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Theorem inss1 3209
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 3172 . . 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 3018 1  |-  ( A  i^i  B )  C_  A
Colors of variables: wff set class
Syntax hints:    e. wcel 1436    i^i cin 2987    C_ wss 2988
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-in 2994  df-ss 3001
This theorem is referenced by:  inss2  3210  ssinss1  3217  unabs  3219  inssddif  3229  inv1  3307  disjdif  3343  inundifss  3348  relin1  4525  resss  4706  resmpt3  4730  cnvcnvss  4853  funin  5052  funimass2  5059  fnresin1  5095  fnres  5097  fresin  5154  ssimaex  5330  fneqeql2  5373  isoini2  5561  ofrfval  5823  fnofval  5824  ofrval  5825  off  5827  ofres  5828  ofco  5832  smores  6013  smores2  6015  tfrlem5  6035  pmresg  6387  unfiin  6590  sbthlem7  6619  peano5nnnn  7374  peano5nni  8363  rexanuz  10338
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