Theorem elinel2 3416

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 3412	. 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 )

This proof depends on syntax axioms:   -> wi 4   e. wcel 2209   i^i cin 3219

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226

This theorem is used by:  elin2d  3419  ressuppss  6494  fival  7304  hashfibclem  11282  ballotfilemofi  13219  ballotfilem2  13228  subrngpropd  14524  subrgpropd  14561  sralmod  14787  blres  15535  limcresi  15767  elply2  15836  pilem3  15884  uhgrspansubgrlem  16517  taupi  17123