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Theorem List for Metamath Proof Explorer - 3101-3200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsbceq1g 3101* Move proper substitution to first argument of an equality. (Contributed by NM, 30-Nov-2005.)

Theoremsbcel2g 3102* Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.)

Theoremsbceq2g 3103* Move proper substitution to second argument of an equality. (Contributed by NM, 30-Nov-2005.)

Theoremcsbcomg 3104* Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.)

Theoremcsbeq2d 3105 Formula-building deduction rule for class substitution. (Contributed by NM, 22-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)

Theoremcsbeq2dv 3106* Formula-building deduction rule for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)

Theoremcsbeq2i 3107 Formula-building inference rule for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)

Theoremcsbvarg 3108 The proper substitution of a class for set variable results in the class (if the class exists). (Contributed by NM, 10-Nov-2005.)

Theoremsbccsbg 3109* Substitution into a wff expressed in terms of substitution into a class. (Contributed by NM, 15-Aug-2007.)

Theoremsbccsb2g 3110 Substitution into a wff expressed in using substitution into a class. (Contributed by NM, 27-Nov-2005.)

Theoremnfcsb1d 3111 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)

Theoremnfcsb1 3112 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)

Theoremnfcsb1v 3113* Bound-variable hypothesis builder for substitution into a class. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)

Theoremnfcsbd 3114 Deduction version of nfcsb 3115. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)

Theoremnfcsb 3115 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)

Theoremcsbhypf 3116* Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2833 for class substitution version. (Contributed by NM, 19-Dec-2008.)

Theoremcsbiebt 3117* Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3121.) (Contributed by NM, 11-Nov-2005.)

Theoremcsbiedf 3118* Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 13-Oct-2016.)

Theoremcsbieb 3119* Bidirectional conversion between an implicit class substitution hypothesis and its explicit substitution equivalent. (Contributed by NM, 2-Mar-2008.)

Theoremcsbiebg 3120* Bidirectional conversion between an implicit class substitution hypothesis and its explicit substitution equivalent. (Contributed by NM, 24-Mar-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremcsbiegf 3121* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 11-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremcsbief 3122* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremcsbied 3123* Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremcsbied2 3124* Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremcsbie2t 3125* Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3126). (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremcsbie2 3126* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 27-Aug-2007.)

Theoremcsbie2g 3127* Conversion of implicit substitution to explicit class substitution. This version of sbcie 3025 avoids a disjointness condition on by substituting twice. (Contributed by Mario Carneiro, 11-Nov-2016.)

Theoremsbcnestgf 3128 Nest the composition of two substitutions. (Contributed by Mario Carneiro, 11-Nov-2016.)

Theoremcsbnestgf 3129 Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)

Theoremsbcnestg 3130* Nest the composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)

Theoremcsbnestg 3131* Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)

TheoremcsbnestgOLD 3132* Nest the composition of two substitutions. (New usage is discouraged.) (Contributed by NM, 23-Nov-2005.)

Theoremcsbnest1g 3133 Nest the composition of two substitutions. (Contributed by NM, 23-May-2006.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)

Theoremcsbnest1gOLD 3134* Nest the composition of two substitutions. Obsolete as of 11-Nov-2016. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)

Theoremcsbidmg 3135* Idempotent law for class substitutions. (Contributed by NM, 1-Mar-2008.)

Theoremsbcco3g 3136* Composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)

Theoremsbcco3gOLD 3137* Composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (New usage is discouraged.)

Theoremcsbco3g 3138* Composition of two class substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)

Theoremcsbco3gOLD 3139* Composition of two class substitutions. Obsolete as of 11-Nov-2016. (Contributed by NM, 27-Nov-2005.) (New usage is discouraged.)

Theoremrspcsbela 3140* Special case related to rspsbc 3069. (Contributed by NM, 10-Dec-2005.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)

Theoremsbnfc2 3141* Two ways of expressing " is (effectively) not free in ." (Contributed by Mario Carneiro, 14-Oct-2016.)

Theoremcsbabg 3142* Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremcbvralcsf 3143 A more general version of cbvralf 2758 that doesn't require and to be distinct from or . Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.)

Theoremcbvrexcsf 3144 A more general version of cbvrexf 2759 that has no distinct variable restrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.) (Proof shortened by Mario Carneiro, 7-Dec-2014.)

Theoremcbvreucsf 3145 A more general version of cbvreuv 2766 that has no distinct variable rextrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.)

Theoremcbvrabcsf 3146 A more general version of cbvrab 2786 with no distinct variable restrictions. (Contributed by Andrew Salmon, 13-Jul-2011.)

Theoremcbvralv2 3147* Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. (Contributed by David Moews, 1-May-2017.)

Theoremcbvrexv2 3148* Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. (Contributed by David Moews, 1-May-2017.)

2.1.11  Define basic set operations and relations

Syntaxcdif 3149 Extend class notation to include class difference (read: " minus ").

Syntaxcun 3150 Extend class notation to include union of two classes (read: " union ").

Syntaxcin 3151 Extend class notation to include the intersection of two classes (read: " intersect ").

Syntaxwss 3152 Extend wff notation to include the subclass relation. This is read " is a subclass of " or " includes ." When exists as a set, it is also read " is a subset of ."

Syntaxwpss 3153 Extend wff notation with proper subclass relation.

Theoremdifjust 3154* Soundness justification theorem for df-dif 3155. (Contributed by Rodolfo Medina, 27-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Definitiondf-dif 3155* Define class difference, also called relative complement. Definition 5.12 of [TakeutiZaring] p. 20. For example, (ex-dif 20810). Contrast this operation with union (df-un 3157) and intersection (df-in 3159). Several notations are used in the literature; we chose the convention used in Definition 5.3 of [Eisenberg] p. 67 instead of the more common minus sign to reserve the latter for later use in, e.g., arithmetic. We will use the terminology " excludes " to mean . We will use " is removed from " to mean i.e. the removal of an element or equivalently the exclusion of a singleton. (Contributed by NM, 29-Apr-1994.)

Theoremunjust 3156* Soundness justification theorem for df-un 3157. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Definitiondf-un 3157* Define the union of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, (ex-un 20811). Contrast this operation with difference (df-dif 3155) and intersection (df-in 3159). For an alternate definition in terms of class difference, requiring no dummy variables, see dfun2 3404. For union defined in terms of intersection, see dfun3 3407. (Contributed by NM, 23-Aug-1993.)

Theoreminjust 3158* Soundness justification theorem for df-in 3159. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Definitiondf-in 3159* Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. For example, (ex-in 20812). Contrast this operation with union (df-un 3157) and difference (df-dif 3155). For alternate definitions in terms of class difference, requiring no dummy variables, see dfin2 3405 and dfin4 3409. For intersection defined in terms of union, see dfin3 3408. (Contributed by NM, 29-Apr-1994.)

Theoremdfin5 3160* Alternate definition for the intersection of two classes. (Contributed by NM, 6-Jul-2005.)

Theoremdfdif2 3161* Alternate definition of class difference. (Contributed by NM, 25-Mar-2004.)

Theoremeldif 3162 Expansion of membership in a class difference. (Contributed by NM, 29-Apr-1994.)

Theoremeldifd 3163 If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 3162. (Contributed by David Moews, 1-May-2017.)

Theoremeldifad 3164 If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 3162. (Contributed by David Moews, 1-May-2017.)

Theoremeldifbd 3165 If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 3162. (Contributed by David Moews, 1-May-2017.)

2.1.12  Subclasses and subsets

Definitiondf-ss 3166 Define the subclass relationship. Exercise 9 of [TakeutiZaring] p. 18. For example, (ex-ss 20814). Note that (proved in ssid 3197). Contrast this relationship with the relationship (as will be defined in df-pss 3168). For a more traditional definition, but requiring a dummy variable, see dfss2 3169. Other possible definitions are given by dfss3 3170, dfss4 3403, sspss 3275, ssequn1 3345, ssequn2 3348, sseqin2 3388, and ssdif0 3513. (Contributed by NM, 27-Apr-1994.)

Theoremdfss 3167 Variant of subclass definition df-ss 3166. (Contributed by NM, 3-Sep-2004.)

Definitiondf-pss 3168 Define proper subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. For example, (ex-pss 20815). Note that (proved in pssirr 3276). Contrast this relationship with the relationship (as defined in df-ss 3166). Other possible definitions are given by dfpss2 3261 and dfpss3 3262. (Contributed by NM, 7-Feb-1996.)

Theoremdfss2 3169* Alternate definition of the subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Jan-2002.)

Theoremdfss3 3170* Alternate definition of subclass relationship. (Contributed by NM, 14-Oct-1999.)

Theoremdfss2f 3171 Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 3-Jul-1994.) (Revised by Andrew Salmon, 27-Aug-2011.)

Theoremdfss3f 3172 Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 20-Mar-2004.)

Theoremnfss 3173 If is not free in and , it is not free in . (Contributed by NM, 27-Dec-1996.)

Theoremssel 3174 Membership relationships follow from a subclass relationship. (Contributed by NM, 5-Aug-1993.)

Theoremssel2 3175 Membership relationships follow from a subclass relationship. (Contributed by NM, 7-Jun-2004.)

Theoremsseli 3176 Membership inference from subclass relationship. (Contributed by NM, 5-Aug-1993.)

Theoremsselii 3177 Membership inference from subclass relationship. (Contributed by NM, 31-May-1999.)

Theoremsseldi 3178 Membership inference from subclass relationship. (Contributed by NM, 25-Jun-2014.)

Theoremsseld 3179 Membership deduction from subclass relationship. (Contributed by NM, 15-Nov-1995.)

Theoremsselda 3180 Membership deduction from subclass relationship. (Contributed by NM, 26-Jun-2014.)

Theoremsseldd 3181 Membership inference from subclass relationship. (Contributed by NM, 14-Dec-2004.)

Theoremssneld 3182 If a class is not in another class, it is also not in a subclass of that class. Deduction form. (Contributed by David Moews, 1-May-2017.)

Theoremssneldd 3183 If an element is not in a class, it is also not in a subclass of that class. Deduction form. (Contributed by David Moews, 1-May-2017.)

Theoremssriv 3184* Inference rule based on subclass definition. (Contributed by NM, 5-Aug-1993.)

Theoremssrdv 3185* Deduction rule based on subclass definition. (Contributed by NM, 15-Nov-1995.)

Theoremsstr2 3186 Transitivity of subclasses. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Theoremsstr 3187 Transitivity of subclasses. Theorem 6 of [Suppes] p. 23. (Contributed by NM, 5-Sep-2003.)

Theoremsstri 3188 Subclass transitivity inference. (Contributed by NM, 5-May-2000.)

Theoremsstrd 3189 Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.)

Theoremsyl5ss 3190 Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.)

Theoremsyl6ss 3191 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremsylan9ss 3192 A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Theoremsylan9ssr 3193 A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)

Theoremeqss 3194 The subclass relationship is antisymmetric. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 5-Aug-1993.)

Theoremeqssi 3195 Infer equality from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 9-Sep-1993.)

Theoremeqssd 3196 Equality deduction from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 27-Jun-2004.)

Theoremssid 3197 Any class is a subclass of itself. Exercise 10 of [TakeutiZaring] p. 18. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Theoremssv 3198 Any class is a subclass of the universal class. (Contributed by NM, 31-Oct-1995.)

Theoremsseq1 3199 Equality theorem for subclasses. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)

Theoremsseq2 3200 Equality theorem for the subclass relationship. (Contributed by NM, 25-Jun-1998.)

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