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Theorem brrpss 6556
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 6555 . 2  |-  ( B  e.  _V  ->  ( A [ C.]  B  <->  A  C.  B
) )
31, 2ax-mp 5 1  |-  ( A [ C.]  B  <->  A  C.  B )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    e. wcel 1823   _Vcvv 3106    C. wpss 3462   class class class wbr 4439   [ C.] crpss 6552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-rpss 6553
This theorem is referenced by:  porpss  6557  sorpss  6558  fin23lem40  8722  compssiso  8745  isfin1-3  8757  fin12  8784  zorng  8875  fin2solem  30279  psshepw  38261
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