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Theorem List for Intuitionistic Logic Explorer - 3101-3200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremssun2 3101 Subclass relationship for union of classes. (Contributed by NM, 30-Aug-1993.)
 C_  u.
 
Theoremssun3 3102 Subclass law for union of classes. (Contributed by NM, 5-Aug-1993.)
 C_  C_  u.  C
 
Theoremssun4 3103 Subclass law for union of classes. (Contributed by NM, 14-Aug-1994.)
 C_  C_  C  u.
 
Theoremelun1 3104 Membership law for union of classes. (Contributed by NM, 5-Aug-1993.)
 u.  C
 
Theoremelun2 3105 Membership law for union of classes. (Contributed by NM, 30-Aug-1993.)
 C  u.
 
Theoremunss1 3106 Subclass law for union of classes. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 C_  u.  C  C_  u.  C
 
Theoremssequn1 3107 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.)
 C_  u.
 
Theoremunss2 3108 Subclass law for union of classes. Exercise 7 of [TakeutiZaring] p. 18. (Contributed by NM, 14-Oct-1999.)
 C_  C  u.  C_  C  u.
 
Theoremunss12 3109 Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.)
 C_  C  C_  D  u.  C  C_  u.  D
 
Theoremssequn2 3110 A relationship between subclass and union. (Contributed by NM, 13-Jun-1994.)
 C_  u.
 
Theoremunss 3111 The union of two subclasses is a subclass. Theorem 27 of [Suppes] p. 27 and its converse. (Contributed by NM, 11-Jun-2004.)
 C_  C  C_  C  u.  C_  C
 
Theoremunssi 3112 An inference showing the union of two subclasses is a subclass. (Contributed by Raph Levien, 10-Dec-2002.)
 C_  C   &     C_  C   =>     u.  C_  C
 
Theoremunssd 3113 A deduction showing the union of two subclasses is a subclass. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 C_  C   &     C_  C   =>     u.  C_  C
 
Theoremunssad 3114 If  u. is contained in  C, so is . One-way deduction form of unss 3111. Partial converse of unssd 3113. (Contributed by David Moews, 1-May-2017.)
 u.  C_  C   =>     C_  C
 
Theoremunssbd 3115 If  u. is contained in  C, so is . One-way deduction form of unss 3111. Partial converse of unssd 3113. (Contributed by David Moews, 1-May-2017.)
 u.  C_  C   =>     C_  C
 
Theoremssun 3116 A condition that implies inclusion in the union of two classes. (Contributed by NM, 23-Nov-2003.)
 C_  C_  C  C_  u.  C
 
Theoremrexun 3117 Restricted existential quantification over union. (Contributed by Jeff Madsen, 5-Jan-2011.)
 u.
 
Theoremralunb 3118 Restricted quantification over a union. (Contributed by Scott Fenton, 12-Apr-2011.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 u.
 
Theoremralun 3119 Restricted quantification over union. (Contributed by Jeff Madsen, 2-Sep-2009.)
 u.
 
2.1.13.3  The intersection of two classes
 
Theoremelin 3120 Expansion of membership in an intersection of two classes. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 29-Apr-1994.)
 i^i  C  C
 
Theoremelin2 3121 Membership in a class defined as an intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 X 
 i^i  C   =>     X  C
 
Theoremelin3 3122 Membership in a class defined as a ternary intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 X  i^i  C 
 i^i  D   =>     X  C  D
 
Theoremincom 3123 Commutative law for intersection of classes. Exercise 7 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.)
 i^i  i^i
 
Theoremineqri 3124* Inference from membership to intersection. (Contributed by NM, 5-Aug-1993.)
 C   =>     i^i  C
 
Theoremineq1 3125 Equality theorem for intersection of two classes. (Contributed by NM, 14-Dec-1993.)
 i^i  C  i^i  C
 
Theoremineq2 3126 Equality theorem for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
 C  i^i  C  i^i
 
Theoremineq12 3127 Equality theorem for intersection of two classes. (Contributed by NM, 8-May-1994.)
 C  D  i^i  C 
 i^i  D
 
Theoremineq1i 3128 Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
   =>     i^i  C  i^i  C
 
Theoremineq2i 3129 Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
   =>     C  i^i  C  i^i
 
Theoremineq12i 3130 Equality inference for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
   &     C  D   =>     i^i  C  i^i  D
 
Theoremineq1d 3131 Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)
   =>     i^i  C  i^i  C
 
Theoremineq2d 3132 Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)
   =>     C  i^i  C  i^i
 
Theoremineq12d 3133 Equality deduction for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
   &     C  D   =>     i^i  C  i^i  D
 
Theoremineqan12d 3134 Equality deduction for intersection of two classes. (Contributed by NM, 7-Feb-2007.)
   &     C  D   =>     i^i  C 
 i^i  D
 
Theoremdfss1 3135 A frequently-used variant of subclass definition df-ss 2925. (Contributed by NM, 10-Jan-2015.)
 C_  i^i
 
Theoremdfss5 3136 Another definition of subclasshood. Similar to df-ss 2925, dfss 2926, and dfss1 3135. (Contributed by David Moews, 1-May-2017.)
 C_  i^i
 
Theoremnfin 3137 Bound-variable hypothesis builder for the intersection of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)
 F/_   &     F/_   =>     F/_ 
 i^i
 
Theoremcsbing 3138 Distribute proper substitution through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.)
 [_  ]_ C  i^i  D  [_  ]_ C  i^i  [_  ]_ D
 
Theoremrabbi2dva 3139* Deduction from a wff to a restricted class abstraction. (Contributed by NM, 14-Jan-2014.)
   =>     i^i 
 {  |  }
 
Theoreminidm 3140 Idempotent law for intersection of classes. Theorem 15 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.)
 i^i
 
Theoreminass 3141 Associative law for intersection of classes. Exercise 9 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.)
 i^i  i^i  C  i^i  i^i  C
 
Theoremin12 3142 A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.)
 i^i  i^i  C  i^i  i^i  C
 
Theoremin32 3143 A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 i^i  i^i  C 
 i^i  C  i^i
 
Theoremin13 3144 A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
 i^i  i^i  C  C  i^i  i^i
 
Theoremin31 3145 A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
 i^i  i^i  C  C  i^i  i^i
 
Theoreminrot 3146 Rotate the intersection of 3 classes. (Contributed by NM, 27-Aug-2012.)
 i^i  i^i  C  C  i^i  i^i
 
Theoremin4 3147 Rearrangement of intersection of 4 classes. (Contributed by NM, 21-Apr-2001.)
 i^i  i^i  C  i^i  D 
 i^i  C  i^i  i^i  D
 
Theoreminindi 3148 Intersection distributes over itself. (Contributed by NM, 6-May-1994.)
 i^i  i^i  C 
 i^i  i^i  i^i  C
 
Theoreminindir 3149 Intersection distributes over itself. (Contributed by NM, 17-Aug-2004.)
 i^i  i^i  C 
 i^i  C  i^i  i^i  C
 
Theoremsseqin2 3150 A relationship between subclass and intersection. Similar to Exercise 9 of [TakeutiZaring] p. 18. (Contributed by NM, 17-May-1994.)
 C_  i^i
 
Theoreminss1 3151 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.)
 i^i  C_
 
Theoreminss2 3152 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.)
 i^i  C_
 
Theoremssin 3153 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.)
 C_  C_  C  C_  i^i  C
 
Theoremssini 3154 An inference showing that a subclass of two classes is a subclass of their intersection. (Contributed by NM, 24-Nov-2003.)
 C_    &     C_  C   =>     C_  i^i  C
 
Theoremssind 3155 A deduction showing that a subclass of two classes is a subclass of their intersection. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 C_    &     C_  C   =>     C_  i^i  C
 
Theoremssrin 3156 Add right intersection to subclass relation. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 C_  i^i  C  C_  i^i  C
 
Theoremsslin 3157 Add left intersection to subclass relation. (Contributed by NM, 19-Oct-1999.)
 C_  C  i^i  C_  C  i^i
 
Theoremss2in 3158 Intersection of subclasses. (Contributed by NM, 5-May-2000.)
 C_  C  C_  D  i^i  C  C_  i^i  D
 
Theoremssinss1 3159 Intersection preserves subclass relationship. (Contributed by NM, 14-Sep-1999.)
 C_  C  i^i  C_  C
 
Theoreminss 3160 Inclusion of an intersection of two classes. (Contributed by NM, 30-Oct-2014.)
 C_  C  C_  C  i^i  C_  C
 
2.1.13.4  Combinations of difference, union, and intersection of two classes
 
Theoremunabs 3161 Absorption law for union. (Contributed by NM, 16-Apr-2006.)
 u.  i^i
 
Theoreminabs 3162 Absorption law for intersection. (Contributed by NM, 16-Apr-2006.)
 i^i  u.
 
Theoremnssinpss 3163 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.)
 C_  i^i  C.
 
Theoremnsspssun 3164 Negation of subclass expressed in terms of proper subclass and union. (Contributed by NM, 15-Sep-2004.)
 C_  C.  u.
 
Theoremssddif 3165 Double complement and subset. Similar to ddifss 3169 but inside a class instead of the universal class  _V. In classical logic the subset operation on the right hand side could be an equality (that is,  C_  \  \ ). (Contributed by Jim Kingdon, 24-Jul-2018.)
 C_  C_  \  \
 
Theoremunssdif 3166 Union of two classes and class difference. In classical logic this would be an equality. (Contributed by Jim Kingdon, 24-Jul-2018.)
 u.  C_  _V  \  _V  \ 
 \
 
Theoreminssdif 3167 Intersection of two classes and class difference. In classical logic this would be an equality. (Contributed by Jim Kingdon, 24-Jul-2018.)
 i^i  C_  \  _V  \
 
Theoremdifin 3168 Difference with intersection. Theorem 33 of [Suppes] p. 29. (Contributed by NM, 31-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 \  i^i  \
 
Theoremddifss 3169 Double complement under universal class. In classical logic (or given an additional hypothesis, as in ddifnel 3069), this is equality rather than subset. (Contributed by Jim Kingdon, 24-Jul-2018.)
 C_  _V  \  _V  \
 
Theoremunssin 3170 Union as a subset of class complement and intersection (De Morgan's law). One direction of the definition of union in [Mendelson] p. 231. This would be an equality, rather than subset, in classical logic. (Contributed by Jim Kingdon, 25-Jul-2018.)
 u.  C_  _V  \  _V  \ 
 i^i  _V  \
 
Theoreminssun 3171 Intersection in terms of class difference and union (De Morgan's law). Similar to Exercise 4.10(n) of [Mendelson] p. 231. This would be an equality, rather than subset, in classical logic. (Contributed by Jim Kingdon, 25-Jul-2018.)
 i^i  C_  _V  \  _V  \  u.  _V  \
 
Theoreminssddif 3172 Intersection of two classes and class difference. In classical logic, such as Exercise 4.10(q) of [Mendelson] p. 231, this is an equality rather than subset. (Contributed by Jim Kingdon, 26-Jul-2018.)
 i^i  C_  \  \
 
Theoreminvdif 3173 Intersection with universal complement. Remark in [Stoll] p. 20. (Contributed by NM, 17-Aug-2004.)
 i^i  _V  \  \
 
Theoremindif 3174 Intersection with class difference. Theorem 34 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
 i^i  \  \
 
Theoremindif2 3175 Bring an intersection in and out of a class difference. (Contributed by Jeff Hankins, 15-Jul-2009.)
 i^i  \  C 
 i^i  \  C
 
Theoremindif1 3176 Bring an intersection in and out of a class difference. (Contributed by Mario Carneiro, 15-May-2015.)
 \  C  i^i 
 i^i  \  C
 
Theoremindifcom 3177 Commutation law for intersection and difference. (Contributed by Scott Fenton, 18-Feb-2013.)
 i^i  \  C  i^i  \  C
 
Theoremindi 3178 Distributive law for intersection over union. Exercise 10 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 i^i  u.  C 
 i^i  u.  i^i  C
 
Theoremundi 3179 Distributive law for union over intersection. Exercise 11 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 u.  i^i  C  u.  i^i  u.  C
 
Theoremindir 3180 Distributive law for intersection over union. Theorem 28 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
 u.  i^i  C 
 i^i  C  u.  i^i  C
 
Theoremundir 3181 Distributive law for union over intersection. Theorem 29 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
 i^i  u.  C  u.  C  i^i  u.  C
 
Theoremuneqin 3182 Equality of union and intersection implies equality of their arguments. (Contributed by NM, 16-Apr-2006.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 u.  i^i
 
Theoremdifundi 3183 Distributive law for class difference. Theorem 39 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
 \  u.  C 
 \  i^i  \  C
 
Theoremdifundir 3184 Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
 u.  \  C 
 \  C  u.  \  C
 
Theoremdifindiss 3185 Distributive law for class difference. In classical logic, for example, theorem 40 of [Suppes] p. 29, this is an equality instead of subset. (Contributed by Jim Kingdon, 26-Jul-2018.)
 \  u.  \  C  C_  \  i^i  C
 
Theoremdifindir 3186 Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
 i^i  \  C 
 \  C  i^i  \  C
 
Theoremindifdir 3187 Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011.)
 \  i^i  C 
 i^i  C  \  i^i  C
 
Theoremdifdif2ss 3188 Set difference with a set difference. In classical logic this would be equality rather than subset. (Contributed by Jim Kingdon, 27-Jul-2018.)
 \  u.  i^i  C  C_  \  \  C
 
Theoremundm 3189 De Morgan's law for union. Theorem 5.2(13) of [Stoll] p. 19. (Contributed by NM, 18-Aug-2004.)
 _V  \  u.  _V  \  i^i  _V  \
 
Theoremindmss 3190 De Morgan's law for intersection. In classical logic, this would be equality rather than subset, as in Theorem 5.2(13') of [Stoll] p. 19. (Contributed by Jim Kingdon, 27-Jul-2018.)
 _V  \  u.  _V  \  C_  _V  \  i^i
 
Theoremdifun1 3191 A relationship involving double difference and union. (Contributed by NM, 29-Aug-2004.)
 \  u.  C 
 \  \  C
 
Theoremundif3ss 3192 A subset relationship involving class union and class difference. In classical logic, this would be equality rather than subset, as in the first equality of Exercise 13 of [TakeutiZaring] p. 22. (Contributed by Jim Kingdon, 28-Jul-2018.)
 u.  \  C  C_  u.  \  C 
 \
 
Theoremdifin2 3193 Represent a set difference as an intersection with a larger difference. (Contributed by Jeff Madsen, 2-Sep-2009.)
 C_  C  \  C  \  i^i
 
Theoremdif32 3194 Swap second and third argument of double difference. (Contributed by NM, 18-Aug-2004.)
 \  \  C 
 \  C  \
 
Theoremdifabs 3195 Absorption-like law for class difference: you can remove a class only once. (Contributed by FL, 2-Aug-2009.)
 \  \  \
 
Theoremsymdif1 3196 Two ways to express symmetric difference. This theorem shows the equivalence of the definition of symmetric difference in [Stoll] p. 13 and the restated definition in Example 4.1 of [Stoll] p. 262. (Contributed by NM, 17-Aug-2004.)
 \  u.  \  u.  \  i^i
 
2.1.13.5  Class abstractions with difference, union, and intersection of two classes
 
Theoremsymdifxor 3197* Expressing symmetric difference with exclusive-or or two differences. (Contributed by Jim Kingdon, 28-Jul-2018.)
 \  u.  \  {  |  \/_  }
 
Theoremunab 3198 Union of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 {  | 
 }  u.  {  |  } 
 {  |  }
 
Theoreminab 3199 Intersection of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 {  | 
 }  i^i  {  |  } 
 {  |  }
 
Theoremdifab 3200 Difference of two class abstractions. (Contributed by NM, 23-Oct-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 {  | 
 }  \  {  |  } 
 {  |  }
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