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Theorem brrpss 6280
Description: The proper subset relation on sets is the same as class proper subsethood. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Hypothesis
Ref Expression
brrpss.a  |-  B  e. 
_V
Assertion
Ref Expression
brrpss  |-  ( A [
C.]  B  <->  A  C.  B )

Proof of Theorem brrpss
StepHypRef Expression
1 brrpss.a . 2  |-  B  e. 
_V
2 brrpssg 6279 . 2  |-  ( B  e.  _V  ->  ( A [ C.]  B  <->  A  C.  B ) )
31, 2ax-mp 8 1  |-  ( A [
C.]  B  <->  A  C.  B )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    e. wcel 1684   _Vcvv 2788    C. wpss 3153   class class class wbr 4023   [ C.] crpss 6276
This theorem is referenced by:  porpss  6281  sorpss  6282  fin23lem40  7977  compssiso  8000  isfin1-3  8012  fin12  8039  zorng  8131
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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