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Theorem brrpss 6468
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 6467 . 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 1758   _Vcvv 3072    C. wpss 3432   class class class wbr 4395   [ C.] crpss 6464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-br 4396  df-opab 4454  df-xp 4949  df-rel 4950  df-rpss 6465
This theorem is referenced by:  porpss  6469  sorpss  6470  fin23lem40  8626  compssiso  8649  isfin1-3  8661  fin12  8688  zorng  8779  fin2solem  28558
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