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Theorem clel3g 2730
Description: An alternate definition of class membership when the class is a set. (Contributed by NM, 13-Aug-2005.)
Assertion
Ref Expression
clel3g  |-  ( B  e.  V  ->  ( A  e.  B  <->  E. x
( x  =  B  /\  A  e.  x
) ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    V( x)

Proof of Theorem clel3g
StepHypRef Expression
1 eleq2 2143 . . 3  |-  ( x  =  B  ->  ( A  e.  x  <->  A  e.  B ) )
21ceqsexgv 2725 . 2  |-  ( B  e.  V  ->  ( E. x ( x  =  B  /\  A  e.  x )  <->  A  e.  B ) )
32bicomd 139 1  |-  ( B  e.  V  ->  ( A  e.  B  <->  E. x
( x  =  B  /\  A  e.  x
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285   E.wex 1422    e. wcel 1434
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604
This theorem is referenced by:  clel3  2731  dfiun2g  3718
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