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Theorem elin2 3491
Description: Membership in a class defined as an intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Hypothesis
Ref Expression
elin2.x  |-  X  =  ( B  i^i  C
)
Assertion
Ref Expression
elin2  |-  ( A  e.  X  <->  ( A  e.  B  /\  A  e.  C ) )

Proof of Theorem elin2
StepHypRef Expression
1 elin2.x . . 3  |-  X  =  ( B  i^i  C
)
21eleq2i 2468 . 2  |-  ( A  e.  X  <->  A  e.  ( B  i^i  C ) )
3 elin 3490 . 2  |-  ( A  e.  ( B  i^i  C )  <->  ( A  e.  B  /\  A  e.  C ) )
42, 3bitri 241 1  |-  ( A  e.  X  <->  ( A  e.  B  /\  A  e.  C ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    i^i cin 3279
This theorem is referenced by:  elin3  3492  fnres  5520  funfvima  5932  fnwelem  6420  fz1isolem  11665  isabl  15371  isfld  15799  2idlcpbl  16260  divs1  16261  divsrhm  16263  isidom  16319  lmres  17318  isnvc  18683  ishl  19269  ply1pid  20055  rplogsum  21174  isphg  22271  ishlo  22342  hhsscms  22732  mayete3i  23183  elpredim  25390  elpred  25391  elpredg  25392  sltres  25532  nofulllem5  25574  caures  26356  iscrngo  26497  fldcrng  26504  isdmn  26554  isolat  29695
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 2385
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 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-in 3287
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