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

Theoremsseldd 3101 Membership inference from subclass relationship. (Contributed by NM, 14-Dec-2004.)

Theoremssneld 3102 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 3103 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 3104* Inference based on subclass definition. (Contributed by NM, 5-Aug-1993.)

Theoremssrd 3105 Deduction based on subclass definition. (Contributed by Thierry Arnoux, 8-Mar-2017.)

Theoremssrdv 3106* Deduction based on subclass definition. (Contributed by NM, 15-Nov-1995.)

Theoremsstr2 3107 Transitivity of subclasses. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Theoremsstr 3108 Transitivity of subclasses. Theorem 6 of [Suppes] p. 23. (Contributed by NM, 5-Sep-2003.)

Theoremsstri 3109 Subclass transitivity inference. (Contributed by NM, 5-May-2000.)

Theoremsstrd 3110 Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.)

Theoremsstrid 3111 Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.)

Theoremsstrdi 3112 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremsylan9ss 3113 A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Theoremsylan9ssr 3114 A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)

Theoremeqss 3115 The subclass relationship is antisymmetric. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 5-Aug-1993.)

Theoremeqssi 3116 Infer equality from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 9-Sep-1993.)

Theoremeqssd 3117 Equality deduction from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 27-Jun-2004.)

Theoremeqrd 3118 Deduce equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 21-Mar-2017.)

Theoremeqelssd 3119* Equality deduction from subclass relationship and membership. (Contributed by AV, 21-Aug-2022.)

Theoremssid 3120 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.)

Theoremssidd 3121 Weakening of ssid 3120. (Contributed by BJ, 1-Sep-2022.)

Theoremssv 3122 Any class is a subclass of the universal class. (Contributed by NM, 31-Oct-1995.)

Theoremsseq1 3123 Equality theorem for subclasses. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)

Theoremsseq2 3124 Equality theorem for the subclass relationship. (Contributed by NM, 25-Jun-1998.)

Theoremsseq12 3125 Equality theorem for the subclass relationship. (Contributed by NM, 31-May-1999.)

Theoremsseq1i 3126 An equality inference for the subclass relationship. (Contributed by NM, 18-Aug-1993.)

Theoremsseq2i 3127 An equality inference for the subclass relationship. (Contributed by NM, 30-Aug-1993.)

Theoremsseq12i 3128 An equality inference for the subclass relationship. (Contributed by NM, 31-May-1999.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Theoremsseq1d 3129 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)

Theoremsseq2d 3130 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)

Theoremsseq12d 3131 An equality deduction for the subclass relationship. (Contributed by NM, 31-May-1999.)

Theoremeqsstri 3132 Substitution of equality into a subclass relationship. (Contributed by NM, 16-Jul-1995.)

Theoremeqsstrri 3133 Substitution of equality into a subclass relationship. (Contributed by NM, 19-Oct-1999.)

Theoremsseqtri 3134 Substitution of equality into a subclass relationship. (Contributed by NM, 28-Jul-1995.)

Theoremsseqtrri 3135 Substitution of equality into a subclass relationship. (Contributed by NM, 4-Apr-1995.)

Theoremeqsstrd 3136 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Theoremeqsstrrd 3137 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Theoremsseqtrd 3138 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Theoremsseqtrrd 3139 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Theorem3sstr3i 3140 Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Theorem3sstr4i 3141 Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Theorem3sstr3g 3142 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)

Theorem3sstr4g 3143 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Theorem3sstr3d 3144 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)

Theorem3sstr4d 3145 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 30-Nov-1995.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Theoremeqsstrid 3146 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)

Theoremeqsstrrid 3147 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)

Theoremsseqtrdi 3148 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)

Theoremsseqtrrdi 3149 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)

Theoremsseqtrid 3150 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremsseqtrrid 3151 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremeqsstrdi 3152 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremeqsstrrdi 3153 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremeqimss 3154 Equality implies the subclass relation. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)

Theoremeqimss2 3155 Equality implies the subclass relation. (Contributed by NM, 23-Nov-2003.)

Theoremeqimssi 3156 Infer subclass relationship from equality. (Contributed by NM, 6-Jan-2007.)

Theoremeqimss2i 3157 Infer subclass relationship from equality. (Contributed by NM, 7-Jan-2007.)

Theoremnssne1 3158 Two classes are different if they don't include the same class. (Contributed by NM, 23-Apr-2015.)

Theoremnssne2 3159 Two classes are different if they are not subclasses of the same class. (Contributed by NM, 23-Apr-2015.)

Theoremnssr 3160* Negation of subclass relationship. One direction of Exercise 13 of [TakeutiZaring] p. 18. (Contributed by Jim Kingdon, 15-Jul-2018.)

Theoremnelss 3161 Demonstrate by witnesses that two classes lack a subclass relation. (Contributed by Stefan O'Rear, 5-Feb-2015.)

Theoremssrexf 3162 Restricted existential quantification follows from a subclass relationship. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremssrmof 3163 "At most one" existential quantification restricted to a subclass. (Contributed by Thierry Arnoux, 8-Oct-2017.)

Theoremssralv 3164* Quantification restricted to a subclass. (Contributed by NM, 11-Mar-2006.)

Theoremssrexv 3165* Existential quantification restricted to a subclass. (Contributed by NM, 11-Jan-2007.)

Theoremralss 3166* Restricted universal quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Theoremrexss 3167* Restricted existential quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Theoremss2ab 3168 Class abstractions in a subclass relationship. (Contributed by NM, 3-Jul-1994.)

Theoremabss 3169* Class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)

Theoremssab 3170* Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.)

Theoremssabral 3171* The relation for a subclass of a class abstraction is equivalent to restricted quantification. (Contributed by NM, 6-Sep-2006.)

Theoremss2abi 3172 Inference of abstraction subclass from implication. (Contributed by NM, 31-Mar-1995.)

Theoremss2abdv 3173* Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011.)

Theoremabssdv 3174* Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)

Theoremabssi 3175* Inference of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)

Theoremss2rab 3176 Restricted abstraction classes in a subclass relationship. (Contributed by NM, 30-May-1999.)

Theoremrabss 3177* Restricted class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)

Theoremssrab 3178* Subclass of a restricted class abstraction. (Contributed by NM, 16-Aug-2006.)

Theoremssrabdv 3179* Subclass of a restricted class abstraction (deduction form). (Contributed by NM, 31-Aug-2006.)

Theoremrabssdv 3180* Subclass of a restricted class abstraction (deduction form). (Contributed by NM, 2-Feb-2015.)

Theoremss2rabdv 3181* Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006.)

Theoremss2rabi 3182 Inference of restricted abstraction subclass from implication. (Contributed by NM, 14-Oct-1999.)

Theoremrabss2 3183* Subclass law for restricted abstraction. (Contributed by NM, 18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremssab2 3184* Subclass relation for the restriction of a class abstraction. (Contributed by NM, 31-Mar-1995.)

Theoremssrab2 3185* Subclass relation for a restricted class. (Contributed by NM, 19-Mar-1997.)

Theoremssrabeq 3186* If the restricting class of a restricted class abstraction is a subset of this restricted class abstraction, it is equal to this restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)

Theoremrabssab 3187 A restricted class is a subclass of the corresponding unrestricted class. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremuniiunlem 3188* A subset relationship useful for converting union to indexed union using dfiun2 or dfiun2g and intersection to indexed intersection using dfiin2 . (Contributed by NM, 5-Oct-2006.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)

2.1.13  The difference, union, and intersection of two classes

2.1.13.1  The difference of two classes

Theoremdfdif3 3189* Alternate definition of class difference. Definition of relative set complement in Section 2.3 of [Pierik], p. 10. (Contributed by BJ and Jim Kingdon, 16-Jun-2022.)

Theoremdifeq1 3190 Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdifeq2 3191 Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdifeq12 3192 Equality theorem for class difference. (Contributed by FL, 31-Aug-2009.)

Theoremdifeq1i 3193 Inference adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)

Theoremdifeq2i 3194 Inference adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)

Theoremdifeq12i 3195 Equality inference for class difference. (Contributed by NM, 29-Aug-2004.)

Theoremdifeq1d 3196 Deduction adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)

Theoremdifeq2d 3197 Deduction adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)

Theoremdifeq12d 3198 Equality deduction for class difference. (Contributed by FL, 29-May-2014.)

Theoremdifeqri 3199* Inference from membership to difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremnfdif 3200 Bound-variable hypothesis builder for class difference. (Contributed by NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)

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