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Theorem elini 3260
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 3259 . 2  |-  ( A  e.  ( B  i^i  C )  <->  ( A  e.  B  /\  A  e.  C ) )
41, 2, 3mpbir2an 926 1  |-  A  e.  ( B  i^i  C
)
Colors of variables: wff set class
Syntax hints:    e. wcel 1480    i^i cin 3070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077
This theorem is referenced by:  exmidonfinlem  7054  taupi  13330
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