Theorem elinel1 3415

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 3412	. 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 )

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:  elin1d  3418  ressuppss  6498  fival  7309  fi0  7314  hashfibclem  11304  ballotfilem2  13286  ballotfilemfp1  13289  resscntz  14166  gsummptfidmsplit  14254  2idlval  14948  blbas  15659  blres  15660  pilem3  16010  ppiprm  16259  chtprm  16261  prmorcht  16282  uhgrspansubgrlem  16722  taupi  17330