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Theorem elriin 3974
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 3358 . 2  |-  ( B  e.  ( A  i^i  |^|_
x  e.  X  S
)  <->  ( B  e.  A  /\  B  e. 
|^|_ x  e.  X  S ) )
2 eliin 3910 . . 3  |-  ( B  e.  A  ->  ( B  e.  |^|_ x  e.  X  S  <->  A. x  e.  X  B  e.  S ) )
32pm5.32i 618 . 2  |-  ( ( B  e.  A  /\  B  e.  |^|_ x  e.  X  S )  <->  ( B  e.  A  /\  A. x  e.  X  B  e.  S ) )
41, 3bitri 240 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 176    /\ wa 358    e. wcel 1684   A.wral 2543    i^i cin 3151   |^|_ciin 3906
This theorem is referenced by:  limciun  19244  limcun  19245
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-iin 3908
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