ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  inss1 Unicode version

Theorem inss1 3193
Description: The intersection of two classes is a subset of one of them. Part of Exercise 12 of [TakeutiZaring] p. 18. (Contributed by NM, 27-Apr-1994.)
Assertion
Ref Expression
inss1  |-  ( A  i^i  B )  C_  A

Proof of Theorem inss1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elin 3156 . . 3  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
21simplbi 268 . 2  |-  ( x  e.  ( A  i^i  B )  ->  x  e.  A )
32ssriv 3004 1  |-  ( A  i^i  B )  C_  A
Colors of variables: wff set class
Syntax hints:    e. wcel 1434    i^i cin 2973    C_ wss 2974
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-in 2980  df-ss 2987
This theorem is referenced by:  inss2  3194  ssinss1  3201  unabs  3203  inssddif  3212  inv1  3287  disjdif  3323  inundifss  3328  relin1  4483  resss  4663  resmpt3  4687  cnvcnvss  4805  funin  5001  funimass2  5008  fnresin1  5044  fnres  5046  fresin  5099  ssimaex  5266  fneqeql2  5308  isoini2  5489  ofrfval  5751  fnofval  5752  ofrval  5753  off  5755  ofres  5756  ofco  5760  smores  5941  smores2  5943  tfrlem5  5963  unfiin  6444  peano5nnnn  7120  peano5nni  8109  rexanuz  10012
  Copyright terms: Public domain W3C validator