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Theorem clel2 3240
Description: An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
Hypothesis
Ref Expression
clel2.1  |-  A  e. 
_V
Assertion
Ref Expression
clel2  |-  ( A  e.  B  <->  A. x
( x  =  A  ->  x  e.  B
) )
Distinct variable groups:    x, A    x, B

Proof of Theorem clel2
StepHypRef Expression
1 clel2.1 . . 3  |-  A  e. 
_V
2 eleq1 2539 . . 3  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
31, 2ceqsalv 3141 . 2  |-  ( A. x ( x  =  A  ->  x  e.  B )  <->  A  e.  B )
43bicomi 202 1  |-  ( A  e.  B  <->  A. x
( x  =  A  ->  x  e.  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1377    = wceq 1379    e. wcel 1767   _Vcvv 3113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-12 1803  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-v 3115
This theorem is referenced by:  snss  4151  mptelee  23874
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