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Theorem elint2 3869
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 3868 . 2  |-  ( A  e.  |^| B  <->  A. x
( x  e.  B  ->  A  e.  x ) )
3 df-ral 2548 . 2  |-  ( A. x  e.  B  A  e.  x  <->  A. x ( x  e.  B  ->  A  e.  x ) )
42, 3bitr4i 243 1  |-  ( A  e.  |^| B  <->  A. x  e.  B  A  e.  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527    e. wcel 1684   A.wral 2543   _Vcvv 2788   |^|cint 3862
This theorem is referenced by:  elintg  3870  ssint  3878  intssuni  3884  iinuni  3985  trint  4128  trintss  4129  onint  4586  intwun  8357  inttsk  8396  intgru  8436  subgint  14641  subrgint  15567  lssintcl  15721  toponmre  16830  alexsubALTlem3  17743  shintcli  21908  chintcli  21910  intidl  26654  mzpincl  26812
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-v 2790  df-int 3863
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