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Theorem elin2 3338
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 2256 . 2  |-  ( A  e.  X  <->  A  e.  ( B  i^i  C ) )
3 elin 3333 . 2  |-  ( A  e.  ( B  i^i  C )  <->  ( A  e.  B  /\  A  e.  C ) )
42, 3bitri 184 1  |-  ( A  e.  X  <->  ( A  e.  B  /\  A  e.  C ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    = wceq 1364    e. wcel 2160    i^i cin 3143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150
This theorem is referenced by:  elin3  3341  fnres  5351  funfvima  5769  isabl  13244  lmres  14225
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