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

Theorem3sstr3i 3101 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 3102 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 3103 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)

Theorem3sstr4g 3104 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 3105 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)

Theorem3sstr4d 3106 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 3107 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)

Theoremsyl5eqssr 3108 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)

Theoremsyl6sseq 3109 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)

Theoremsyl6sseqr 3110 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)

Theoremsseqtrid 3111 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

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

Theoremsyl6eqss 3113 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremsyl6eqssr 3114 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremeqimss 3115 Equality implies the subclass relation. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)

Theoremeqimss2 3116 Equality implies the subclass relation. (Contributed by NM, 23-Nov-2003.)

Theoremeqimssi 3117 Infer subclass relationship from equality. (Contributed by NM, 6-Jan-2007.)

Theoremeqimss2i 3118 Infer subclass relationship from equality. (Contributed by NM, 7-Jan-2007.)

Theoremnssne1 3119 Two classes are different if they don't include the same class. (Contributed by NM, 23-Apr-2015.)

Theoremnssne2 3120 Two classes are different if they are not subclasses of the same class. (Contributed by NM, 23-Apr-2015.)

Theoremnssr 3121* Negation of subclass relationship. One direction of Exercise 13 of [TakeutiZaring] p. 18. (Contributed by Jim Kingdon, 15-Jul-2018.)

Theoremnelss 3122 Demonstrate by witnesses that two classes lack a subclass relation. (Contributed by Stefan O'Rear, 5-Feb-2015.)

Theoremssrexf 3123 Restricted existential quantification follows from a subclass relationship. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremssrmof 3124 "At most one" existential quantification restricted to a subclass. (Contributed by Thierry Arnoux, 8-Oct-2017.)

Theoremssralv 3125* Quantification restricted to a subclass. (Contributed by NM, 11-Mar-2006.)

Theoremssrexv 3126* Existential quantification restricted to a subclass. (Contributed by NM, 11-Jan-2007.)

Theoremralss 3127* Restricted universal quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Theoremrexss 3128* Restricted existential quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Theoremss2ab 3129 Class abstractions in a subclass relationship. (Contributed by NM, 3-Jul-1994.)

Theoremabss 3130* Class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)

Theoremssab 3131* Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.)

Theoremssabral 3132* The relation for a subclass of a class abstraction is equivalent to restricted quantification. (Contributed by NM, 6-Sep-2006.)

Theoremss2abi 3133 Inference of abstraction subclass from implication. (Contributed by NM, 31-Mar-1995.)

Theoremss2abdv 3134* Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011.)

Theoremabssdv 3135* Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)

Theoremabssi 3136* Inference of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)

Theoremss2rab 3137 Restricted abstraction classes in a subclass relationship. (Contributed by NM, 30-May-1999.)

Theoremrabss 3138* Restricted class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)

Theoremssrab 3139* Subclass of a restricted class abstraction. (Contributed by NM, 16-Aug-2006.)

Theoremssrabdv 3140* Subclass of a restricted class abstraction (deduction form). (Contributed by NM, 31-Aug-2006.)

Theoremrabssdv 3141* Subclass of a restricted class abstraction (deduction form). (Contributed by NM, 2-Feb-2015.)

Theoremss2rabdv 3142* Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006.)

Theoremss2rabi 3143 Inference of restricted abstraction subclass from implication. (Contributed by NM, 14-Oct-1999.)

Theoremrabss2 3144* Subclass law for restricted abstraction. (Contributed by NM, 18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremssab2 3145* Subclass relation for the restriction of a class abstraction. (Contributed by NM, 31-Mar-1995.)

Theoremssrab2 3146* Subclass relation for a restricted class. (Contributed by NM, 19-Mar-1997.)

Theoremssrabeq 3147* 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 3148 A restricted class is a subclass of the corresponding unrestricted class. (Contributed by Mario Carneiro, 23-Dec-2016.)

Theoremuniiunlem 3149* 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 3150* 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 3151 Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdifeq2 3152 Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremdifeq12 3153 Equality theorem for class difference. (Contributed by FL, 31-Aug-2009.)

Theoremdifeq1i 3154 Inference adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)

Theoremdifeq2i 3155 Inference adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)

Theoremdifeq12i 3156 Equality inference for class difference. (Contributed by NM, 29-Aug-2004.)

Theoremdifeq1d 3157 Deduction adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)

Theoremdifeq2d 3158 Deduction adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)

Theoremdifeq12d 3159 Equality deduction for class difference. (Contributed by FL, 29-May-2014.)

Theoremdifeqri 3160* Inference from membership to difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremnfdif 3161 Bound-variable hypothesis builder for class difference. (Contributed by NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)

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

Theoremeldifn 3163 Implication of membership in a class difference. (Contributed by NM, 3-May-1994.)

Theoremelndif 3164 A set does not belong to a class excluding it. (Contributed by NM, 27-Jun-1994.)

Theoremdifdif 3165 Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.)

Theoremdifss 3166 Subclass relationship for class difference. Exercise 14 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)

Theoremdifssd 3167 A difference of two classes is contained in the minuend. Deduction form of difss 3166. (Contributed by David Moews, 1-May-2017.)

Theoremdifss2 3168 If a class is contained in a difference, it is contained in the minuend. (Contributed by David Moews, 1-May-2017.)

Theoremdifss2d 3169 If a class is contained in a difference, it is contained in the minuend. Deduction form of difss2 3168. (Contributed by David Moews, 1-May-2017.)

Theoremssdifss 3170 Preservation of a subclass relationship by class difference. (Contributed by NM, 15-Feb-2007.)

Theoremddifnel 3171* Double complement under universal class. The hypothesis corresponds to stability of membership in , which is weaker than decidability (see dcstab 812). Actually, the conclusion is a characterization of stability of membership in a class (see ddifstab 3172) . Exercise 4.10(s) of [Mendelson] p. 231, but with an additional hypothesis. For a version without a hypothesis, but which only states that is a subset of , see ddifss 3278. (Contributed by Jim Kingdon, 21-Jul-2018.)

Theoremddifstab 3172* A class is equal to its double complement if and only if it is stable (that is, membership in it is a stable property). (Contributed by BJ, 12-Dec-2021.)
STAB

Theoremssconb 3173 Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)

Theoremsscon 3174 Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22. (Contributed by NM, 22-Mar-1998.)

Theoremssdif 3175 Difference law for subsets. (Contributed by NM, 28-May-1998.)

Theoremssdifd 3176 If is contained in , then is contained in . Deduction form of ssdif 3175. (Contributed by David Moews, 1-May-2017.)

Theoremsscond 3177 If is contained in , then is contained in . Deduction form of sscon 3174. (Contributed by David Moews, 1-May-2017.)

Theoremssdifssd 3178 If is contained in , then is also contained in . Deduction form of ssdifss 3170. (Contributed by David Moews, 1-May-2017.)

Theoremssdif2d 3179 If is contained in and is contained in , then is contained in . Deduction form. (Contributed by David Moews, 1-May-2017.)

Theoremraldifb 3180 Restricted universal quantification on a class difference in terms of an implication. (Contributed by Alexander van der Vekens, 3-Jan-2018.)

2.1.13.2  The union of two classes

Theoremelun 3181 Expansion of membership in class union. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 7-Aug-1994.)

Theoremuneqri 3182* Inference from membership to union. (Contributed by NM, 5-Aug-1993.)

Theoremunidm 3183 Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)

Theoremuncom 3184 Commutative law for union of classes. Exercise 6 of [TakeutiZaring] p. 17. (Contributed by NM, 25-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremequncom 3185 If a class equals the union of two other classes, then it equals the union of those two classes commuted. (Contributed by Alan Sare, 18-Feb-2012.)

Theoremequncomi 3186 Inference form of equncom 3185. (Contributed by Alan Sare, 18-Feb-2012.)

Theoremuneq1 3187 Equality theorem for union of two classes. (Contributed by NM, 5-Aug-1993.)

Theoremuneq2 3188 Equality theorem for the union of two classes. (Contributed by NM, 5-Aug-1993.)

Theoremuneq12 3189 Equality theorem for union of two classes. (Contributed by NM, 29-Mar-1998.)

Theoremuneq1i 3190 Inference adding union to the right in a class equality. (Contributed by NM, 30-Aug-1993.)

Theoremuneq2i 3191 Inference adding union to the left in a class equality. (Contributed by NM, 30-Aug-1993.)

Theoremuneq12i 3192 Equality inference for union of two classes. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Theoremuneq1d 3193 Deduction adding union to the right in a class equality. (Contributed by NM, 29-Mar-1998.)

Theoremuneq2d 3194 Deduction adding union to the left in a class equality. (Contributed by NM, 29-Mar-1998.)

Theoremuneq12d 3195 Equality deduction for union of two classes. (Contributed by NM, 29-Sep-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremnfun 3196 Bound-variable hypothesis builder for the union of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)

Theoremunass 3197 Associative law for union of classes. Exercise 8 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremun12 3198 A rearrangement of union. (Contributed by NM, 12-Aug-2004.)

Theoremun23 3199 A rearrangement of union. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremun4 3200 A rearrangement of the union of 4 classes. (Contributed by NM, 12-Aug-2004.)

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