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Theorem clel3 2906
Description: An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
Hypothesis
Ref Expression
clel3.1  |-  B  e. 
_V
Assertion
Ref Expression
clel3  |-  ( A  e.  B  <->  E. x
( x  =  B  /\  A  e.  x
) )
Distinct variable groups:    x, A    x, B

Proof of Theorem clel3
StepHypRef Expression
1 clel3.1 . 2  |-  B  e. 
_V
2 clel3g 2905 . 2  |-  ( B  e.  _V  ->  ( A  e.  B  <->  E. x
( x  =  B  /\  A  e.  x
) ) )
31, 2ax-mp 8 1  |-  ( A  e.  B  <->  E. x
( x  =  B  /\  A  e.  x
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788
This theorem is referenced by:  unipr  3841
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
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