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Theorem compel 27743
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 2804 . 2  |-  x  e. 
_V
2 eldif 3175 . 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 1696   _Vcvv 2801    \ cdif 3162
This theorem is referenced by:  compeq  27744  compab  27747  conss34  27748
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-dif 3168
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