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

Theorem elind 3265
Description: Deduce membership in an intersection of two classes. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
Hypotheses
Ref Expression
elind.1  |-  ( ph  ->  X  e.  A )
elind.2  |-  ( ph  ->  X  e.  B )
Assertion
Ref Expression
elind  |-  ( ph  ->  X  e.  ( A  i^i  B ) )

Proof of Theorem elind
StepHypRef Expression
1 elind.1 . 2  |-  ( ph  ->  X  e.  A )
2 elind.2 . 2  |-  ( ph  ->  X  e.  B )
3 elin 3263 . 2  |-  ( X  e.  ( A  i^i  B )  <->  ( X  e.  A  /\  X  e.  B ) )
41, 2, 3sylanbrc 414 1  |-  ( ph  ->  X  e.  ( A  i^i  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1481    i^i cin 3074
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 3081
This theorem is referenced by:  elfir  6868  infpwfidom  7070  strslfv2d  12038  baspartn  12254  bastg  12267  isopn3  12331  restbasg  12374  lmss  12452  metrest  12712  tgioo  12752  dvmulxxbr  12872  pilem3  12910
  Copyright terms: Public domain W3C validator