Theorem elinel1 3335

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

Step	Hyp	Ref	Expression
1		elin 3332	. 2 |- ( A e. ( B i^i C ) <-> ( A e. B /\ A e. C ) )
2	1	simplbi 274	1 |- ( A e. ( B i^i C ) -> A e. B )

Syntax hints:   -> wi 4   e. wcel 2159   i^i cin 3142

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-v 2753  df-in 3149

This theorem is referenced by:  elin1d  3338  fival  6986  fi0  6991  blbas  14316  blres  14317  pilem3  14587  taupi  15205