ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elinel2 Unicode version

Theorem elinel2 3268
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
StepHypRef Expression
1 elin 3264 . 2  |-  ( A  e.  ( B  i^i  C )  <->  ( A  e.  B  /\  A  e.  C ) )
21simprbi 273 1  |-  ( A  e.  ( B  i^i  C )  ->  A  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1481    i^i cin 3075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-in 3082
This theorem is referenced by:  elin2d  3271  fival  6866  blres  12642  limcresi  12843  pilem3  12912  taupi  13430
  Copyright terms: Public domain W3C validator