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Theorem clel3 2750
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 2749 . 2  |-  ( B  e.  _V  ->  ( A  e.  B  <->  E. x
( x  =  B  /\  A  e.  x
) ) )
31, 2ax-mp 7 1  |-  ( A  e.  B  <->  E. x
( x  =  B  /\  A  e.  x
) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   _Vcvv 2619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621
This theorem is referenced by:  unipr  3662
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