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Theorem List for Metamath Proof Explorer - 3301-3400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremeldifbd 3301 If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 3298. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  ( B  \  C ) )   =>    |-  ( ph  ->  -.  A  e.  C )
 
2.1.12  Subclasses and subsets
 
Definitiondf-ss 3302 Define the subclass relationship. Exercise 9 of [TakeutiZaring] p. 18. For example,  { 1 ,  2 }  C_  { 1 ,  2 ,  3 } (ex-ss 21696). Note that  A  C_  A (proved in ssid 3335). Contrast this relationship with the relationship  A  C.  B (as will be defined in df-pss 3304). For a more traditional definition, but requiring a dummy variable, see dfss2 3305. Other possible definitions are given by dfss3 3306, dfss4 3543, sspss 3414, ssequn1 3485, ssequn2 3488, sseqin2 3528, and ssdif0 3654. (Contributed by NM, 27-Apr-1994.)
 |-  ( A  C_  B  <->  ( A  i^i  B )  =  A )
 
Theoremdfss 3303 Variant of subclass definition df-ss 3302. (Contributed by NM, 3-Sep-2004.)
 |-  ( A  C_  B  <->  A  =  ( A  i^i  B ) )
 
Definitiondf-pss 3304 Define proper subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. For example,  { 1 ,  2 }  C.  {
1 ,  2 ,  3 } (ex-pss 21697). Note that  -.  A  C.  A (proved in pssirr 3415). Contrast this relationship with the relationship  A 
C_  B (as defined in df-ss 3302). Other possible definitions are given by dfpss2 3400 and dfpss3 3401. (Contributed by NM, 7-Feb-1996.)
 |-  ( A  C.  B  <->  ( A  C_  B  /\  A  =/=  B ) )
 
Theoremdfss2 3305* Alternate definition of the subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Jan-2002.)
 |-  ( A  C_  B  <->  A. x ( x  e.  A  ->  x  e.  B ) )
 
Theoremdfss3 3306* Alternate definition of subclass relationship. (Contributed by NM, 14-Oct-1999.)
 |-  ( A  C_  B  <->  A. x  e.  A  x  e.  B )
 
Theoremdfss2f 3307 Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 3-Jul-1994.) (Revised by Andrew Salmon, 27-Aug-2011.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  ( A  C_  B 
 <-> 
 A. x ( x  e.  A  ->  x  e.  B ) )
 
Theoremdfss3f 3308 Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 20-Mar-2004.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  ( A  C_  B 
 <-> 
 A. x  e.  A  x  e.  B )
 
Theoremnfss 3309 If  x is not free in  A and  B, it is not free in  A  C_  B. (Contributed by NM, 27-Dec-1996.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/ x  A  C_  B
 
Theoremssel 3310 Membership relationships follow from a subclass relationship. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  C_  B  ->  ( C  e.  A  ->  C  e.  B ) )
 
Theoremssel2 3311 Membership relationships follow from a subclass relationship. (Contributed by NM, 7-Jun-2004.)
 |-  ( ( A  C_  B  /\  C  e.  A )  ->  C  e.  B )
 
Theoremsseli 3312 Membership inference from subclass relationship. (Contributed by NM, 5-Aug-1993.)
 |-  A  C_  B   =>    |-  ( C  e.  A  ->  C  e.  B )
 
Theoremsselii 3313 Membership inference from subclass relationship. (Contributed by NM, 31-May-1999.)
 |-  A  C_  B   &    |-  C  e.  A   =>    |-  C  e.  B
 
Theoremsseldi 3314 Membership inference from subclass relationship. (Contributed by NM, 25-Jun-2014.)
 |-  A  C_  B   &    |-  ( ph  ->  C  e.  A )   =>    |-  ( ph  ->  C  e.  B )
 
Theoremsseld 3315 Membership deduction from subclass relationship. (Contributed by NM, 15-Nov-1995.)
 |-  ( ph  ->  A  C_  B )   =>    |-  ( ph  ->  ( C  e.  A  ->  C  e.  B ) )
 
Theoremsselda 3316 Membership deduction from subclass relationship. (Contributed by NM, 26-Jun-2014.)
 |-  ( ph  ->  A  C_  B )   =>    |-  ( ( ph  /\  C  e.  A )  ->  C  e.  B )
 
Theoremsseldd 3317 Membership inference from subclass relationship. (Contributed by NM, 14-Dec-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  C  e.  A )   =>    |-  ( ph  ->  C  e.  B )
 
Theoremssneld 3318 If a class is not in another class, it is also not in a subclass of that class. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   =>    |-  ( ph  ->  ( -.  C  e.  B  ->  -.  C  e.  A ) )
 
Theoremssneldd 3319 If an element is not in a class, it is also not in a subclass of that class. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  -.  C  e.  B )   =>    |-  ( ph  ->  -.  C  e.  A )
 
Theoremssriv 3320* Inference rule based on subclass definition. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  e.  A  ->  x  e.  B )   =>    |-  A  C_  B
 
Theoremssrd 3321 Deduction rule based on subclass definition. (Contributed by Thierry Arnoux, 8-Mar-2017.)
 |- 
 F/ x ph   &    |-  F/_ x A   &    |-  F/_ x B   &    |-  ( ph  ->  ( x  e.  A  ->  x  e.  B ) )   =>    |-  ( ph  ->  A  C_  B )
 
Theoremssrdv 3322* Deduction rule based on subclass definition. (Contributed by NM, 15-Nov-1995.)
 |-  ( ph  ->  ( x  e.  A  ->  x  e.  B ) )   =>    |-  ( ph  ->  A  C_  B )
 
Theoremsstr2 3323 Transitivity of subclasses. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
 |-  ( A  C_  B  ->  ( B  C_  C  ->  A  C_  C )
 )
 
Theoremsstr 3324 Transitivity of subclasses. Theorem 6 of [Suppes] p. 23. (Contributed by NM, 5-Sep-2003.)
 |-  ( ( A  C_  B  /\  B  C_  C )  ->  A  C_  C )
 
Theoremsstri 3325 Subclass transitivity inference. (Contributed by NM, 5-May-2000.)
 |-  A  C_  B   &    |-  B  C_  C   =>    |-  A  C_  C
 
Theoremsstrd 3326 Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  B 
 C_  C )   =>    |-  ( ph  ->  A 
 C_  C )
 
Theoremsyl5ss 3327 Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.)
 |-  A  C_  B   &    |-  ( ph  ->  B 
 C_  C )   =>    |-  ( ph  ->  A 
 C_  C )
 
Theoremsyl6ss 3328 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  ->  A  C_  B )   &    |-  B  C_  C   =>    |-  ( ph  ->  A  C_  C )
 
Theoremsylan9ss 3329 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 3330 A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ps  ->  B 
 C_  C )   =>    |-  ( ( ps 
 /\  ph )  ->  A  C_  C )
 
Theoremeqss 3331 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 3332 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 3333 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 )
 
Theoremeqrd 3334 Deduce equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 21-Mar-2017.)
 |- 
 F/ x ph   &    |-  F/_ x A   &    |-  F/_ x B   &    |-  ( ph  ->  ( x  e.  A  <->  x  e.  B ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremssid 3335 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 3336 Any class is a subclass of the universal class. (Contributed by NM, 31-Oct-1995.)
 |-  A  C_  _V
 
Theoremsseq1 3337 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 3338 Equality theorem for the subclass relationship. (Contributed by NM, 25-Jun-1998.)
 |-  ( A  =  B  ->  ( C  C_  A  <->  C 
 C_  B ) )
 
Theoremsseq12 3339 Equality theorem for the subclass relationship. (Contributed by NM, 31-May-1999.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  C_  C 
 <->  B  C_  D )
 )
 
Theoremsseq1i 3340 An equality inference for the subclass relationship. (Contributed by NM, 18-Aug-1993.)
 |-  A  =  B   =>    |-  ( A  C_  C 
 <->  B  C_  C )
 
Theoremsseq2i 3341 An equality inference for the subclass relationship. (Contributed by NM, 30-Aug-1993.)
 |-  A  =  B   =>    |-  ( C  C_  A 
 <->  C  C_  B )
 
Theoremsseq12i 3342 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 3343 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  C_  C  <->  B  C_  C ) )
 
Theoremsseq2d 3344 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( C  C_  A  <->  C  C_  B ) )
 
Theoremsseq12d 3345 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 3346 Substitution of equality into a subclass relationship. (Contributed by NM, 16-Jul-1995.)
 |-  A  =  B   &    |-  B  C_  C   =>    |-  A  C_  C
 
Theoremeqsstr3i 3347 Substitution of equality into a subclass relationship. (Contributed by NM, 19-Oct-1999.)
 |-  B  =  A   &    |-  B  C_  C   =>    |-  A  C_  C
 
Theoremsseqtri 3348 Substitution of equality into a subclass relationship. (Contributed by NM, 28-Jul-1995.)
 |-  A  C_  B   &    |-  B  =  C   =>    |-  A  C_  C
 
Theoremsseqtr4i 3349 Substitution of equality into a subclass relationship. (Contributed by NM, 4-Apr-1995.)
 |-  A  C_  B   &    |-  C  =  B   =>    |-  A  C_  C
 
Theoremeqsstrd 3350 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 3351 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 3352 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 3353 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 3354 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 3355 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 3356 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 3357 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 3358 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 3359 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 3360 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
 |-  A  =  B   &    |-  ( ph  ->  B  C_  C )   =>    |-  ( ph  ->  A  C_  C )
 
Theoremsyl5eqssr 3361 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
 |-  B  =  A   &    |-  ( ph  ->  B  C_  C )   =>    |-  ( ph  ->  A  C_  C )
 
Theoremsyl6sseq 3362 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  B  =  C   =>    |-  ( ph  ->  A  C_  C )
 
Theoremsyl6sseqr 3363 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
 |-  ( ph  ->  A  C_  B )   &    |-  C  =  B   =>    |-  ( ph  ->  A  C_  C )
 
Theoremsyl5sseq 3364 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  B  C_  A   &    |-  ( ph  ->  A  =  C )   =>    |-  ( ph  ->  B 
 C_  C )
 
Theoremsyl5sseqr 3365 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  B  C_  A   &    |-  ( ph  ->  C  =  A )   =>    |-  ( ph  ->  B 
 C_  C )
 
Theoremsyl6eqss 3366 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  A  =  B )   &    |-  B  C_  C   =>    |-  ( ph  ->  A  C_  C )
 
Theoremsyl6eqssr 3367 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  B  =  A )   &    |-  B  C_  C   =>    |-  ( ph  ->  A  C_  C )
 
Theoremeqimss 3368 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 3369 Equality implies the subclass relation. (Contributed by NM, 23-Nov-2003.)
 |-  ( B  =  A  ->  A  C_  B )
 
Theoremeqimssi 3370 Infer subclass relationship from equality. (Contributed by NM, 6-Jan-2007.)
 |-  A  =  B   =>    |-  A  C_  B
 
Theoremeqimss2i 3371 Infer subclass relationship from equality. (Contributed by NM, 7-Jan-2007.)
 |-  A  =  B   =>    |-  B  C_  A
 
Theoremnssne1 3372 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 3373 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 3374* 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 3375* 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 3376* 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 3377* 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 3378* 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 3379 Class abstractions in a subclass relationship. (Contributed by NM, 3-Jul-1994.)
 |-  ( { x  |  ph
 }  C_  { x  |  ps }  <->  A. x ( ph  ->  ps ) )
 
Theoremabss 3380* Class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
 |-  ( { x  |  ph
 }  C_  A  <->  A. x ( ph  ->  x  e.  A ) )
 
Theoremssab 3381* Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.)
 |-  ( A  C_  { x  |  ph }  <->  A. x ( x  e.  A  ->  ph )
 )
 
Theoremssabral 3382* 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 3383 Inference of abstraction subclass from implication. (Contributed by NM, 31-Mar-1995.)
 |-  ( ph  ->  ps )   =>    |-  { x  |  ph }  C_  { x  |  ps }
 
Theoremss2abdv 3384* Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  { x  |  ps }  C_ 
 { x  |  ch } )
 
Theoremabssdv 3385* Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
 |-  ( ph  ->  ( ps  ->  x  e.  A ) )   =>    |-  ( ph  ->  { x  |  ps }  C_  A )
 
Theoremabssi 3386* Inference of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
 |-  ( ph  ->  x  e.  A )   =>    |- 
 { x  |  ph } 
 C_  A
 
Theoremss2rab 3387 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 3388* 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 3389* 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 3390* 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 3391* 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 3392* 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 3393 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 3394* 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 3395* Subclass relation for the restriction of a class abstraction. (Contributed by NM, 31-Mar-1995.)
 |- 
 { x  |  ( x  e.  A  /\  ph ) }  C_  A
 
Theoremssrab2 3396* Subclass relation for a restricted class. (Contributed by NM, 19-Mar-1997.)
 |- 
 { x  e.  A  |  ph }  C_  A
 
Theoremssrabeq 3397* 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 3398 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 3399* A subset relationship useful for converting union to indexed union using dfiun2 4093 or dfiun2g 4091 and intersection to indexed intersection using dfiin2 4094. (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 3400 Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.)
 |-  ( A  C.  B  <->  ( A  C_  B  /\  -.  A  =  B ) )
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