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

Theoremsseldi 3101 Membership inference from subclass relationship. (Contributed by NM, 25-Jun-2014.)

Theoremsseld 3102 Membership deduction from subclass relationship. (Contributed by NM, 15-Nov-1995.)

Theoremsselda 3103 Membership deduction from subclass relationship. (Contributed by NM, 26-Jun-2014.)

Theoremsseldd 3104 Membership inference from subclass relationship. (Contributed by NM, 14-Dec-2004.)

Theoremssriv 3105* Inference rule based on subclass definition. (Contributed by NM, 5-Aug-1993.)

Theoremssrdv 3106* Deduction rule 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.)

Theoremsyl5ss 3111 Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.)

Theoremsyl6ss 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.)

Theoremssid 3118 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 3119 Any class is a subclass of the universal class. (Contributed by NM, 31-Oct-1995.)

Theoremsseq1 3120 Equality theorem for subclasses. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)

Theoremsseq2 3121 Equality theorem for the subclass relationship. (Contributed by NM, 25-Jun-1998.)

Theoremsseq12 3122 Equality theorem for the subclass relationship. (Contributed by NM, 31-May-1999.)

Theoremsseq1i 3123 An equality inference for the subclass relationship. (Contributed by NM, 18-Aug-1993.)

Theoremsseq2i 3124 An equality inference for the subclass relationship. (Contributed by NM, 30-Aug-1993.)

Theoremsseq12i 3125 An equality inference for the subclass relationship. (Contributed by NM, 31-May-1999.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Theoremsseq1d 3126 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)

Theoremsseq2d 3127 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)

Theoremsseq12d 3128 An equality deduction for the subclass relationship. (Contributed by NM, 31-May-1999.)

Theoremeqsstri 3129 Substitution of equality into a subclass relationship. (Contributed by NM, 16-Jul-1995.)

Theoremeqsstr3i 3130 Substitution of equality into a subclass relationship. (Contributed by NM, 19-Oct-1999.)

Theoremsseqtri 3131 Substitution of equality into a subclass relationship. (Contributed by NM, 28-Jul-1995.)

Theoremsseqtr4i 3132 Substitution of equality into a subclass relationship. (Contributed by NM, 4-Apr-1995.)

Theoremeqsstrd 3133 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Theoremeqsstr3d 3134 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Theoremsseqtrd 3135 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Theoremsseqtr4d 3136 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Theorem3sstr3i 3137 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 3138 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 3139 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)

Theorem3sstr4g 3140 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 3141 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)

Theorem3sstr4d 3142 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 30-Nov-1995.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Theoremsyl5eqss 3143 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)

Theoremsyl5eqssr 3144 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)

Theoremsyl6sseq 3145 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)

Theoremsyl6sseqr 3146 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)

Theoremsyl5sseq 3147 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremsyl5sseqr 3148 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremsyl6eqss 3149 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremsyl6eqssr 3150 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremeqimss 3151 Equality implies the subclass relation. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)

Theoremeqimss2 3152 Equality implies the subclass relation. (Contributed by NM, 23-Nov-2003.)

Theoremeqimssi 3153 Infer subclass relationship from equality. (Contributed by NM, 6-Jan-2007.)

Theoremeqimss2i 3154 Infer subclass relationship from equality. (Contributed by NM, 7-Jan-2007.)

Theoremnssne1 3155 Two classes are different if they don't include the same class. (Contributed by NM, 23-Apr-2015.)

Theoremnssne2 3156 Two classes are different if they are not subclasses of the same class. (Contributed by NM, 23-Apr-2015.)

Theoremnss 3157* Negation of subclass relationship. Exercise 13 of [TakeutiZaring] p. 18. (Contributed by NM, 25-Feb-1996.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)

Theoremssralv 3158* Quantification restricted to a subclass. (Contributed by NM, 11-Mar-2006.)

Theoremssrexv 3159* Existential quantification restricted to a subclass. (Contributed by NM, 11-Jan-2007.)

Theoremralss 3160* Restricted universal quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Theoremrexss 3161* Restricted existential quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Theoremss2ab 3162 Class abstractions in a subclass relationship. (Contributed by NM, 3-Jul-1994.)

Theoremabss 3163* Class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)

Theoremssab 3164* Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.)

Theoremssabral 3165* The relation for a subclass of a class abstraction is equivalent to restricted quantification. (Contributed by NM, 6-Sep-2006.)

Theoremss2abi 3166 Inference of abstraction subclass from implication. (Contributed by NM, 31-Mar-1995.)

Theoremss2abdv 3167* Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011.)

Theoremabssdv 3168* Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)

Theoremabssi 3169* Inference of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)

Theoremss2rab 3170 Restricted abstraction classes in a subclass relationship. (Contributed by NM, 30-May-1999.)

Theoremrabss 3171* Restricted class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)

Theoremssrab 3172* Subclass of a restricted class abstraction. (Contributed by NM, 16-Aug-2006.)

Theoremssrabdv 3173* Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 31-Aug-2006.)

Theoremrabssdv 3174* Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 2-Feb-2015.)

Theoremss2rabdv 3175* Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006.)

Theoremss2rabi 3176 Inference of restricted abstraction subclass from implication. (Contributed by NM, 14-Oct-1999.)

Theoremrabss2 3177* Subclass law for restricted abstraction. (Contributed by NM, 18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremssab2 3178* Subclass relation for the restriction of a class abstraction. (Contributed by NM, 31-Mar-1995.)

Theoremssrab2 3179* Subclass relation for a restricted class. (Contributed by NM, 19-Mar-1997.)

Theoremrabssab 3180 A restricted class is a subclass of the corresponding unrestricted class. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremuniiunlem 3181* A subset relationship useful for converting union to indexed union using dfiun2 3835 or dfiun2g 3833 and intersection to indexed intersection using dfiin2 3836. (Contributed by NM, 5-Oct-2006.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)

Theoremdfpss2 3182 Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.)

Theoremdfpss3 3183 Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theorempsseq1 3184 Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)

Theorempsseq2 3185 Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)

Theorempsseq1i 3186 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)

Theorempsseq2i 3187 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)

Theorempsseq12i 3188 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)

Theorempsseq1d 3189 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)

Theorempsseq2d 3190 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)

Theorempsseq12d 3191 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)

Theorempssss 3192 A proper subclass is a subclass. Theorem 10 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)

Theorempssne 3193 Two classes in a proper subclass relationship are not equal. (Contributed by NM, 16-Feb-2015.)

Theorempssssd 3194 Deduce subclass from proper subclass. (Contributed by NM, 29-Feb-1996.)

Theoremsspss 3195 Subclass in terms of proper subclass. (Contributed by NM, 25-Feb-1996.)

Theorempssirr 3196 Proper subclass is irreflexive. Theorem 7 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)

Theorempssn2lp 3197 Proper subclass has no 2-cycle loops. Compare Theorem 8 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremsspsstri 3198 Two ways of stating trichotomy with respect to inclusion. (Contributed by NM, 12-Aug-2004.)

Theoremssnpss 3199 Partial trichotomy law for subclasses. (Contributed by NM, 16-May-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theorempsstr 3200 Transitive law for proper subclass. Theorem 9 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)

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