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Theorem List for Intuitionistic Logic Explorer - 3101-3200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsseqtri 3101 Substitution of equality into a subclass relationship. (Contributed by NM, 28-Jul-1995.)
 |-  A  C_  B   &    |-  B  =  C   =>    |-  A  C_  C
 
Theoremsseqtrri 3102 Substitution of equality into a subclass relationship. (Contributed by NM, 4-Apr-1995.)
 |-  A  C_  B   &    |-  C  =  B   =>    |-  A  C_  C
 
Theoremeqsstrd 3103 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  B 
 C_  C )   =>    |-  ( ph  ->  A 
 C_  C )
 
Theoremeqsstrrd 3104 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
 |-  ( ph  ->  B  =  A )   &    |-  ( ph  ->  B 
 C_  C )   =>    |-  ( ph  ->  A 
 C_  C )
 
Theoremsseqtrd 3105 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  A 
 C_  C )
 
Theoremsseqtrrd 3106 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  C  =  B )   =>    |-  ( ph  ->  A 
 C_  C )
 
Theorem3sstr3i 3107 Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
 |-  A  C_  B   &    |-  A  =  C   &    |-  B  =  D   =>    |-  C  C_  D
 
Theorem3sstr4i 3108 Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
 |-  A  C_  B   &    |-  C  =  A   &    |-  D  =  B   =>    |-  C  C_  D
 
Theorem3sstr3g 3109 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)
 |-  ( ph  ->  A  C_  B )   &    |-  A  =  C   &    |-  B  =  D   =>    |-  ( ph  ->  C  C_  D )
 
Theorem3sstr4g 3110 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
 |-  ( ph  ->  A  C_  B )   &    |-  C  =  A   &    |-  D  =  B   =>    |-  ( ph  ->  C  C_  D )
 
Theorem3sstr3d 3111 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  A  =  C )   &    |-  ( ph  ->  B  =  D )   =>    |-  ( ph  ->  C  C_  D )
 
Theorem3sstr4d 3112 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 30-Nov-1995.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  C  =  A )   &    |-  ( ph  ->  D  =  B )   =>    |-  ( ph  ->  C  C_  D )
 
Theoremeqsstrid 3113 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
 |-  A  =  B   &    |-  ( ph  ->  B  C_  C )   =>    |-  ( ph  ->  A  C_  C )
 
Theoremeqsstrrid 3114 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
 |-  B  =  A   &    |-  ( ph  ->  B  C_  C )   =>    |-  ( ph  ->  A  C_  C )
 
Theoremsseqtrdi 3115 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  B  =  C   =>    |-  ( ph  ->  A  C_  C )
 
Theoremsseqtrrdi 3116 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  C  =  B   =>    |-  ( ph  ->  A  C_  C )
 
Theoremsseqtrid 3117 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  B  C_  A   &    |-  ( ph  ->  A  =  C )   =>    |-  ( ph  ->  B 
 C_  C )
 
Theoremsseqtrrid 3118 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  B  C_  A   &    |-  ( ph  ->  C  =  A )   =>    |-  ( ph  ->  B 
 C_  C )
 
Theoremeqsstrdi 3119 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  A  =  B )   &    |-  B  C_  C   =>    |-  ( ph  ->  A  C_  C )
 
Theoremeqsstrrdi 3120 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  B  =  A )   &    |-  B  C_  C   =>    |-  ( ph  ->  A  C_  C )
 
Theoremeqimss 3121 Equality implies the subclass relation. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
 |-  ( A  =  B  ->  A  C_  B )
 
Theoremeqimss2 3122 Equality implies the subclass relation. (Contributed by NM, 23-Nov-2003.)
 |-  ( B  =  A  ->  A  C_  B )
 
Theoremeqimssi 3123 Infer subclass relationship from equality. (Contributed by NM, 6-Jan-2007.)
 |-  A  =  B   =>    |-  A  C_  B
 
Theoremeqimss2i 3124 Infer subclass relationship from equality. (Contributed by NM, 7-Jan-2007.)
 |-  A  =  B   =>    |-  B  C_  A
 
Theoremnssne1 3125 Two classes are different if they don't include the same class. (Contributed by NM, 23-Apr-2015.)
 |-  ( ( A  C_  B  /\  -.  A  C_  C )  ->  B  =/=  C )
 
Theoremnssne2 3126 Two classes are different if they are not subclasses of the same class. (Contributed by NM, 23-Apr-2015.)
 |-  ( ( A  C_  C  /\  -.  B  C_  C )  ->  A  =/=  B )
 
Theoremnssr 3127* Negation of subclass relationship. One direction of Exercise 13 of [TakeutiZaring] p. 18. (Contributed by Jim Kingdon, 15-Jul-2018.)
 |-  ( E. x ( x  e.  A  /\  -.  x  e.  B ) 
 ->  -.  A  C_  B )
 
Theoremnelss 3128 Demonstrate by witnesses that two classes lack a subclass relation. (Contributed by Stefan O'Rear, 5-Feb-2015.)
 |-  ( ( A  e.  B  /\  -.  A  e.  C )  ->  -.  B  C_  C )
 
Theoremssrexf 3129 Restricted existential quantification follows from a subclass relationship. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  ( A  C_  B  ->  ( E. x  e.  A  ph  ->  E. x  e.  B  ph ) )
 
Theoremssrmof 3130 "At most one" existential quantification restricted to a subclass. (Contributed by Thierry Arnoux, 8-Oct-2017.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  ( A  C_  B  ->  ( E* x  e.  B  ph  ->  E* x  e.  A  ph ) )
 
Theoremssralv 3131* Quantification restricted to a subclass. (Contributed by NM, 11-Mar-2006.)
 |-  ( A  C_  B  ->  ( A. x  e.  B  ph  ->  A. x  e.  A  ph ) )
 
Theoremssrexv 3132* Existential quantification restricted to a subclass. (Contributed by NM, 11-Jan-2007.)
 |-  ( A  C_  B  ->  ( E. x  e.  A  ph  ->  E. x  e.  B  ph ) )
 
Theoremralss 3133* Restricted universal quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( A  C_  B  ->  ( A. x  e.  A  ph  <->  A. x  e.  B  ( x  e.  A  -> 
 ph ) ) )
 
Theoremrexss 3134* Restricted existential quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)
 |-  ( A  C_  B  ->  ( E. x  e.  A  ph  <->  E. x  e.  B  ( x  e.  A  /\  ph ) ) )
 
Theoremss2ab 3135 Class abstractions in a subclass relationship. (Contributed by NM, 3-Jul-1994.)
 |-  ( { x  |  ph
 }  C_  { x  |  ps }  <->  A. x ( ph  ->  ps ) )
 
Theoremabss 3136* Class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
 |-  ( { x  |  ph
 }  C_  A  <->  A. x ( ph  ->  x  e.  A ) )
 
Theoremssab 3137* Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.)
 |-  ( A  C_  { x  |  ph }  <->  A. x ( x  e.  A  ->  ph )
 )
 
Theoremssabral 3138* The relation for a subclass of a class abstraction is equivalent to restricted quantification. (Contributed by NM, 6-Sep-2006.)
 |-  ( A  C_  { x  |  ph }  <->  A. x  e.  A  ph )
 
Theoremss2abi 3139 Inference of abstraction subclass from implication. (Contributed by NM, 31-Mar-1995.)
 |-  ( ph  ->  ps )   =>    |-  { x  |  ph }  C_  { x  |  ps }
 
Theoremss2abdv 3140* Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  { x  |  ps }  C_ 
 { x  |  ch } )
 
Theoremabssdv 3141* Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
 |-  ( ph  ->  ( ps  ->  x  e.  A ) )   =>    |-  ( ph  ->  { x  |  ps }  C_  A )
 
Theoremabssi 3142* Inference of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
 |-  ( ph  ->  x  e.  A )   =>    |- 
 { x  |  ph } 
 C_  A
 
Theoremss2rab 3143 Restricted abstraction classes in a subclass relationship. (Contributed by NM, 30-May-1999.)
 |-  ( { x  e.  A  |  ph }  C_  { x  e.  A  |  ps }  <->  A. x  e.  A  ( ph  ->  ps )
 )
 
Theoremrabss 3144* Restricted class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
 |-  ( { x  e.  A  |  ph }  C_  B 
 <-> 
 A. x  e.  A  ( ph  ->  x  e.  B ) )
 
Theoremssrab 3145* Subclass of a restricted class abstraction. (Contributed by NM, 16-Aug-2006.)
 |-  ( B  C_  { x  e.  A  |  ph }  <->  ( B  C_  A  /\  A. x  e.  B  ph ) )
 
Theoremssrabdv 3146* Subclass of a restricted class abstraction (deduction form). (Contributed by NM, 31-Aug-2006.)
 |-  ( ph  ->  B  C_  A )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  ps )   =>    |-  ( ph  ->  B  C_ 
 { x  e.  A  |  ps } )
 
Theoremrabssdv 3147* Subclass of a restricted class abstraction (deduction form). (Contributed by NM, 2-Feb-2015.)
 |-  ( ( ph  /\  x  e.  A  /\  ps )  ->  x  e.  B )   =>    |-  ( ph  ->  { x  e.  A  |  ps }  C_  B )
 
Theoremss2rabdv 3148* Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006.)
 |-  ( ( ph  /\  x  e.  A )  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  { x  e.  A  |  ps }  C_  { x  e.  A  |  ch }
 )
 
Theoremss2rabi 3149 Inference of restricted abstraction subclass from implication. (Contributed by NM, 14-Oct-1999.)
 |-  ( x  e.  A  ->  ( ph  ->  ps )
 )   =>    |- 
 { x  e.  A  |  ph }  C_  { x  e.  A  |  ps }
 
Theoremrabss2 3150* Subclass law for restricted abstraction. (Contributed by NM, 18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( A  C_  B  ->  { x  e.  A  |  ph }  C_  { x  e.  B  |  ph } )
 
Theoremssab2 3151* Subclass relation for the restriction of a class abstraction. (Contributed by NM, 31-Mar-1995.)
 |- 
 { x  |  ( x  e.  A  /\  ph ) }  C_  A
 
Theoremssrab2 3152* Subclass relation for a restricted class. (Contributed by NM, 19-Mar-1997.)
 |- 
 { x  e.  A  |  ph }  C_  A
 
Theoremssrabeq 3153* 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.)
 |-  ( V  C_  { x  e.  V  |  ph }  <->  V  =  { x  e.  V  |  ph
 } )
 
Theoremrabssab 3154 A restricted class is a subclass of the corresponding unrestricted class. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |- 
 { x  e.  A  |  ph }  C_  { x  |  ph }
 
Theoremuniiunlem 3155* 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.)
 |-  ( A. x  e.  A  B  e.  D  ->  ( A. x  e.  A  B  e.  C  <->  { y  |  E. x  e.  A  y  =  B }  C_  C ) )
 
2.1.13  The difference, union, and intersection of two classes
 
2.1.13.1  The difference of two classes
 
Theoremdfdif3 3156* 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.)
 |-  ( A  \  B )  =  { x  e.  A  |  A. y  e.  B  x  =/=  y }
 
Theoremdifeq1 3157 Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( A  =  B  ->  ( A  \  C )  =  ( B  \  C ) )
 
Theoremdifeq2 3158 Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( A  =  B  ->  ( C  \  A )  =  ( C  \  B ) )
 
Theoremdifeq12 3159 Equality theorem for class difference. (Contributed by FL, 31-Aug-2009.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  \  C )  =  ( B  \  D ) )
 
Theoremdifeq1i 3160 Inference adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)
 |-  A  =  B   =>    |-  ( A  \  C )  =  ( B  \  C )
 
Theoremdifeq2i 3161 Inference adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
 |-  A  =  B   =>    |-  ( C  \  A )  =  ( C  \  B )
 
Theoremdifeq12i 3162 Equality inference for class difference. (Contributed by NM, 29-Aug-2004.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  \  C )  =  ( B  \  D )
 
Theoremdifeq1d 3163 Deduction adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  \  C )  =  ( B  \  C ) )
 
Theoremdifeq2d 3164 Deduction adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  \  A )  =  ( C  \  B ) )
 
Theoremdifeq12d 3165 Equality deduction for class difference. (Contributed by FL, 29-May-2014.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  \  C )  =  ( B  \  D ) )
 
Theoremdifeqri 3166* Inference from membership to difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ( x  e.  A  /\  -.  x  e.  B )  <->  x  e.  C )   =>    |-  ( A  \  B )  =  C
 
Theoremnfdif 3167 Bound-variable hypothesis builder for class difference. (Contributed by NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x ( A 
 \  B )
 
Theoremeldifi 3168 Implication of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
 |-  ( A  e.  ( B  \  C )  ->  A  e.  B )
 
Theoremeldifn 3169 Implication of membership in a class difference. (Contributed by NM, 3-May-1994.)
 |-  ( A  e.  ( B  \  C )  ->  -.  A  e.  C )
 
Theoremelndif 3170 A set does not belong to a class excluding it. (Contributed by NM, 27-Jun-1994.)
 |-  ( A  e.  B  ->  -.  A  e.  ( C  \  B ) )
 
Theoremdifdif 3171 Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.)
 |-  ( A  \  ( B  \  A ) )  =  A
 
Theoremdifss 3172 Subclass relationship for class difference. Exercise 14 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)
 |-  ( A  \  B )  C_  A
 
Theoremdifssd 3173 A difference of two classes is contained in the minuend. Deduction form of difss 3172. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  ( A  \  B )  C_  A )
 
Theoremdifss2 3174 If a class is contained in a difference, it is contained in the minuend. (Contributed by David Moews, 1-May-2017.)
 |-  ( A  C_  ( B  \  C )  ->  A  C_  B )
 
Theoremdifss2d 3175 If a class is contained in a difference, it is contained in the minuend. Deduction form of difss2 3174. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  ( B  \  C ) )   =>    |-  ( ph  ->  A  C_  B )
 
Theoremssdifss 3176 Preservation of a subclass relationship by class difference. (Contributed by NM, 15-Feb-2007.)
 |-  ( A  C_  B  ->  ( A  \  C )  C_  B )
 
Theoremddifnel 3177* Double complement under universal class. The hypothesis corresponds to stability of membership in 
A, which is weaker than decidability (see dcstab 814). Actually, the conclusion is a characterization of stability of membership in a class (see ddifstab 3178) . 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 3284. (Contributed by Jim Kingdon, 21-Jul-2018.)
 |-  ( -.  x  e.  ( _V  \  A )  ->  x  e.  A )   =>    |-  ( _V  \  ( _V  \  A ) )  =  A
 
Theoremddifstab 3178* 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.)
 |-  ( ( _V  \  ( _V  \  A ) )  =  A  <->  A. xSTAB  x  e.  A )
 
Theoremssconb 3179 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 3180 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 3181 Difference law for subsets. (Contributed by NM, 28-May-1998.)
 |-  ( A  C_  B  ->  ( A  \  C )  C_  ( B  \  C ) )
 
Theoremssdifd 3182 If  A is contained in  B, then  ( A 
\  C ) is contained in  ( B  \  C ). Deduction form of ssdif 3181. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   =>    |-  ( ph  ->  ( A  \  C )  C_  ( B  \  C ) )
 
Theoremsscond 3183 If  A is contained in  B, then  ( C 
\  B ) is contained in  ( C  \  A ). Deduction form of sscon 3180. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   =>    |-  ( ph  ->  ( C  \  B )  C_  ( C  \  A ) )
 
Theoremssdifssd 3184 If  A is contained in  B, then  ( A 
\  C ) is also contained in  B. Deduction form of ssdifss 3176. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   =>    |-  ( ph  ->  ( A  \  C )  C_  B )
 
Theoremssdif2d 3185 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 3186 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 3187 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 3188* 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 3189 Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  u.  A )  =  A
 
Theoremuncom 3190 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 3191 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 3192 Inference form of equncom 3191. (Contributed by Alan Sare, 18-Feb-2012.)
 |-  A  =  ( B  u.  C )   =>    |-  A  =  ( C  u.  B )
 
Theoremuneq1 3193 Equality theorem for union of two classes. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  ->  ( A  u.  C )  =  ( B  u.  C ) )
 
Theoremuneq2 3194 Equality theorem for the union of two classes. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  ->  ( C  u.  A )  =  ( C  u.  B ) )
 
Theoremuneq12 3195 Equality theorem for union of two classes. (Contributed by NM, 29-Mar-1998.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  u.  C )  =  ( B  u.  D ) )
 
Theoremuneq1i 3196 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 3197 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 3198 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 3199 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 3200 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 ) )
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