MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  clel4 Unicode version

Theorem clel4 3019
Description: An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)
Hypothesis
Ref Expression
clel4.1  |-  B  e. 
_V
Assertion
Ref Expression
clel4  |-  ( A  e.  B  <->  A. x
( x  =  B  ->  A  e.  x
) )
Distinct variable groups:    x, A    x, B

Proof of Theorem clel4
StepHypRef Expression
1 clel4.1 . . 3  |-  B  e. 
_V
2 eleq2 2449 . . 3  |-  ( x  =  B  ->  ( A  e.  x  <->  A  e.  B ) )
31, 2ceqsalv 2926 . 2  |-  ( A. x ( x  =  B  ->  A  e.  x )  <->  A  e.  B )
43bicomi 194 1  |-  ( A  e.  B  <->  A. x
( x  =  B  ->  A  e.  x
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1546    = wceq 1649    e. wcel 1717   _Vcvv 2900
This theorem is referenced by:  intpr  4026
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-11 1753  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-v 2902
  Copyright terms: Public domain W3C validator