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Theorem elint2 4133
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 4132 . 2  |-  ( A  e.  |^| B  <->  A. x
( x  e.  B  ->  A  e.  x ) )
3 df-ral 2718 . 2  |-  ( A. x  e.  B  A  e.  x  <->  A. x ( x  e.  B  ->  A  e.  x ) )
42, 3bitr4i 252 1  |-  ( A  e.  |^| B  <->  A. x  e.  B  A  e.  x )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1367    e. wcel 1756   A.wral 2713   _Vcvv 2970   |^|cint 4126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ral 2718  df-v 2972  df-int 4127
This theorem is referenced by:  elintg  4134  ssint  4142  intssuni  4148  iinuni  4252  trint  4398  trintss  4399  onint  6404  intwun  8900  inttsk  8939  intgru  8979  subgint  15703  subrgint  16885  lssintcl  17043  toponmre  18695  alexsubALTlem3  19619  shintcli  24730  chintcli  24732  fin2so  28413  intidl  28826  mzpincl  29067
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