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Theorem elint 4048
Description: Membership in class intersection. (Contributed by NM, 21-May-1994.)
Hypothesis
Ref Expression
elint.1  |-  A  e. 
_V
Assertion
Ref Expression
elint  |-  ( A  e.  |^| B  <->  A. x
( x  e.  B  ->  A  e.  x ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem elint
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 elint.1 . 2  |-  A  e. 
_V
2 eleq1 2495 . . . 4  |-  ( y  =  A  ->  (
y  e.  x  <->  A  e.  x ) )
32imbi2d 308 . . 3  |-  ( y  =  A  ->  (
( x  e.  B  ->  y  e.  x )  <-> 
( x  e.  B  ->  A  e.  x ) ) )
43albidv 1635 . 2  |-  ( y  =  A  ->  ( A. x ( x  e.  B  ->  y  e.  x )  <->  A. x
( x  e.  B  ->  A  e.  x ) ) )
5 df-int 4043 . 2  |-  |^| B  =  { y  |  A. x ( x  e.  B  ->  y  e.  x ) }
61, 4, 5elab2 3077 1  |-  ( A  e.  |^| B  <->  A. x
( x  e.  B  ->  A  e.  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549    = wceq 1652    e. wcel 1725   _Vcvv 2948   |^|cint 4042
This theorem is referenced by:  elint2  4049  elintab  4053  intss1  4057  intss  4063  intun  4074  intpr  4075  cssmre  16910  dfom5b  25722
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-int 4043
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