Theorem elinel2 3406

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 3402	. 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 2203   i^i cin 3210

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-in 3217

This theorem is referenced by:  elin2d  3409  ressuppss  6454  fival  7257  hashfibclem  11206  ballotfilemofi  13138  ballotfilem2  13142  subrngpropd  14361  subrgpropd  14398  sralmod  14598  blres  15299  limcresi  15531  elply2  15600  pilem3  15648  uhgrspansubgrlem  16271  taupi  16859