2015.)

Theoremerexb 6131 An equivalence relation is a set if and only if its domain is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremiserd 6132* A reflexive, symmetric, transitive relation is an equivalence relation on its domain. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theorembrdifun 6133 Evaluate the incomparability relation. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremswoer 6134* Incomparability under a strict weak partial order is an equivalence relation. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremswoord1 6135* The incomparability equivalence relation is compatible with the original order. (Contributed by Mario Carneiro, 31-Dec-2014.)

Theoremswoord2 6136* The incomparability equivalence relation is compatible with the original order. (Contributed by Mario Carneiro, 31-Dec-2014.)

Theoremeqerlem 6137* Lemma for eqer 6138. (Contributed by NM, 17-Mar-2008.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)

Theoremeqer 6138* Equivalence relation involving equality of dependent classes and . (Contributed by NM, 17-Mar-2008.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremider 6139 The identity relation is an equivalence relation. (Contributed by NM, 10-May-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Proof shortened by Mario Carneiro, 9-Jul-2014.)

Theorem0er 6140 The empty set is an equivalence relation on the empty set. (Contributed by Mario Carneiro, 5-Sep-2015.)

Theoremeceq1 6141 Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)

Theoremeceq1d 6142 Equality theorem for equivalence class (deduction form). (Contributed by Jim Kingdon, 31-Dec-2019.)

Theoremeceq2 6143 Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)

Theoremelecg 6144 Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremelec 6145 Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.)

Theoremrelelec 6146 Membership in an equivalence class when is a relation. (Contributed by Mario Carneiro, 11-Sep-2015.)

Theoremecss 6147 An equivalence class is a subset of the domain. (Contributed by NM, 6-Aug-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremecdmn0m 6148* A representative of an inhabited equivalence class belongs to the domain of the equivalence relation. (Contributed by Jim Kingdon, 21-Aug-2019.)

Theoremereldm 6149 Equality of equivalence classes implies equivalence of domain membership. (Contributed by NM, 28-Jan-1996.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremerth 6150 Basic property of equivalence relations. Theorem 73 of [Suppes] p. 82. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)

Theoremerth2 6151 Basic property of equivalence relations. Compare Theorem 73 of [Suppes] p. 82. Assumes membership of the second argument in the domain. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)

Theoremerthi 6152 Basic property of equivalence relations. Part of Lemma 3N of [Enderton] p. 57. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremecidsn 6153 An equivalence class modulo the identity relation is a singleton. (Contributed by NM, 24-Oct-2004.)

Theoremqseq1 6154 Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)

Theoremqseq2 6155 Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.)

Theoremelqsg 6156* Closed form of elqs 6157. (Contributed by Rodolfo Medina, 12-Oct-2010.)

Theoremelqs 6157* Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)

Theoremelqsi 6158* Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)

Theoremecelqsg 6159 Membership of an equivalence class in a quotient set. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremecelqsi 6160 Membership of an equivalence class in a quotient set. (Contributed by NM, 25-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremecopqsi 6161 "Closure" law for equivalence class of ordered pairs. (Contributed by NM, 25-Mar-1996.)

Theoremqsexg 6162 A quotient set exists. (Contributed by FL, 19-May-2007.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremqsex 6163 A quotient set exists. (Contributed by NM, 14-Aug-1995.)

Theoremuniqs 6164 The union of a quotient set. (Contributed by NM, 9-Dec-2008.)

Theoremqsss 6165 A quotient set is a set of subsets of the base set. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremuniqs2 6166 The union of a quotient set. (Contributed by Mario Carneiro, 11-Jul-2014.)

Theoremsnec 6167 The singleton of an equivalence class. (Contributed by NM, 29-Jan-1999.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremecqs 6168 Equivalence class in terms of quotient set. (Contributed by NM, 29-Jan-1999.)

Theoremecid 6169 A set is equal to its converse epsilon coset. (Note: converse epsilon is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremecidg 6170 A set is equal to its converse epsilon coset. (Note: converse epsilon is not an equivalence relation.) (Contributed by Jim Kingdon, 8-Jan-2020.)

Theoremqsid 6171 A set is equal to its quotient set mod converse epsilon. (Note: converse epsilon is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremectocld 6172* Implicit substitution of class for equivalence class. (Contributed by Mario Carneiro, 9-Jul-2014.)

Theoremectocl 6173* Implicit substitution of class for equivalence class. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremelqsn0m 6174* An element of a quotient set is inhabited. (Contributed by Jim Kingdon, 21-Aug-2019.)

Theoremelqsn0 6175 A quotient set doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)

Theoremecelqsdm 6176 Membership of an equivalence class in a quotient set. (Contributed by NM, 30-Jul-1995.)

Theoremxpiderm 6177* A square Cartesian product is an equivalence relation (in general it's not a poset). (Contributed by Jim Kingdon, 22-Aug-2019.)

Theoremiinerm 6178* The intersection of a nonempty family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)

Theoremriinerm 6179* The relative intersection of a family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)

Theoremerinxp 6180 A restricted equivalence relation is an equivalence relation. (Contributed by Mario Carneiro, 10-Jul-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)

Theoremecinxp 6181 Restrict the relation in an equivalence class to a base set. (Contributed by Mario Carneiro, 10-Jul-2015.)

Theoremqsinxp 6182 Restrict the equivalence relation in a quotient set to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)

Theoremqsel 6183 If an element of a quotient set contains a given element, it is equal to the equivalence class of the element. (Contributed by Mario Carneiro, 12-Aug-2015.)

Theoremqliftlem 6184* , a function lift, is a subset of . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftrel 6185* , a function lift, is a subset of . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftel 6186* Elementhood in the relation . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftel1 6187* Elementhood in the relation . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftfun 6188* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftfund 6189* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftfuns 6190* The function is the unique function defined by , provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftf 6191* The domain and range of the function . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremqliftval 6192* The value of the function . (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremecoptocl 6193* Implicit substitution of class for equivalence class of ordered pair. (Contributed by NM, 23-Jul-1995.)

Theorem2ecoptocl 6194* Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 23-Jul-1995.)

Theorem3ecoptocl 6195* Implicit substitution of classes for equivalence classes of ordered pairs. (Contributed by NM, 9-Aug-1995.)

Theorembrecop 6196* Binary relation on a quotient set. Lemma for real number construction. (Contributed by NM, 29-Jan-1996.)

Theoremeroveu 6197* Lemma for eroprf 6199. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)

Theoremerovlem 6198* Lemma for eroprf 6199. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theoremeroprf 6199* Functionality of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.)

Theoremeroprf2 6200* Functionality of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.)

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