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Theorem elint2 3999
Description: Membership in class intersection. (Contributed by NM, 14-Oct-1999.)
Hypothesis
Ref Expression
elint2.1  |-  A  e. 
_V
Assertion
Ref Expression
elint2  |-  ( A  e.  |^| B  <->  A. x  e.  B  A  e.  x )
Distinct variable groups:    x, A    x, B

Proof of Theorem elint2
StepHypRef Expression
1 elint2.1 . . 3  |-  A  e. 
_V
21elint 3998 . 2  |-  ( A  e.  |^| B  <->  A. x
( x  e.  B  ->  A  e.  x ) )
3 df-ral 2654 . 2  |-  ( A. x  e.  B  A  e.  x  <->  A. x ( x  e.  B  ->  A  e.  x ) )
42, 3bitr4i 244 1  |-  ( A  e.  |^| B  <->  A. x  e.  B  A  e.  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1546    e. wcel 1717   A.wral 2649   _Vcvv 2899   |^|cint 3992
This theorem is referenced by:  elintg  4000  ssint  4008  intssuni  4014  iinuni  4115  trint  4258  trintss  4259  onint  4715  intwun  8543  inttsk  8582  intgru  8622  subgint  14891  subrgint  15817  lssintcl  15967  toponmre  17080  alexsubALTlem3  18001  shintcli  22679  chintcli  22681  intidl  26330  mzpincl  26482
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-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-v 2901  df-int 3993
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