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Theorem inss2 3292
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 3263 . 2  |-  ( B  i^i  A )  =  ( A  i^i  B
)
2 inss1 3291 . 2  |-  ( B  i^i  A )  C_  B
31, 2eqsstrri 3125 1  |-  ( A  i^i  B )  C_  B
Colors of variables: wff set class
Syntax hints:    i^i cin 3065    C_ wss 3066
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072  df-ss 3079
This theorem is referenced by:  difin0  3431  bnd2  4092  ordin  4302  relin2  4653  relres  4842  ssrnres  4976  cnvcnv  4986  funinsn  5167  funimaexg  5202  fnresin2  5233  ssimaex  5475  ffvresb  5576  ofrfval  5983  ofvalg  5984  ofrval  5985  off  5987  ofres  5989  ofco  5993  offres  6026  tpostpos  6154  smores3  6183  tfrlem5  6204  tfrexlem  6224  erinxp  6496  pmresg  6563  unfiin  6807  ltrelpi  7125  peano5nnnn  7693  peano5nni  8716  rexanuz  10753  structcnvcnv  11964  restsspw  12119  eltg4i  12213  ntrss2  12279  ntrin  12282  isopn3  12283  resttopon  12329  restuni2  12335  cnrest2r  12395  cnptopresti  12396  cnptoprest  12397  lmss  12404  metrest  12664  tgioo  12704  peano5set  13127
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