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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.)
 |-  ( ph  ->  A  C_  ( B  \  C ) )   =>    |-  ( ph  ->  A  C_  B )
 
Theoremssdifss 3102 Preservation of a subclass relationship by class difference. (Contributed by NM, 15-Feb-2007.)
 |-  ( A  C_  B  ->  ( A  \  C )  C_  B )
 
Theoremddifnel 3103* Double complement under universal class. The hypothesis is one way of expressing the idea that membership in  A 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  A is a subset of  _V  \  ( _V  \  A ), see ddifss 3203. (Contributed by Jim Kingdon, 21-Jul-2018.)
 |-  ( -.  x  e.  ( _V  \  A )  ->  x  e.  A )   =>    |-  ( _V  \  ( _V  \  A ) )  =  A
 
Theoremssconb 3104 Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)
 |-  ( ( A  C_  C  /\  B  C_  C )  ->  ( A  C_  ( C  \  B )  <->  B  C_  ( C  \  A ) ) )
 
Theoremsscon 3105 Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22. (Contributed by NM, 22-Mar-1998.)
 |-  ( A  C_  B  ->  ( C  \  B )  C_  ( C  \  A ) )
 
Theoremssdif 3106 Difference law for subsets. (Contributed by NM, 28-May-1998.)
 |-  ( A  C_  B  ->  ( A  \  C )  C_  ( B  \  C ) )
 
Theoremssdifd 3107 If  A is contained in  B, then  ( A 
\  C ) is contained in  ( B  \  C ). Deduction form of ssdif 3106. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   =>    |-  ( ph  ->  ( A  \  C )  C_  ( B  \  C ) )
 
Theoremsscond 3108 If  A is contained in  B, then  ( C 
\  B ) is contained in  ( C  \  A ). Deduction form of sscon 3105. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   =>    |-  ( ph  ->  ( C  \  B )  C_  ( C  \  A ) )
 
Theoremssdifssd 3109 If  A is contained in  B, then  ( A 
\  C ) is also contained in  B. Deduction form of ssdifss 3102. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   =>    |-  ( ph  ->  ( A  \  C )  C_  B )
 
Theoremssdif2d 3110 If  A is contained in  B and  C is contained in  D, then  ( A  \  D ) is contained in  ( B  \  C ). Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  C 
 C_  D )   =>    |-  ( ph  ->  ( A  \  D ) 
 C_  ( B  \  C ) )
 
Theoremraldifb 3111 Restricted universal quantification on a class difference in terms of an implication. (Contributed by Alexander van der Vekens, 3-Jan-2018.)
 |-  ( A. x  e.  A  ( x  e/  B  ->  ph )  <->  A. x  e.  ( A  \  B ) ph )
 
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.)
 |-  ( A  e.  ( B  u.  C )  <->  ( A  e.  B  \/  A  e.  C ) )
 
Theoremuneqri 3113* Inference from membership to union. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( x  e.  A  \/  x  e.  B )  <->  x  e.  C )   =>    |-  ( A  u.  B )  =  C
 
Theoremunidm 3114 Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  u.  A )  =  A
 
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.)
 |-  ( A  u.  B )  =  ( B  u.  A )
 
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.)
 |-  ( A  =  ( B  u.  C )  <->  A  =  ( C  u.  B ) )
 
Theoremequncomi 3117 Inference form of equncom 3116. (Contributed by Alan Sare, 18-Feb-2012.)
 |-  A  =  ( B  u.  C )   =>    |-  A  =  ( C  u.  B )
 
Theoremuneq1 3118 Equality theorem for union of two classes. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  ->  ( A  u.  C )  =  ( B  u.  C ) )
 
Theoremuneq2 3119 Equality theorem for the union of two classes. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  ->  ( C  u.  A )  =  ( C  u.  B ) )
 
Theoremuneq12 3120 Equality theorem for union of two classes. (Contributed by NM, 29-Mar-1998.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  u.  C )  =  ( B  u.  D ) )
 
Theoremuneq1i 3121 Inference adding union to the right in a class equality. (Contributed by NM, 30-Aug-1993.)
 |-  A  =  B   =>    |-  ( A  u.  C )  =  ( B  u.  C )
 
Theoremuneq2i 3122 Inference adding union to the left in a class equality. (Contributed by NM, 30-Aug-1993.)
 |-  A  =  B   =>    |-  ( C  u.  A )  =  ( C  u.  B )
 
Theoremuneq12i 3123 Equality inference for union of two classes. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  u.  C )  =  ( B  u.  D )
 
Theoremuneq1d 3124 Deduction adding union to the right in a class equality. (Contributed by NM, 29-Mar-1998.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  u.  C )  =  ( B  u.  C ) )
 
Theoremuneq2d 3125 Deduction adding union to the left in a class equality. (Contributed by NM, 29-Mar-1998.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  u.  A )  =  ( C  u.  B ) )
 
Theoremuneq12d 3126 Equality deduction for union of two classes. (Contributed by NM, 29-Sep-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  u.  C )  =  ( B  u.  D ) )
 
Theoremnfun 3127 Bound-variable hypothesis builder for the union of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x ( A  u.  B )
 
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.)
 |-  ( ( A  u.  B )  u.  C )  =  ( A  u.  ( B  u.  C ) )
 
Theoremun12 3129 A rearrangement of union. (Contributed by NM, 12-Aug-2004.)
 |-  ( A  u.  ( B  u.  C ) )  =  ( B  u.  ( A  u.  C ) )
 
Theoremun23 3130 A rearrangement of union. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ( A  u.  B )  u.  C )  =  ( ( A  u.  C )  u.  B )
 
Theoremun4 3131 A rearrangement of the union of 4 classes. (Contributed by NM, 12-Aug-2004.)
 |-  ( ( A  u.  B )  u.  ( C  u.  D ) )  =  ( ( A  u.  C )  u.  ( B  u.  D ) )
 
Theoremunundi 3132 Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
 |-  ( A  u.  ( B  u.  C ) )  =  ( ( A  u.  B )  u.  ( A  u.  C ) )
 
Theoremunundir 3133 Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
 |-  ( ( A  u.  B )  u.  C )  =  ( ( A  u.  C )  u.  ( B  u.  C ) )
 
Theoremssun1 3134 Subclass relationship for union of classes. Theorem 25 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
 |-  A  C_  ( A  u.  B )
 
Theoremssun2 3135 Subclass relationship for union of classes. (Contributed by NM, 30-Aug-1993.)
 |-  A  C_  ( B  u.  A )
 
Theoremssun3 3136 Subclass law for union of classes. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  C_  B  ->  A  C_  ( B  u.  C ) )
 
Theoremssun4 3137 Subclass law for union of classes. (Contributed by NM, 14-Aug-1994.)
 |-  ( A  C_  B  ->  A  C_  ( C  u.  B ) )
 
Theoremelun1 3138 Membership law for union of classes. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  e.  B  ->  A  e.  ( B  u.  C ) )
 
Theoremelun2 3139 Membership law for union of classes. (Contributed by NM, 30-Aug-1993.)
 |-  ( A  e.  B  ->  A  e.  ( C  u.  B ) )
 
Theoremunss1 3140 Subclass law for union of classes. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( A  C_  B  ->  ( A  u.  C )  C_  ( B  u.  C ) )
 
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.)
 |-  ( A  C_  B  <->  ( A  u.  B )  =  B )
 
Theoremunss2 3142 Subclass law for union of classes. Exercise 7 of [TakeutiZaring] p. 18. (Contributed by NM, 14-Oct-1999.)
 |-  ( A  C_  B  ->  ( C  u.  A )  C_  ( C  u.  B ) )
 
Theoremunss12 3143 Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.)
 |-  ( ( A  C_  B  /\  C  C_  D )  ->  ( A  u.  C )  C_  ( B  u.  D ) )
 
Theoremssequn2 3144 A relationship between subclass and union. (Contributed by NM, 13-Jun-1994.)
 |-  ( A  C_  B  <->  ( B  u.  A )  =  B )
 
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.)
 |-  ( ( A  C_  C  /\  B  C_  C ) 
 <->  ( A  u.  B )  C_  C )
 
Theoremunssi 3146 An inference showing the union of two subclasses is a subclass. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  A  C_  C   &    |-  B  C_  C   =>    |-  ( A  u.  B )  C_  C
 
Theoremunssd 3147 A deduction showing the union of two subclasses is a subclass. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  ->  A  C_  C )   &    |-  ( ph  ->  B 
 C_  C )   =>    |-  ( ph  ->  ( A  u.  B ) 
 C_  C )
 
Theoremunssad 3148 If  ( A  u.  B ) is contained in  C, so is  A. One-way deduction form of unss 3145. Partial converse of unssd 3147. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  ( A  u.  B )  C_  C )   =>    |-  ( ph  ->  A  C_  C )
 
Theoremunssbd 3149 If  ( A  u.  B ) is contained in  C, so is  B. One-way deduction form of unss 3145. Partial converse of unssd 3147. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  ( A  u.  B )  C_  C )   =>    |-  ( ph  ->  B  C_  C )
 
Theoremssun 3150 A condition that implies inclusion in the union of two classes. (Contributed by NM, 23-Nov-2003.)
 |-  ( ( A  C_  B  \/  A  C_  C )  ->  A  C_  ( B  u.  C ) )
 
Theoremrexun 3151 Restricted existential quantification over union. (Contributed by Jeff Madsen, 5-Jan-2011.)
 |-  ( E. x  e.  ( A  u.  B ) ph  <->  ( E. x  e.  A  ph  \/  E. x  e.  B  ph ) )
 
Theoremralunb 3152 Restricted quantification over a union. (Contributed by Scott Fenton, 12-Apr-2011.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( A. x  e.  ( A  u.  B ) ph  <->  ( A. x  e.  A  ph  /\  A. x  e.  B  ph ) )
 
Theoremralun 3153 Restricted quantification over union. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( A. x  e.  A  ph  /\  A. x  e.  B  ph )  ->  A. x  e.  ( A  u.  B ) ph )
 
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.)
 |-  ( A  e.  ( B  i^i  C )  <->  ( A  e.  B  /\  A  e.  C ) )
 
Theoremelin2 3155 Membership in a class defined as an intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  X  =  ( B  i^i  C )   =>    |-  ( A  e.  X 
 <->  ( A  e.  B  /\  A  e.  C ) )
 
Theoremelin3 3156 Membership in a class defined as a ternary intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  X  =  ( ( B  i^i  C )  i^i  D )   =>    |-  ( A  e.  X 
 <->  ( A  e.  B  /\  A  e.  C  /\  A  e.  D )
 )
 
Theoremincom 3157 Commutative law for intersection of classes. Exercise 7 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  i^i  B )  =  ( B  i^i  A )
 
Theoremineqri 3158* Inference from membership to intersection. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( x  e.  A  /\  x  e.  B )  <->  x  e.  C )   =>    |-  ( A  i^i  B )  =  C
 
Theoremineq1 3159 Equality theorem for intersection of two classes. (Contributed by NM, 14-Dec-1993.)
 |-  ( A  =  B  ->  ( A  i^i  C )  =  ( B  i^i  C ) )
 
Theoremineq2 3160 Equality theorem for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
 |-  ( A  =  B  ->  ( C  i^i  A )  =  ( C  i^i  B ) )
 
Theoremineq12 3161 Equality theorem for intersection of two classes. (Contributed by NM, 8-May-1994.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  i^i  C )  =  ( B  i^i  D ) )
 
Theoremineq1i 3162 Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
 |-  A  =  B   =>    |-  ( A  i^i  C )  =  ( B  i^i  C )
 
Theoremineq2i 3163 Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
 |-  A  =  B   =>    |-  ( C  i^i  A )  =  ( C  i^i  B )
 
Theoremineq12i 3164 Equality inference for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  i^i  C )  =  ( B  i^i  D )
 
Theoremineq1d 3165 Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  i^i  C )  =  ( B  i^i  C ) )
 
Theoremineq2d 3166 Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  i^i  A )  =  ( C  i^i  B ) )
 
Theoremineq12d 3167 Equality deduction for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  i^i  C )  =  ( B  i^i  D ) )
 
Theoremineqan12d 3168 Equality deduction for intersection of two classes. (Contributed by NM, 7-Feb-2007.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ps  ->  C  =  D )   =>    |-  ( ( ph  /\ 
 ps )  ->  ( A  i^i  C )  =  ( B  i^i  D ) )
 
Theoremdfss1 3169 A frequently-used variant of subclass definition df-ss 2959. (Contributed by NM, 10-Jan-2015.)
 |-  ( A  C_  B  <->  ( B  i^i  A )  =  A )
 
Theoremdfss5 3170 Another definition of subclasshood. Similar to df-ss 2959, dfss 2960, and dfss1 3169. (Contributed by David Moews, 1-May-2017.)
 |-  ( A  C_  B  <->  A  =  ( B  i^i  A ) )
 
Theoremnfin 3171 Bound-variable hypothesis builder for the intersection of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x ( A  i^i  B )
 
Theoremcsbing 3172 Distribute proper substitution through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.)
 |-  ( A  e.  B  -> 
 [_ A  /  x ]_ ( C  i^i  D )  =  ( [_ A  /  x ]_ C  i^i  [_ A  /  x ]_ D ) )
 
Theoremrabbi2dva 3173* Deduction from a wff to a restricted class abstraction. (Contributed by NM, 14-Jan-2014.)
 |-  ( ( ph  /\  x  e.  A )  ->  ( x  e.  B  <->  ps ) )   =>    |-  ( ph  ->  ( A  i^i  B )  =  { x  e.  A  |  ps }
 )
 
Theoreminidm 3174 Idempotent law for intersection of classes. Theorem 15 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  i^i  A )  =  A
 
Theoreminass 3175 Associative law for intersection of classes. Exercise 9 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.)
 |-  ( ( A  i^i  B )  i^i  C )  =  ( A  i^i  ( B  i^i  C ) )
 
Theoremin12 3176 A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.)
 |-  ( A  i^i  ( B  i^i  C ) )  =  ( B  i^i  ( A  i^i  C ) )
 
Theoremin32 3177 A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ( A  i^i  B )  i^i  C )  =  ( ( A  i^i  C )  i^i 
 B )
 
Theoremin13 3178 A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
 |-  ( A  i^i  ( B  i^i  C ) )  =  ( C  i^i  ( B  i^i  A ) )
 
Theoremin31 3179 A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
 |-  ( ( A  i^i  B )  i^i  C )  =  ( ( C  i^i  B )  i^i 
 A )
 
Theoreminrot 3180 Rotate the intersection of 3 classes. (Contributed by NM, 27-Aug-2012.)
 |-  ( ( A  i^i  B )  i^i  C )  =  ( ( C  i^i  A )  i^i 
 B )
 
Theoremin4 3181 Rearrangement of intersection of 4 classes. (Contributed by NM, 21-Apr-2001.)
 |-  ( ( A  i^i  B )  i^i  ( C  i^i  D ) )  =  ( ( A  i^i  C )  i^i  ( B  i^i  D ) )
 
Theoreminindi 3182 Intersection distributes over itself. (Contributed by NM, 6-May-1994.)
 |-  ( A  i^i  ( B  i^i  C ) )  =  ( ( A  i^i  B )  i^i  ( A  i^i  C ) )
 
Theoreminindir 3183 Intersection distributes over itself. (Contributed by NM, 17-Aug-2004.)
 |-  ( ( A  i^i  B )  i^i  C )  =  ( ( A  i^i  C )  i^i  ( B  i^i  C ) )
 
Theoremsseqin2 3184 A relationship between subclass and intersection. Similar to Exercise 9 of [TakeutiZaring] p. 18. (Contributed by NM, 17-May-1994.)
 |-  ( A  C_  B  <->  ( B  i^i  A )  =  A )
 
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.)
 |-  ( A  i^i  B )  C_  A
 
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.)
 |-  ( A  i^i  B )  C_  B
 
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.)
 |-  ( ( A  C_  B  /\  A  C_  C ) 
 <->  A  C_  ( B  i^i  C ) )
 
Theoremssini 3188 An inference showing that a subclass of two classes is a subclass of their intersection. (Contributed by NM, 24-Nov-2003.)
 |-  A  C_  B   &    |-  A  C_  C   =>    |-  A  C_  ( B  i^i  C )
 
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.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  A 
 C_  C )   =>    |-  ( ph  ->  A 
 C_  ( B  i^i  C ) )
 
Theoremssrin 3190 Add right intersection to subclass relation. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( A  C_  B  ->  ( A  i^i  C )  C_  ( B  i^i  C ) )
 
Theoremsslin 3191 Add left intersection to subclass relation. (Contributed by NM, 19-Oct-1999.)
 |-  ( A  C_  B  ->  ( C  i^i  A )  C_  ( C  i^i  B ) )
 
Theoremss2in 3192 Intersection of subclasses. (Contributed by NM, 5-May-2000.)
 |-  ( ( A  C_  B  /\  C  C_  D )  ->  ( A  i^i  C )  C_  ( B  i^i  D ) )
 
Theoremssinss1 3193 Intersection preserves subclass relationship. (Contributed by NM, 14-Sep-1999.)
 |-  ( A  C_  C  ->  ( A  i^i  B )  C_  C )
 
Theoreminss 3194 Inclusion of an intersection of two classes. (Contributed by NM, 30-Oct-2014.)
 |-  ( ( A  C_  C  \/  B  C_  C )  ->  ( A  i^i  B )  C_  C )
 
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.)
 |-  ( A  u.  ( A  i^i  B ) )  =  A
 
Theoreminabs 3196 Absorption law for intersection. (Contributed by NM, 16-Apr-2006.)
 |-  ( A  i^i  ( A  u.  B ) )  =  A
 
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.)
 |-  ( -.  A  C_  B 
 <->  ( A  i^i  B )  C.  A )
 
Theoremnsspssun 3198 Negation of subclass expressed in terms of proper subclass and union. (Contributed by NM, 15-Sep-2004.)
 |-  ( -.  A  C_  B 
 <->  B  C.  ( A  u.  B ) )
 
Theoremssddif 3199 Double complement and subset. Similar to ddifss 3203 but inside a class  B instead of the universal class  _V. In classical logic the subset operation on the right hand side could be an equality (that is,  A  C_  B  <->  ( B  \  ( B 
\  A ) )  =  A). (Contributed by Jim Kingdon, 24-Jul-2018.)
 |-  ( A  C_  B  <->  A 
 C_  ( B  \  ( B  \  A ) ) )
 
Theoremunssdif 3200 Union of two classes and class difference. In classical logic this would be an equality. (Contributed by Jim Kingdon, 24-Jul-2018.)
 |-  ( A  u.  B )  C_  ( _V  \  (
 ( _V  \  A )  \  B ) )
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