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Theorem List for Metamath Proof Explorer - 3201-3300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsstri 3201 Subclass transitivity inference. (Contributed by NM, 5-May-2000.)
 |-  A  C_  B   &    |-  B  C_  C   =>    |-  A  C_  C
 
Theoremsstrd 3202 Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  B 
 C_  C )   =>    |-  ( ph  ->  A 
 C_  C )
 
Theoremsyl5ss 3203 Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.)
 |-  A  C_  B   &    |-  ( ph  ->  B 
 C_  C )   =>    |-  ( ph  ->  A 
 C_  C )
 
Theoremsyl6ss 3204 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  ->  A  C_  B )   &    |-  B  C_  C   =>    |-  ( ph  ->  A  C_  C )
 
Theoremsylan9ss 3205 A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ps  ->  B 
 C_  C )   =>    |-  ( ( ph  /\ 
 ps )  ->  A  C_  C )
 
Theoremsylan9ssr 3206 A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ps  ->  B 
 C_  C )   =>    |-  ( ( ps 
 /\  ph )  ->  A  C_  C )
 
Theoremeqss 3207 The subclass relationship is antisymmetric. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
 
Theoremeqssi 3208 Infer equality from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 9-Sep-1993.)
 |-  A  C_  B   &    |-  B  C_  A   =>    |-  A  =  B
 
Theoremeqssd 3209 Equality deduction from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 27-Jun-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  B 
 C_  A )   =>    |-  ( ph  ->  A  =  B )
 
Theoremssid 3210 Any class is a subclass of itself. Exercise 10 of [TakeutiZaring] p. 18. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
 |-  A  C_  A
 
Theoremssv 3211 Any class is a subclass of the universal class. (Contributed by NM, 31-Oct-1995.)
 |-  A  C_  _V
 
Theoremsseq1 3212 Equality theorem for subclasses. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
 |-  ( A  =  B  ->  ( A  C_  C  <->  B 
 C_  C ) )
 
Theoremsseq2 3213 Equality theorem for the subclass relationship. (Contributed by NM, 25-Jun-1998.)
 |-  ( A  =  B  ->  ( C  C_  A  <->  C 
 C_  B ) )
 
Theoremsseq12 3214 Equality theorem for the subclass relationship. (Contributed by NM, 31-May-1999.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  C_  C 
 <->  B  C_  D )
 )
 
Theoremsseq1i 3215 An equality inference for the subclass relationship. (Contributed by NM, 18-Aug-1993.)
 |-  A  =  B   =>    |-  ( A  C_  C 
 <->  B  C_  C )
 
Theoremsseq2i 3216 An equality inference for the subclass relationship. (Contributed by NM, 30-Aug-1993.)
 |-  A  =  B   =>    |-  ( C  C_  A 
 <->  C  C_  B )
 
Theoremsseq12i 3217 An equality inference for the subclass relationship. (Contributed by NM, 31-May-1999.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
 |-  A  =  B   &    |-  C  =  D   =>    |-  ( A  C_  C  <->  B 
 C_  D )
 
Theoremsseq1d 3218 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  C_  C  <->  B  C_  C ) )
 
Theoremsseq2d 3219 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  C_  A  <->  C  C_  B ) )
 
Theoremsseq12d 3220 An equality deduction for the subclass relationship. (Contributed by NM, 31-May-1999.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  C_  C  <->  B  C_  D ) )
 
Theoremeqsstri 3221 Substitution of equality into a subclass relationship. (Contributed by NM, 16-Jul-1995.)
 |-  A  =  B   &    |-  B  C_  C   =>    |-  A  C_  C
 
Theoremeqsstr3i 3222 Substitution of equality into a subclass relationship. (Contributed by NM, 19-Oct-1999.)
 |-  B  =  A   &    |-  B  C_  C   =>    |-  A  C_  C
 
Theoremsseqtri 3223 Substitution of equality into a subclass relationship. (Contributed by NM, 28-Jul-1995.)
 |-  A  C_  B   &    |-  B  =  C   =>    |-  A  C_  C
 
Theoremsseqtr4i 3224 Substitution of equality into a subclass relationship. (Contributed by NM, 4-Apr-1995.)
 |-  A  C_  B   &    |-  C  =  B   =>    |-  A  C_  C
 
Theoremeqsstrd 3225 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  B 
 C_  C )   =>    |-  ( ph  ->  A 
 C_  C )
 
Theoremeqsstr3d 3226 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 3227 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  B  =  C )   =>    |-  ( ph  ->  A 
 C_  C )
 
Theoremsseqtr4d 3228 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 3229 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 3230 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 3231 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 3232 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 3233 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 3234 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 )
 
Theoremsyl5eqss 3235 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
 |-  A  =  B   &    |-  ( ph  ->  B  C_  C )   =>    |-  ( ph  ->  A  C_  C )
 
Theoremsyl5eqssr 3236 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
 |-  B  =  A   &    |-  ( ph  ->  B  C_  C )   =>    |-  ( ph  ->  A  C_  C )
 
Theoremsyl6sseq 3237 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  B  =  C   =>    |-  ( ph  ->  A  C_  C )
 
Theoremsyl6sseqr 3238 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  C  =  B   =>    |-  ( ph  ->  A  C_  C )
 
Theoremsyl5sseq 3239 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  B  C_  A   &    |-  ( ph  ->  A  =  C )   =>    |-  ( ph  ->  B 
 C_  C )
 
Theoremsyl5sseqr 3240 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  B  C_  A   &    |-  ( ph  ->  C  =  A )   =>    |-  ( ph  ->  B 
 C_  C )
 
Theoremsyl6eqss 3241 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  A  =  B )   &    |-  B  C_  C   =>    |-  ( ph  ->  A  C_  C )
 
Theoremsyl6eqssr 3242 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  B  =  A )   &    |-  B  C_  C   =>    |-  ( ph  ->  A  C_  C )
 
Theoremeqimss 3243 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 3244 Equality implies the subclass relation. (Contributed by NM, 23-Nov-2003.)
 |-  ( B  =  A  ->  A  C_  B )
 
Theoremeqimssi 3245 Infer subclass relationship from equality. (Contributed by NM, 6-Jan-2007.)
 |-  A  =  B   =>    |-  A  C_  B
 
Theoremeqimss2i 3246 Infer subclass relationship from equality. (Contributed by NM, 7-Jan-2007.)
 |-  A  =  B   =>    |-  B  C_  A
 
Theoremnssne1 3247 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 3248 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 3249* 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 3250* 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 3251* 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 3252* 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 3253* 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 3254 Class abstractions in a subclass relationship. (Contributed by NM, 3-Jul-1994.)
 |-  ( { x  |  ph
 }  C_  { x  |  ps }  <->  A. x ( ph  ->  ps ) )
 
Theoremabss 3255* Class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
 |-  ( { x  |  ph
 }  C_  A  <->  A. x ( ph  ->  x  e.  A ) )
 
Theoremssab 3256* Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.)
 |-  ( A  C_  { x  |  ph }  <->  A. x ( x  e.  A  ->  ph )
 )
 
Theoremssabral 3257* 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 3258 Inference of abstraction subclass from implication. (Contributed by NM, 31-Mar-1995.)
 |-  ( ph  ->  ps )   =>    |-  { x  |  ph }  C_  { x  |  ps }
 
Theoremss2abdv 3259* Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  { x  |  ps }  C_ 
 { x  |  ch } )
 
Theoremabssdv 3260* Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
 |-  ( ph  ->  ( ps  ->  x  e.  A ) )   =>    |-  ( ph  ->  { x  |  ps }  C_  A )
 
Theoremabssi 3261* Inference of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
 |-  ( ph  ->  x  e.  A )   =>    |- 
 { x  |  ph } 
 C_  A
 
Theoremss2rab 3262 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 3263* 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 3264* 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 3265* 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 3266* 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 3267* 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 3268 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 3269* 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 3270* Subclass relation for the restriction of a class abstraction. (Contributed by NM, 31-Mar-1995.)
 |- 
 { x  |  ( x  e.  A  /\  ph ) }  C_  A
 
Theoremssrab2 3271* Subclass relation for a restricted class. (Contributed by NM, 19-Mar-1997.)
 |- 
 { x  e.  A  |  ph }  C_  A
 
Theoremrabssab 3272 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 3273* A subset relationship useful for converting union to indexed union using dfiun2 3953 or dfiun2g 3951 and intersection to indexed intersection using dfiin2 3954. (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 3274 Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.)
 |-  ( A  C.  B  <->  ( A  C_  B  /\  -.  A  =  B ) )
 
Theoremdfpss3 3275 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 3276 Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)
 |-  ( A  =  B  ->  ( A  C.  C  <->  B 
 C.  C ) )
 
Theorempsseq2 3277 Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)
 |-  ( A  =  B  ->  ( C  C.  A  <->  C 
 C.  B ) )
 
Theorempsseq1i 3278 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
 |-  A  =  B   =>    |-  ( A  C.  C 
 <->  B  C.  C )
 
Theorempsseq2i 3279 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
 |-  A  =  B   =>    |-  ( C  C.  A 
 <->  C  C.  B )
 
Theorempsseq12i 3280 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 3281 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 3282 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 3283 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 3284 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 3285 Two classes in a proper subclass relationship are not equal. (Contributed by NM, 16-Feb-2015.)
 |-  ( A  C.  B  ->  A  =/=  B )
 
Theorempssssd 3286 Deduce subclass from proper subclass. (Contributed by NM, 29-Feb-1996.)
 |-  ( ph  ->  A  C.  B )   =>    |-  ( ph  ->  A  C_  B )
 
Theorempssned 3287 Proper subclasses are unequal. Deduction form of pssne 3285. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C.  B )   =>    |-  ( ph  ->  A  =/=  B )
 
Theoremsspss 3288 Subclass in terms of proper subclass. (Contributed by NM, 25-Feb-1996.)
 |-  ( A  C_  B  <->  ( A  C.  B  \/  A  =  B )
 )
 
Theorempssirr 3289 Proper subclass is irreflexive. Theorem 7 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)
 |- 
 -.  A  C.  A
 
Theorempssn2lp 3290 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 3291 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 3292 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 3293 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 3294 Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
 |-  ( ( A  C_  B  /\  B  C.  C )  ->  A  C.  C )
 
Theorempsssstr 3295 Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
 |-  ( ( A  C.  B  /\  B  C_  C )  ->  A  C.  C )
 
Theorempsstrd 3296 Proper subclass inclusion is transitive. Deduction form of psstr 3293. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C.  B )   &    |-  ( ph  ->  B 
 C.  C )   =>    |-  ( ph  ->  A 
 C.  C )
 
Theoremsspsstrd 3297 Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 3294. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  B 
 C.  C )   =>    |-  ( ph  ->  A 
 C.  C )
 
Theorempsssstrd 3298 Transitivity involving subclass and proper subclass inclusion. Deduction form of psssstr 3295. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C.  B )   &    |-  ( ph  ->  B 
 C_  C )   =>    |-  ( ph  ->  A 
 C.  C )
 
Theoremnpss 3299 A class is not a proper subclass of another iff it satisfies a one-directional form of eqss 3207. (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 3300 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 ) )
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