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Theorem elrint 4083
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 3522 . 2  |-  ( X  e.  ( A  i^i  |^| B )  <->  ( X  e.  A  /\  X  e. 
|^| B ) )
2 elintg 4050 . . 3  |-  ( X  e.  A  ->  ( X  e.  |^| B  <->  A. y  e.  B  X  e.  y ) )
32pm5.32i 619 . 2  |-  ( ( X  e.  A  /\  X  e.  |^| B )  <-> 
( X  e.  A  /\  A. y  e.  B  X  e.  y )
)
41, 3bitri 241 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 177    /\ wa 359    e. wcel 1725   A.wral 2697    i^i cin 3311   |^|cint 4042
This theorem is referenced by:  elrint2  4084  ptcnplem  17643  tmdgsum2  18116  limciun  19771
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-in 3319  df-int 4043
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