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Theorem brrpss 6296
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 6295 . 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 1696   _Vcvv 2801    C. wpss 3166   class class class wbr 4039   [ C.] crpss 6292
This theorem is referenced by:  porpss  6297  sorpss  6298  fin23lem40  7993  compssiso  8016  isfin1-3  8028  fin12  8055  zorng  8147
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-rpss 6293
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