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Theorem elinel1 3307
Description: Membership in an intersection implies membership in the first set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Assertion
Ref Expression
elinel1  |-  ( A  e.  ( B  i^i  C )  ->  A  e.  B )

Proof of Theorem elinel1
StepHypRef Expression
1 elin 3304 . 2  |-  ( A  e.  ( B  i^i  C )  <->  ( A  e.  B  /\  A  e.  C ) )
21simplbi 272 1  |-  ( A  e.  ( B  i^i  C )  ->  A  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2136    i^i cin 3114
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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147
This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-v 2727  df-in 3121
This theorem is referenced by:  elin1d  3310  fival  6931  fi0  6936  blbas  13033  blres  13034  pilem3  13304  taupi  13896
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