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Theorem elriin 4166
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 3532 . 2  |-  ( B  e.  ( A  i^i  |^|_
x  e.  X  S
)  <->  ( B  e.  A  /\  B  e. 
|^|_ x  e.  X  S ) )
2 eliin 4100 . . 3  |-  ( B  e.  A  ->  ( B  e.  |^|_ x  e.  X  S  <->  A. x  e.  X  B  e.  S ) )
32pm5.32i 620 . 2  |-  ( ( B  e.  A  /\  B  e.  |^|_ x  e.  X  S )  <->  ( B  e.  A  /\  A. x  e.  X  B  e.  S ) )
41, 3bitri 242 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 178    /\ wa 360    e. wcel 1726   A.wral 2707    i^i cin 3321   |^|_ciin 4096
This theorem is referenced by:  limciun  19786  limcun  19787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-v 2960  df-in 3329  df-iin 4098
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