MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  clel2 Structured version   Unicode version

Theorem clel2 3085
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 2493 . . 3  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
31, 2ceqsalv 2989 . 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 1360    = wceq 1362    e. wcel 1755   _Vcvv 2962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-10 1774  ax-12 1791  ax-ext 2414
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1590  df-nf 1593  df-sb 1700  df-clab 2420  df-cleq 2426  df-clel 2429  df-v 2964
This theorem is referenced by:  snss  3987  mptelee  22964
  Copyright terms: Public domain W3C validator