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Theorem elinel1 3184
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 3181 . 2  |-  ( A  e.  ( B  i^i  C )  <->  ( A  e.  B  /\  A  e.  C ) )
21simplbi 268 1  |-  ( A  e.  ( B  i^i  C )  ->  A  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1438    i^i cin 2996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3003
This theorem is referenced by:  elin1d  3187
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