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  6498  fival  7309  hashfibclem  11303  ballotfilemofi  13276  ballotfilem2  13285  gsumresfi  14262  subrngpropd  14632  subrgpropd  14669  sralmod  14895  blres  15677  limcresi  15909  elply2  15978  pilem3  16027  uhgrspansubgrlem  16739  taupi  17347