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Theorem elriin 4370
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 3650 . 2  |-  ( B  e.  ( A  i^i  |^|_
x  e.  X  S
)  <->  ( B  e.  A  /\  B  e. 
|^|_ x  e.  X  S ) )
2 eliin 4303 . . 3  |-  ( B  e.  A  ->  ( B  e.  |^|_ x  e.  X  S  <->  A. x  e.  X  B  e.  S ) )
32pm5.32i 642 . 2  |-  ( ( B  e.  A  /\  B  e.  |^|_ x  e.  X  S )  <->  ( B  e.  A  /\  A. x  e.  X  B  e.  S ) )
41, 3bitri 253 1  |-  ( B  e.  ( A  i^i  |^|_
x  e.  X  S
)  <->  ( B  e.  A  /\  A. x  e.  X  B  e.  S ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    /\ wa 371    e. wcel 1869   A.wral 2776    i^i cin 3436   |^|_ciin 4298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ral 2781  df-v 3084  df-in 3444  df-iin 4300
This theorem is referenced by:  limciun  22841  limcun  22842
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