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Theorem elint2 4049
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 4048 . 2  |-  ( A  e.  |^| B  <->  A. x
( x  e.  B  ->  A  e.  x ) )
3 df-ral 2702 . 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 1549    e. wcel 1725   A.wral 2697   _Vcvv 2948   |^|cint 4042
This theorem is referenced by:  elintg  4050  ssint  4058  intssuni  4064  iinuni  4166  trint  4309  trintss  4310  onint  4767  intwun  8602  inttsk  8641  intgru  8681  subgint  14956  subrgint  15882  lssintcl  16032  toponmre  17149  alexsubALTlem3  18072  shintcli  22823  chintcli  22825  intidl  26630  mzpincl  26782
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-ral 2702  df-v 2950  df-int 4043
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