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Theorem brrpssg 6279
Description: The proper subset relation on sets is the same as class proper subsethood. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Assertion
Ref Expression
brrpssg  |-  ( B  e.  V  ->  ( A [ C.]  B  <->  A  C.  B ) )

Proof of Theorem brrpssg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2796 . . 3  |-  ( B  e.  V  ->  B  e.  _V )
2 relrpss 6278 . . . 4  |-  Rel [ C.]
32brrelexi 4729 . . 3  |-  ( A [
C.]  B  ->  A  e.  _V )
41, 3anim12i 549 . 2  |-  ( ( B  e.  V  /\  A [ C.]  B )  -> 
( B  e.  _V  /\  A  e.  _V )
)
51adantr 451 . . 3  |-  ( ( B  e.  V  /\  A  C.  B )  ->  B  e.  _V )
6 pssss 3271 . . . 4  |-  ( A 
C.  B  ->  A  C_  B )
7 ssexg 4160 . . . 4  |-  ( ( A  C_  B  /\  B  e.  _V )  ->  A  e.  _V )
86, 1, 7syl2anr 464 . . 3  |-  ( ( B  e.  V  /\  A  C.  B )  ->  A  e.  _V )
95, 8jca 518 . 2  |-  ( ( B  e.  V  /\  A  C.  B )  -> 
( B  e.  _V  /\  A  e.  _V )
)
10 psseq1 3263 . . . 4  |-  ( x  =  A  ->  (
x  C.  y  <->  A  C.  y ) )
11 psseq2 3264 . . . 4  |-  ( y  =  B  ->  ( A  C.  y  <->  A  C.  B ) )
12 df-rpss 6277 . . . 4  |- [ C.]  =  { <. x ,  y
>.  |  x  C.  y }
1310, 11, 12brabg 4284 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A [ C.]  B  <->  A 
C.  B ) )
1413ancoms 439 . 2  |-  ( ( B  e.  _V  /\  A  e.  _V )  ->  ( A [ C.]  B  <->  A 
C.  B ) )
154, 9, 14pm5.21nd 868 1  |-  ( B  e.  V  ->  ( A [ C.]  B  <->  A  C.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684   _Vcvv 2788    C_ wss 3152    C. wpss 3153   class class class wbr 4023   [ C.] crpss 6276
This theorem is referenced by:  brrpss  6280  sorpssi  6283
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-rpss 6277
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