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Theorem elrint 3919
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 3371 . 2  |-  ( X  e.  ( A  i^i  |^| B )  <->  ( X  e.  A  /\  X  e. 
|^| B ) )
2 elintg 3886 . . 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 1696   A.wral 2556    i^i cin 3164   |^|cint 3878
This theorem is referenced by:  elrint2  3920  ptcnplem  17331  tmdgsum2  17795  limciun  19260
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-v 2803  df-in 3172  df-int 3879
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