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

Theoremdifss2d 3101 If a class is contained in a difference, it is contained in the minuend. Deduction form of difss2 3100. (Contributed by David Moews, 1-May-2017.)

Theoremssdifss 3102 Preservation of a subclass relationship by class difference. (Contributed by NM, 15-Feb-2007.)

Theoremddifnel 3103* Double complement under universal class. The hypothesis is one way of expressing the idea that membership in is decidable. 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 3203. (Contributed by Jim Kingdon, 21-Jul-2018.)

Theoremssconb 3104 Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)

Theoremsscon 3105 Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22. (Contributed by NM, 22-Mar-1998.)

Theoremssdif 3106 Difference law for subsets. (Contributed by NM, 28-May-1998.)

Theoremssdifd 3107 If is contained in , then is contained in . Deduction form of ssdif 3106. (Contributed by David Moews, 1-May-2017.)

Theoremsscond 3108 If is contained in , then is contained in . Deduction form of sscon 3105. (Contributed by David Moews, 1-May-2017.)

Theoremssdifssd 3109 If is contained in , then is also contained in . Deduction form of ssdifss 3102. (Contributed by David Moews, 1-May-2017.)

Theoremssdif2d 3110 If is contained in and is contained in , then is contained in . Deduction form. (Contributed by David Moews, 1-May-2017.)

Theoremraldifb 3111 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 3112 Expansion of membership in class union. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 7-Aug-1994.)

Theoremuneqri 3113* Inference from membership to union. (Contributed by NM, 5-Aug-1993.)

Theoremunidm 3114 Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)

Theoremuncom 3115 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 3116 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 3117 Inference form of equncom 3116. (Contributed by Alan Sare, 18-Feb-2012.)

Theoremuneq1 3118 Equality theorem for union of two classes. (Contributed by NM, 5-Aug-1993.)

Theoremuneq2 3119 Equality theorem for the union of two classes. (Contributed by NM, 5-Aug-1993.)

Theoremuneq12 3120 Equality theorem for union of two classes. (Contributed by NM, 29-Mar-1998.)

Theoremuneq1i 3121 Inference adding union to the right in a class equality. (Contributed by NM, 30-Aug-1993.)

Theoremuneq2i 3122 Inference adding union to the left in a class equality. (Contributed by NM, 30-Aug-1993.)

Theoremuneq12i 3123 Equality inference for union of two classes. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Theoremuneq1d 3124 Deduction adding union to the right in a class equality. (Contributed by NM, 29-Mar-1998.)

Theoremuneq2d 3125 Deduction adding union to the left in a class equality. (Contributed by NM, 29-Mar-1998.)

Theoremuneq12d 3126 Equality deduction for union of two classes. (Contributed by NM, 29-Sep-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremnfun 3127 Bound-variable hypothesis builder for the union of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)

Theoremunass 3128 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 3129 A rearrangement of union. (Contributed by NM, 12-Aug-2004.)

Theoremun23 3130 A rearrangement of union. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremun4 3131 A rearrangement of the union of 4 classes. (Contributed by NM, 12-Aug-2004.)

Theoremunundi 3132 Union distributes over itself. (Contributed by NM, 17-Aug-2004.)

Theoremunundir 3133 Union distributes over itself. (Contributed by NM, 17-Aug-2004.)

Theoremssun1 3134 Subclass relationship for union of classes. Theorem 25 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)

Theoremssun2 3135 Subclass relationship for union of classes. (Contributed by NM, 30-Aug-1993.)

Theoremssun3 3136 Subclass law for union of classes. (Contributed by NM, 5-Aug-1993.)

Theoremssun4 3137 Subclass law for union of classes. (Contributed by NM, 14-Aug-1994.)

Theoremelun1 3138 Membership law for union of classes. (Contributed by NM, 5-Aug-1993.)

Theoremelun2 3139 Membership law for union of classes. (Contributed by NM, 30-Aug-1993.)

Theoremunss1 3140 Subclass law for union of classes. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremssequn1 3141 A relationship between subclass and union. Theorem 26 of [Suppes] p. 27. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremunss2 3142 Subclass law for union of classes. Exercise 7 of [TakeutiZaring] p. 18. (Contributed by NM, 14-Oct-1999.)

Theoremunss12 3143 Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.)

Theoremssequn2 3144 A relationship between subclass and union. (Contributed by NM, 13-Jun-1994.)

Theoremunss 3145 The union of two subclasses is a subclass. Theorem 27 of [Suppes] p. 27 and its converse. (Contributed by NM, 11-Jun-2004.)

Theoremunssi 3146 An inference showing the union of two subclasses is a subclass. (Contributed by Raph Levien, 10-Dec-2002.)

Theoremunssd 3147 A deduction showing the union of two subclasses is a subclass. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremunssad 3148 If is contained in , so is . One-way deduction form of unss 3145. Partial converse of unssd 3147. (Contributed by David Moews, 1-May-2017.)

Theoremunssbd 3149 If is contained in , so is . One-way deduction form of unss 3145. Partial converse of unssd 3147. (Contributed by David Moews, 1-May-2017.)

Theoremssun 3150 A condition that implies inclusion in the union of two classes. (Contributed by NM, 23-Nov-2003.)

Theoremrexun 3151 Restricted existential quantification over union. (Contributed by Jeff Madsen, 5-Jan-2011.)

Theoremralunb 3152 Restricted quantification over a union. (Contributed by Scott Fenton, 12-Apr-2011.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremralun 3153 Restricted quantification over union. (Contributed by Jeff Madsen, 2-Sep-2009.)

2.1.13.3  The intersection of two classes

Theoremelin 3154 Expansion of membership in an intersection of two classes. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 29-Apr-1994.)

Theoremelin2 3155 Membership in a class defined as an intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)

Theoremelin3 3156 Membership in a class defined as a ternary intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)

Theoremincom 3157 Commutative law for intersection of classes. Exercise 7 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.)

Theoremineqri 3158* Inference from membership to intersection. (Contributed by NM, 5-Aug-1993.)

Theoremineq1 3159 Equality theorem for intersection of two classes. (Contributed by NM, 14-Dec-1993.)

Theoremineq2 3160 Equality theorem for intersection of two classes. (Contributed by NM, 26-Dec-1993.)

Theoremineq12 3161 Equality theorem for intersection of two classes. (Contributed by NM, 8-May-1994.)

Theoremineq1i 3162 Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)

Theoremineq2i 3163 Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)

Theoremineq12i 3164 Equality inference for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Theoremineq1d 3165 Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)

Theoremineq2d 3166 Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)

Theoremineq12d 3167 Equality deduction for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremineqan12d 3168 Equality deduction for intersection of two classes. (Contributed by NM, 7-Feb-2007.)

Theoremdfss1 3169 A frequently-used variant of subclass definition df-ss 2959. (Contributed by NM, 10-Jan-2015.)

Theoremdfss5 3170 Another definition of subclasshood. Similar to df-ss 2959, dfss 2960, and dfss1 3169. (Contributed by David Moews, 1-May-2017.)

Theoremnfin 3171 Bound-variable hypothesis builder for the intersection of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)

Theoremcsbing 3172 Distribute proper substitution through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.)

Theoremrabbi2dva 3173* Deduction from a wff to a restricted class abstraction. (Contributed by NM, 14-Jan-2014.)

Theoreminidm 3174 Idempotent law for intersection of classes. Theorem 15 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.)

Theoreminass 3175 Associative law for intersection of classes. Exercise 9 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.)

Theoremin12 3176 A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.)

Theoremin32 3177 A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremin13 3178 A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)

Theoremin31 3179 A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)

Theoreminrot 3180 Rotate the intersection of 3 classes. (Contributed by NM, 27-Aug-2012.)

Theoremin4 3181 Rearrangement of intersection of 4 classes. (Contributed by NM, 21-Apr-2001.)

Theoreminindi 3182 Intersection distributes over itself. (Contributed by NM, 6-May-1994.)

Theoreminindir 3183 Intersection distributes over itself. (Contributed by NM, 17-Aug-2004.)

Theoremsseqin2 3184 A relationship between subclass and intersection. Similar to Exercise 9 of [TakeutiZaring] p. 18. (Contributed by NM, 17-May-1994.)

Theoreminss1 3185 The intersection of two classes is a subset of one of them. Part of Exercise 12 of [TakeutiZaring] p. 18. (Contributed by NM, 27-Apr-1994.)

Theoreminss2 3186 The intersection of two classes is a subset of one of them. Part of Exercise 12 of [TakeutiZaring] p. 18. (Contributed by NM, 27-Apr-1994.)

Theoremssin 3187 Subclass of intersection. Theorem 2.8(vii) of [Monk1] p. 26. (Contributed by NM, 15-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremssini 3188 An inference showing that a subclass of two classes is a subclass of their intersection. (Contributed by NM, 24-Nov-2003.)

Theoremssind 3189 A deduction showing that a subclass of two classes is a subclass of their intersection. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Theoremssrin 3190 Add right intersection to subclass relation. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremsslin 3191 Add left intersection to subclass relation. (Contributed by NM, 19-Oct-1999.)

Theoremss2in 3192 Intersection of subclasses. (Contributed by NM, 5-May-2000.)

Theoremssinss1 3193 Intersection preserves subclass relationship. (Contributed by NM, 14-Sep-1999.)

Theoreminss 3194 Inclusion of an intersection of two classes. (Contributed by NM, 30-Oct-2014.)

2.1.13.4  Combinations of difference, union, and intersection of two classes

Theoremunabs 3195 Absorption law for union. (Contributed by NM, 16-Apr-2006.)

Theoreminabs 3196 Absorption law for intersection. (Contributed by NM, 16-Apr-2006.)

Theoremnssinpss 3197 Negation of subclass expressed in terms of intersection and proper subclass. (Contributed by NM, 30-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Theoremnsspssun 3198 Negation of subclass expressed in terms of proper subclass and union. (Contributed by NM, 15-Sep-2004.)

Theoremssddif 3199 Double complement and subset. Similar to ddifss 3203 but inside a class instead of the universal class . In classical logic the subset operation on the right hand side could be an equality (that is, ). (Contributed by Jim Kingdon, 24-Jul-2018.)

Theoremunssdif 3200 Union of two classes and class difference. In classical logic this would be an equality. (Contributed by Jim Kingdon, 24-Jul-2018.)

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