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Theorem inss1 3300
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 3263 . . 3  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
21simplbi 272 . 2  |-  ( x  e.  ( A  i^i  B )  ->  x  e.  A )
32ssriv 3105 1  |-  ( A  i^i  B )  C_  A
Colors of variables: wff set class
Syntax hints:    e. wcel 1481    i^i cin 3074    C_ wss 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 3081  df-ss 3088
This theorem is referenced by:  inss2  3301  ssinss1  3309  unabs  3311  inssddif  3321  inv1  3403  disjdif  3439  inundifss  3444  relin1  4664  resss  4850  resmpt3  4875  cnvcnvss  5000  funin  5201  funimass2  5208  fnresin1  5244  fnres  5246  fresin  5308  ssimaex  5489  fneqeql2  5536  isoini2  5727  ofrfval  5997  ofvalg  5998  ofrval  5999  off  6001  ofres  6003  ofco  6007  smores  6196  smores2  6198  tfrlem5  6218  pmresg  6577  unfiin  6821  sbthlem7  6858  peano5nnnn  7723  peano5nni  8746  rexanuz  10791  fvsetsid  12030  tgvalex  12256  tgval2  12257  eltg3  12263  tgcl  12270  tgdom  12278  tgidm  12280  epttop  12296  ntropn  12323  ntrin  12330  cnptopresti  12444  cnptoprest  12445  txcnmpt  12479  xmetres  12588  metres  12589  blin2  12638  metrest  12712  tgioo  12752  limcresi  12841
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