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Theorem List for Intuitionistic Logic Explorer - 2901-3000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcsbcomg 2901* Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.)

Theoremcsbeq2d 2902 Formula-building deduction rule for class substitution. (Contributed by NM, 22-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)

Theoremcsbeq2dv 2903* Formula-building deduction rule for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)

Theoremcsbeq2i 2904 Formula-building inference rule for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)

Theoremcsbvarg 2905 The proper substitution of a class for setvar variable results in the class (if the class exists). (Contributed by NM, 10-Nov-2005.)

Theoremsbccsbg 2906* Substitution into a wff expressed in terms of substitution into a class. (Contributed by NM, 15-Aug-2007.)

Theoremsbccsb2g 2907 Substitution into a wff expressed in using substitution into a class. (Contributed by NM, 27-Nov-2005.)

Theoremnfcsb1d 2908 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)

Theoremnfcsb1 2909 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)

Theoremnfcsb1v 2910* Bound-variable hypothesis builder for substitution into a class. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)

Theoremnfcsbd 2911 Deduction version of nfcsb 2912. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)

Theoremnfcsb 2912 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)

Theoremcsbhypf 2913* Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 2620 for class substitution version. (Contributed by NM, 19-Dec-2008.)

Theoremcsbiebt 2914* Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 2918.) (Contributed by NM, 11-Nov-2005.)

Theoremcsbiedf 2915* Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 13-Oct-2016.)

Theoremcsbieb 2916* Bidirectional conversion between an implicit class substitution hypothesis and its explicit substitution equivalent. (Contributed by NM, 2-Mar-2008.)

Theoremcsbiebg 2917* 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 2918* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 11-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremcsbief 2919* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremcsbied 2920* 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 2921* Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremcsbie2t 2922* Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 2923). (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremcsbie2 2923* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 27-Aug-2007.)

Theoremcsbie2g 2924* Conversion of implicit substitution to explicit class substitution. This version of sbcie 2820 avoids a disjointness condition on and by substituting twice. (Contributed by Mario Carneiro, 11-Nov-2016.)

Theoremsbcnestgf 2925 Nest the composition of two substitutions. (Contributed by Mario Carneiro, 11-Nov-2016.)

Theoremcsbnestgf 2926 Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)

Theoremsbcnestg 2927* Nest the composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)

Theoremcsbnestg 2928* Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)

Theoremcsbnest1g 2929 Nest the composition of two substitutions. (Contributed by NM, 23-May-2006.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)

Theoremcsbidmg 2930* Idempotent law for class substitutions. (Contributed by NM, 1-Mar-2008.)

Theoremsbcco3g 2931* Composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)

Theoremcsbco3g 2932* Composition of two class substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)

Theoremrspcsbela 2933* Special case related to rspsbc 2868. (Contributed by NM, 10-Dec-2005.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)

Theoremsbnfc2 2934* Two ways of expressing " is (effectively) not free in ." (Contributed by Mario Carneiro, 14-Oct-2016.)

Theoremcsbabg 2935* Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremcbvralcsf 2936 A more general version of cbvralf 2544 that doesn't require and to be distinct from or . Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.)

Theoremcbvrexcsf 2937 A more general version of cbvrexf 2545 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 2938 A more general version of cbvreuv 2552 that has no distinct variable rextrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.)

Theoremcbvrabcsf 2939 A more general version of cbvrab 2572 with no distinct variable restrictions. (Contributed by Andrew Salmon, 13-Jul-2011.)

Theoremcbvralv2 2940* 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 2941* 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 2942 Extend class notation to include class difference (read: " minus ").

Syntaxcun 2943 Extend class notation to include union of two classes (read: " union ").

Syntaxcin 2944 Extend class notation to include the intersection of two classes (read: " intersect ").

Syntaxwss 2945 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 2946 Extend wff notation with proper subclass relation.

Theoremdifjust 2947* Soundness justification theorem for df-dif 2948. (Contributed by Rodolfo Medina, 27-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Definitiondf-dif 2948* Define class difference, also called relative complement. Definition 5.12 of [TakeutiZaring] p. 20. Contrast this operation with union (df-un 2950) and intersection (df-in 2952). 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 2949* Soundness justification theorem for df-un 2950. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Definitiondf-un 2950* Define the union of two classes. Definition 5.6 of [TakeutiZaring] p. 16. Contrast this operation with difference (df-dif 2948) and intersection (df-in 2952). (Contributed by NM, 23-Aug-1993.)

Theoreminjust 2951* Soundness justification theorem for df-in 2952. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Definitiondf-in 2952* Define the intersection of two classes. Definition 5.6 of [TakeutiZaring] p. 16. Contrast this operation with union (df-un 2950) and difference (df-dif 2948). (Contributed by NM, 29-Apr-1994.)

Theoremdfin5 2953* Alternate definition for the intersection of two classes. (Contributed by NM, 6-Jul-2005.)

Theoremdfdif2 2954* Alternate definition of class difference. (Contributed by NM, 25-Mar-2004.)

Theoremeldif 2955 Expansion of membership in a class difference. (Contributed by NM, 29-Apr-1994.)

Theoremeldifd 2956 If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 2955. (Contributed by David Moews, 1-May-2017.)

Theoremeldifad 2957 If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 2955. (Contributed by David Moews, 1-May-2017.)

Theoremeldifbd 2958 If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 2955. (Contributed by David Moews, 1-May-2017.)

2.1.12  Subclasses and subsets

Definitiondf-ss 2959 Define the subclass relationship. Exercise 9 of [TakeutiZaring] p. 18. Note that (proved in ssid 2992). Contrast this relationship with the relationship (as will be defined in df-pss 2961). For a more traditional definition, but requiring a dummy variable, see dfss2 2962 (or dfss3 2963 which is similar). (Contributed by NM, 27-Apr-1994.)

Theoremdfss 2960 Variant of subclass definition df-ss 2959. (Contributed by NM, 3-Sep-2004.)

Definitiondf-pss 2961 Define proper subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. Note that (proved in pssirr 3072). Contrast this relationship with the relationship (as defined in df-ss 2959). Other possible definitions are given by dfpss2 3057 and dfpss3 3058. (Contributed by NM, 7-Feb-1996.)

Theoremdfss2 2962* Alternate definition of the subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Jan-2002.)

Theoremdfss3 2963* Alternate definition of subclass relationship. (Contributed by NM, 14-Oct-1999.)

Theoremdfss2f 2964 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 2965 Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 20-Mar-2004.)

Theoremnfss 2966 If is not free in and , it is not free in . (Contributed by NM, 27-Dec-1996.)

Theoremssel 2967 Membership relationships follow from a subclass relationship. (Contributed by NM, 5-Aug-1993.)

Theoremssel2 2968 Membership relationships follow from a subclass relationship. (Contributed by NM, 7-Jun-2004.)

Theoremsseli 2969 Membership inference from subclass relationship. (Contributed by NM, 5-Aug-1993.)

Theoremsselii 2970 Membership inference from subclass relationship. (Contributed by NM, 31-May-1999.)

Theoremsseldi 2971 Membership inference from subclass relationship. (Contributed by NM, 25-Jun-2014.)

Theoremsseld 2972 Membership deduction from subclass relationship. (Contributed by NM, 15-Nov-1995.)

Theoremsselda 2973 Membership deduction from subclass relationship. (Contributed by NM, 26-Jun-2014.)

Theoremsseldd 2974 Membership inference from subclass relationship. (Contributed by NM, 14-Dec-2004.)

Theoremssneld 2975 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 2976 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 2977* Inference rule based on subclass definition. (Contributed by NM, 5-Aug-1993.)

Theoremssrd 2978 Deduction rule based on subclass definition. (Contributed by Thierry Arnoux, 8-Mar-2017.)

Theoremssrdv 2979* Deduction rule based on subclass definition. (Contributed by NM, 15-Nov-1995.)

Theoremsstr2 2980 Transitivity of subclasses. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Theoremsstr 2981 Transitivity of subclasses. Theorem 6 of [Suppes] p. 23. (Contributed by NM, 5-Sep-2003.)

Theoremsstri 2982 Subclass transitivity inference. (Contributed by NM, 5-May-2000.)

Theoremsstrd 2983 Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.)

Theoremsyl5ss 2984 Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.)

Theoremsyl6ss 2985 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremsylan9ss 2986 A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Theoremsylan9ssr 2987 A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)

Theoremeqss 2988 The subclass relationship is antisymmetric. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 5-Aug-1993.)

Theoremeqssi 2989 Infer equality from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 9-Sep-1993.)

Theoremeqssd 2990 Equality deduction from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 27-Jun-2004.)

Theoremeqrd 2991 Deduce equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 21-Mar-2017.)

Theoremssid 2992 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 2993 Any class is a subclass of the universal class. (Contributed by NM, 31-Oct-1995.)

Theoremsseq1 2994 Equality theorem for subclasses. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)

Theoremsseq2 2995 Equality theorem for the subclass relationship. (Contributed by NM, 25-Jun-1998.)

Theoremsseq12 2996 Equality theorem for the subclass relationship. (Contributed by NM, 31-May-1999.)

Theoremsseq1i 2997 An equality inference for the subclass relationship. (Contributed by NM, 18-Aug-1993.)

Theoremsseq2i 2998 An equality inference for the subclass relationship. (Contributed by NM, 30-Aug-1993.)

Theoremsseq12i 2999 An equality inference for the subclass relationship. (Contributed by NM, 31-May-1999.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Theoremsseq1d 3000 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)

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