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Theorem brrpss 6484
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 6483 . 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 177    e. wcel 1721   _Vcvv 2916    C. wpss 3281   class class class wbr 4172   [ C.] crpss 6480
This theorem is referenced by:  porpss  6485  sorpss  6486  fin23lem40  8187  compssiso  8210  isfin1-3  8222  fin12  8249  zorng  8340
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-rpss 6481
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