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Theorem inss2 3329
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
inss2  |-  ( A  i^i  B )  C_  B

Proof of Theorem inss2
StepHypRef Expression
1 incom 3300 . 2  |-  ( B  i^i  A )  =  ( A  i^i  B
)
2 inss1 3328 . 2  |-  ( B  i^i  A )  C_  B
31, 2eqsstrri 3161 1  |-  ( A  i^i  B )  C_  B
Colors of variables: wff set class
Syntax hints:    i^i cin 3101    C_ wss 3102
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108  df-ss 3115
This theorem is referenced by:  difin0  3468  bnd2  4136  ordin  4347  relin2  4707  relres  4896  ssrnres  5030  cnvcnv  5040  funinsn  5221  funimaexg  5256  fnresin2  5287  ssimaex  5531  ffvresb  5632  ofrfval  6042  ofvalg  6043  ofrval  6044  off  6046  ofres  6048  ofco  6052  offres  6085  tpostpos  6213  smores3  6242  tfrlem5  6263  tfrexlem  6283  erinxp  6556  pmresg  6623  unfiin  6872  ltrelpi  7246  peano5nnnn  7814  peano5nni  8841  rexanuz  10899  structcnvcnv  12276  restsspw  12431  eltg4i  12525  ntrss2  12591  ntrin  12594  isopn3  12595  resttopon  12641  restuni2  12647  cnrest2r  12707  cnptopresti  12708  cnptoprest  12709  lmss  12716  metrest  12976  tgioo  13016  peano5set  13586
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