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Theorem elrint 3903
Description: Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
elrint  |-  ( X  e.  ( A  i^i  |^| B )  <->  ( X  e.  A  /\  A. y  e.  B  X  e.  y ) )
Distinct variable groups:    y, B    y, X
Allowed substitution hint:    A( y)

Proof of Theorem elrint
StepHypRef Expression
1 elin 3358 . 2  |-  ( X  e.  ( A  i^i  |^| B )  <->  ( X  e.  A  /\  X  e. 
|^| B ) )
2 elintg 3870 . . 3  |-  ( X  e.  A  ->  ( X  e.  |^| B  <->  A. y  e.  B  X  e.  y ) )
32pm5.32i 618 . 2  |-  ( ( X  e.  A  /\  X  e.  |^| B )  <-> 
( X  e.  A  /\  A. y  e.  B  X  e.  y )
)
41, 3bitri 240 1  |-  ( X  e.  ( A  i^i  |^| B )  <->  ( X  e.  A  /\  A. y  e.  B  X  e.  y ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    e. wcel 1684   A.wral 2543    i^i cin 3151   |^|cint 3862
This theorem is referenced by:  elrint2  3904  ptcnplem  17315  tmdgsum2  17779  limciun  19244
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-in 3159  df-int 3863
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