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Theorem elriin 3800
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 3183 . 2  |-  ( B  e.  ( A  i^i  |^|_
x  e.  X  S
)  <->  ( B  e.  A  /\  B  e. 
|^|_ x  e.  X  S ) )
2 eliin 3735 . . 3  |-  ( B  e.  A  ->  ( B  e.  |^|_ x  e.  X  S  <->  A. x  e.  X  B  e.  S ) )
32pm5.32i 442 . 2  |-  ( ( B  e.  A  /\  B  e.  |^|_ x  e.  X  S )  <->  ( B  e.  A  /\  A. x  e.  X  B  e.  S ) )
41, 3bitri 182 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:    /\ wa 102    <-> wb 103    e. wcel 1438   A.wral 2359    i^i cin 2998   |^|_ciin 3731
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-in 3005  df-iin 3733
This theorem is referenced by: (None)
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