Theorem elinel2 3346

Description: Membership in an intersection implies membership in the second set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Assertion

Ref	Expression
elinel2	|- ( A e. ( B i^i C ) -> A e. C )

Proof of Theorem elinel2

Step	Hyp	Ref	Expression
1		elin 3342	. 2 |- ( A e. ( B i^i C ) <-> ( A e. B /\ A e. C ) )
2	1	simprbi 275	1 |- ( A e. ( B i^i C ) -> A e. C )

Syntax hints:   -> wi 4   e. wcel 2164   i^i cin 3152

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 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3159

This theorem is referenced by:  elin2d  3349  fival  7029  subrngpropd  13712  subrgpropd  13749  sralmod  13946  blres  14602  limcresi  14820  elply2  14881  pilem3  14918  taupi  15563