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Theorem elriin 4131
Description: Elementhood in a relative intersection. (Contributed by Mario Carneiro, 30-Dec-2016.)
Assertion
Ref Expression
elriin  |-  ( B  e.  ( A  i^i  |^|_
x  e.  X  S
)  <->  ( B  e.  A  /\  A. x  e.  X  B  e.  S ) )
Distinct variable groups:    x, A    x, X    x, B
Allowed substitution hint:    S( x)

Proof of Theorem elriin
StepHypRef Expression
1 elin 3498 . 2  |-  ( B  e.  ( A  i^i  |^|_
x  e.  X  S
)  <->  ( B  e.  A  /\  B  e. 
|^|_ x  e.  X  S ) )
2 eliin 4066 . . 3  |-  ( B  e.  A  ->  ( B  e.  |^|_ x  e.  X  S  <->  A. x  e.  X  B  e.  S ) )
32pm5.32i 619 . 2  |-  ( ( B  e.  A  /\  B  e.  |^|_ x  e.  X  S )  <->  ( B  e.  A  /\  A. x  e.  X  B  e.  S ) )
41, 3bitri 241 1  |-  ( B  e.  ( A  i^i  |^|_
x  e.  X  S
)  <->  ( B  e.  A  /\  A. x  e.  X  B  e.  S ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    e. wcel 1721   A.wral 2674    i^i cin 3287   |^|_ciin 4062
This theorem is referenced by:  limciun  19742  limcun  19743
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679  df-v 2926  df-in 3295  df-iin 4064
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