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Theorem elini 3334
Description: Membership in an intersection of two classes. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
elini.1  |-  A  e.  B
elini.2  |-  A  e.  C
Assertion
Ref Expression
elini  |-  A  e.  ( B  i^i  C
)

Proof of Theorem elini
StepHypRef Expression
1 elini.1 . 2  |-  A  e.  B
2 elini.2 . 2  |-  A  e.  C
3 elin 3333 . 2  |-  ( A  e.  ( B  i^i  C )  <->  ( A  e.  B  /\  A  e.  C ) )
41, 2, 3mpbir2an 944 1  |-  A  e.  ( B  i^i  C
)
Colors of variables: wff set class
Syntax hints:    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:  exmidonfinlem  7227  taupi  15309
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