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Theorem clel3 3018
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 3017 . 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 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   _Vcvv 2900
This theorem is referenced by:  unipr  3972
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-v 2902
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