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Theorem elrint 4289
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 3628 . 2  |-  ( X  e.  ( A  i^i  |^| B )  <->  ( X  e.  A  /\  X  e. 
|^| B ) )
2 elintg 4255 . . 3  |-  ( X  e.  A  ->  ( X  e.  |^| B  <->  A. y  e.  B  X  e.  y ) )
32pm5.32i 647 . 2  |-  ( ( X  e.  A  /\  X  e.  |^| B )  <-> 
( X  e.  A  /\  A. y  e.  B  X  e.  y )
)
41, 3bitri 257 1  |-  ( X  e.  ( A  i^i  |^| B )  <->  ( X  e.  A  /\  A. y  e.  B  X  e.  y ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 375    e. wcel 1897   A.wral 2748    i^i cin 3414   |^|cint 4247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ral 2753  df-v 3058  df-in 3422  df-int 4248
This theorem is referenced by:  elrint2  4290  ptcnplem  20684  tmdgsum2  21159  limciun  22897
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