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Theorem compel 27617
Description: Equivalence between two ways of saying "is a member of the complement of  A." (Contributed by Andrew Salmon, 15-Jul-2011.)
Assertion
Ref Expression
compel  |-  ( x  e.  ( _V  \  A )  <->  -.  x  e.  A )

Proof of Theorem compel
StepHypRef Expression
1 vex 2959 . 2  |-  x  e. 
_V
2 eldif 3330 . 2  |-  ( x  e.  ( _V  \  A )  <->  ( x  e.  _V  /\  -.  x  e.  A ) )
31, 2mpbiran 885 1  |-  ( x  e.  ( _V  \  A )  <->  -.  x  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    e. wcel 1725   _Vcvv 2956    \ cdif 3317
This theorem is referenced by:  compeq  27618  compab  27620  conss34  27621
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-dif 3323
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