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Theorem compel 27640
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 2791 . 2  |-  x  e. 
_V
2 eldif 3162 . 2  |-  ( x  e.  ( _V  \  A )  <->  ( x  e.  _V  /\  -.  x  e.  A ) )
31, 2mpbiran 884 1  |-  ( x  e.  ( _V  \  A )  <->  -.  x  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    e. wcel 1684   _Vcvv 2788    \ cdif 3149
This theorem is referenced by:  compeq  27641  compab  27644  conss34  27645
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-dif 3155
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