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Theorem List for Metamath Proof Explorer - 3201-3300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsyl5sseq 3201 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  B  C_  A   &    |-  ( ph  ->  A  =  C )   =>    |-  ( ph  ->  B 
 C_  C )
 
Theoremsyl5sseqr 3202 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  B  C_  A   &    |-  ( ph  ->  C  =  A )   =>    |-  ( ph  ->  B 
 C_  C )
 
Theoremsyl6eqss 3203 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  A  =  B )   &    |-  B  C_  C   =>    |-  ( ph  ->  A  C_  C )
 
Theoremsyl6eqssr 3204 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  B  =  A )   &    |-  B  C_  C   =>    |-  ( ph  ->  A  C_  C )
 
Theoremeqimss 3205 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 3206 Equality implies the subclass relation. (Contributed by NM, 23-Nov-2003.)
 |-  ( B  =  A  ->  A  C_  B )
 
Theoremeqimssi 3207 Infer subclass relationship from equality. (Contributed by NM, 6-Jan-2007.)
 |-  A  =  B   =>    |-  A  C_  B
 
Theoremeqimss2i 3208 Infer subclass relationship from equality. (Contributed by NM, 7-Jan-2007.)
 |-  A  =  B   =>    |-  B  C_  A
 
Theoremnssne1 3209 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 3210 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 )
 
Theoremnss 3211* Negation of subclass relationship. Exercise 13 of [TakeutiZaring] p. 18. (Contributed by NM, 25-Feb-1996.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
 |-  ( -.  A  C_  B 
 <-> 
 E. x ( x  e.  A  /\  -.  x  e.  B )
 )
 
Theoremssralv 3212* 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 3213* 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 3214* 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 3215* 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 3216 Class abstractions in a subclass relationship. (Contributed by NM, 3-Jul-1994.)
 |-  ( { x  |  ph
 }  C_  { x  |  ps }  <->  A. x ( ph  ->  ps ) )
 
Theoremabss 3217* Class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
 |-  ( { x  |  ph
 }  C_  A  <->  A. x ( ph  ->  x  e.  A ) )
 
Theoremssab 3218* Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.)
 |-  ( A  C_  { x  |  ph }  <->  A. x ( x  e.  A  ->  ph )
 )
 
Theoremssabral 3219* 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 3220 Inference of abstraction subclass from implication. (Contributed by NM, 31-Mar-1995.)
 |-  ( ph  ->  ps )   =>    |-  { x  |  ph }  C_  { x  |  ps }
 
Theoremss2abdv 3221* Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  { x  |  ps }  C_ 
 { x  |  ch } )
 
Theoremabssdv 3222* Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
 |-  ( ph  ->  ( ps  ->  x  e.  A ) )   =>    |-  ( ph  ->  { x  |  ps }  C_  A )
 
Theoremabssi 3223* Inference of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
 |-  ( ph  ->  x  e.  A )   =>    |- 
 { x  |  ph } 
 C_  A
 
Theoremss2rab 3224 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 3225* 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 3226* 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 3227* Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 31-Aug-2006.)
 |-  ( ph  ->  B  C_  A )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  ps )   =>    |-  ( ph  ->  B  C_ 
 { x  e.  A  |  ps } )
 
Theoremrabssdv 3228* Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 2-Feb-2015.)
 |-  ( ( ph  /\  x  e.  A  /\  ps )  ->  x  e.  B )   =>    |-  ( ph  ->  { x  e.  A  |  ps }  C_  B )
 
Theoremss2rabdv 3229* 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 3230 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 3231* 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 3232* Subclass relation for the restriction of a class abstraction. (Contributed by NM, 31-Mar-1995.)
 |- 
 { x  |  ( x  e.  A  /\  ph ) }  C_  A
 
Theoremssrab2 3233* Subclass relation for a restricted class. (Contributed by NM, 19-Mar-1997.)
 |- 
 { x  e.  A  |  ph }  C_  A
 
Theoremrabssab 3234 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 3235* A subset relationship useful for converting union to indexed union using dfiun2 3911 or dfiun2g 3909 and intersection to indexed intersection using dfiin2 3912. (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 ) )
 
Theoremdfpss2 3236 Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.)
 |-  ( A  C.  B  <->  ( A  C_  B  /\  -.  A  =  B ) )
 
Theoremdfpss3 3237 Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( A  C.  B  <->  ( A  C_  B  /\  -.  B  C_  A )
 )
 
Theorempsseq1 3238 Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)
 |-  ( A  =  B  ->  ( A  C.  C  <->  B 
 C.  C ) )
 
Theorempsseq2 3239 Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)
 |-  ( A  =  B  ->  ( C  C.  A  <->  C 
 C.  B ) )
 
Theorempsseq1i 3240 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
 |-  A  =  B   =>    |-  ( A  C.  C 
 <->  B  C.  C )
 
Theorempsseq2i 3241 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
 |-  A  =  B   =>    |-  ( C  C.  A 
 <->  C  C.  B )
 
Theorempsseq12i 3242 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  C.  C  <->  B 
 C.  D )
 
Theorempsseq1d 3243 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  C.  C  <->  B  C.  C ) )
 
Theorempsseq2d 3244 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  C.  A  <->  C  C.  B ) )
 
Theorempsseq12d 3245 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  C.  C  <->  B  C.  D ) )
 
Theorempssss 3246 A proper subclass is a subclass. Theorem 10 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)
 |-  ( A  C.  B  ->  A  C_  B )
 
Theorempssne 3247 Two classes in a proper subclass relationship are not equal. (Contributed by NM, 16-Feb-2015.)
 |-  ( A  C.  B  ->  A  =/=  B )
 
Theorempssssd 3248 Deduce subclass from proper subclass. (Contributed by NM, 29-Feb-1996.)
 |-  ( ph  ->  A  C.  B )   =>    |-  ( ph  ->  A  C_  B )
 
Theorempssned 3249 Proper subclasses are unequal. Deduction form of pssne 3247. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C.  B )   =>    |-  ( ph  ->  A  =/=  B )
 
Theoremsspss 3250 Subclass in terms of proper subclass. (Contributed by NM, 25-Feb-1996.)
 |-  ( A  C_  B  <->  ( A  C.  B  \/  A  =  B )
 )
 
Theorempssirr 3251 Proper subclass is irreflexive. Theorem 7 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)
 |- 
 -.  A  C.  A
 
Theorempssn2lp 3252 Proper subclass has no 2-cycle loops. Compare Theorem 8 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |- 
 -.  ( A  C.  B  /\  B  C.  A )
 
Theoremsspsstri 3253 Two ways of stating trichotomy with respect to inclusion. (Contributed by NM, 12-Aug-2004.)
 |-  ( ( A  C_  B  \/  B  C_  A ) 
 <->  ( A  C.  B  \/  A  =  B  \/  B  C.  A ) )
 
Theoremssnpss 3254 Partial trichotomy law for subclasses. (Contributed by NM, 16-May-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( A  C_  B  ->  -.  B  C.  A )
 
Theorempsstr 3255 Transitive law for proper subclass. Theorem 9 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)
 |-  ( ( A  C.  B  /\  B  C.  C )  ->  A  C.  C )
 
Theoremsspsstr 3256 Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
 |-  ( ( A  C_  B  /\  B  C.  C )  ->  A  C.  C )
 
Theorempsssstr 3257 Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
 |-  ( ( A  C.  B  /\  B  C_  C )  ->  A  C.  C )
 
Theorempsstrd 3258 Proper subclass inclusion is transitive. Deduction form of psstr 3255. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C.  B )   &    |-  ( ph  ->  B 
 C.  C )   =>    |-  ( ph  ->  A 
 C.  C )
 
Theoremsspsstrd 3259 Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 3256. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  B 
 C.  C )   =>    |-  ( ph  ->  A 
 C.  C )
 
Theorempsssstrd 3260 Transitivity involving subclass and proper subclass inclusion. Deduction form of psssstr 3257. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C.  B )   &    |-  ( ph  ->  B 
 C_  C )   =>    |-  ( ph  ->  A 
 C.  C )
 
Theoremnpss 3261 A class is not a proper subclass of another iff it satisfies a one-directional form of eqss 3169. (Contributed by Mario Carneiro, 15-May-2015.)
 |-  ( -.  A  C.  B 
 <->  ( A  C_  B  ->  A  =  B ) )
 
2.1.13  The difference, union, and intersection of two classes
 
Theoremdifeq1 3262 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 3263 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 3264 Equality theorem for class difference. (Contributed by FL, 31-Aug-2009.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  \  C )  =  ( B  \  D ) )
 
Theoremdifeq1i 3265 Inference adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)
 |-  A  =  B   =>    |-  ( A  \  C )  =  ( B  \  C )
 
Theoremdifeq2i 3266 Inference adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
 |-  A  =  B   =>    |-  ( C  \  A )  =  ( C  \  B )
 
Theoremdifeq12i 3267 Equality inference for class difference. (Contributed by NM, 29-Aug-2004.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  \  C )  =  ( B  \  D )
 
Theoremdifeq1d 3268 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 3269 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 3270 Equality deduction for class difference. (Contributed by FL, 29-May-2014.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  \  C )  =  ( B  \  D ) )
 
Theoremdifeqri 3271* 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 3272 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 3273 Implication of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
 |-  ( A  e.  ( B  \  C )  ->  A  e.  B )
 
Theoremeldifn 3274 Implication of membership in a class difference. (Contributed by NM, 3-May-1994.)
 |-  ( A  e.  ( B  \  C )  ->  -.  A  e.  C )
 
Theoremelndif 3275 A set does not belong to a class excluding it. (Contributed by NM, 27-Jun-1994.)
 |-  ( A  e.  B  ->  -.  A  e.  ( C  \  B ) )
 
Theoremneldif 3276 Implication of membership in a class difference. (Contributed by NM, 28-Jun-1994.)
 |-  ( ( A  e.  B  /\  -.  A  e.  ( B  \  C ) )  ->  A  e.  C )
 
Theoremdifdif 3277 Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.)
 |-  ( A  \  ( B  \  A ) )  =  A
 
Theoremdifss 3278 Subclass relationship for class difference. Exercise 14 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)
 |-  ( A  \  B )  C_  A
 
Theoremdifssd 3279 A difference of two classes is contained in the minuend. Deduction form of difss 3278. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  ( A  \  B )  C_  A )
 
Theoremdifss2 3280 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 3281 If a class is contained in a difference, it is contained in the minuend. Deduction form of difss2 3280. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  ( B  \  C ) )   =>    |-  ( ph  ->  A  C_  B )
 
Theoremssdifss 3282 Preservation of a subclass relationship by class difference. (Contributed by NM, 15-Feb-2007.)
 |-  ( A  C_  B  ->  ( A  \  C )  C_  B )
 
Theoremddif 3283 Double complement under universal class. Exercise 4.10(s) of [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.)
 |-  ( _V  \  ( _V  \  A ) )  =  A
 
Theoremssconb 3284 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 3285 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 3286 Difference law for subsets. (Contributed by NM, 28-May-1998.)
 |-  ( A  C_  B  ->  ( A  \  C )  C_  ( B  \  C ) )
 
Theoremssdifd 3287 If  A is contained in  B, then  ( A 
\  C ) is contained in  ( B  \  C ). Deduction form of ssdif 3286. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   =>    |-  ( ph  ->  ( A  \  C )  C_  ( B  \  C ) )
 
Theoremsscond 3288 If  A is contained in  B, then  ( C 
\  B ) is contained in  ( C  \  A ). Deduction form of sscon 3285. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   =>    |-  ( ph  ->  ( C  \  B )  C_  ( C  \  A ) )
 
Theoremssdifssd 3289 If  A is contained in  B, then  ( A 
\  C ) is also contained in  B. Deduction form of ssdifss 3282. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   =>    |-  ( ph  ->  ( A  \  C )  C_  B )
 
Theoremssdif2d 3290 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 ) )
 
Theoremelun 3291 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 3292* 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 3293 Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  u.  A )  =  A
 
Theoremuncom 3294 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 3295 If a class equals the union of two other classes, then it equals the union of those two classes commuted. equncom 3295 was automatically derived from equncomVD 27694 using the tools program translatewithout_overwriting.cmd and minimizing. (Contributed by Alan Sare, 18-Feb-2012.)
 |-  ( A  =  ( B  u.  C )  <->  A  =  ( C  u.  B ) )
 
Theoremequncomi 3296 Inference form of equncom 3295. equncomi 3296 was automatically derived from equncomiVD 27695 using the tools program translatewithout_overwriting.cmd and minimizing. (Contributed by Alan Sare, 18-Feb-2012.)
 |-  A  =  ( B  u.  C )   =>    |-  A  =  ( C  u.  B )
 
Theoremuneq1 3297 Equality theorem for union of two classes. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  ->  ( A  u.  C )  =  ( B  u.  C ) )
 
Theoremuneq2 3298 Equality theorem for the union of two classes. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  ->  ( C  u.  A )  =  ( C  u.  B ) )
 
Theoremuneq12 3299 Equality theorem for union of two classes. (Contributed by NM, 29-Mar-1998.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  u.  C )  =  ( B  u.  D ) )
 
Theoremuneq1i 3300 Inference adding union to the right in a class equality. (Contributed by NM, 30-Aug-1993.)
 |-  A  =  B   =>    |-  ( A  u.  C )  =  ( B  u.  C )
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