Theorem inss1 3186

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.

Step	Hyp	Ref	Expression
1		elin 3155	. . 3 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
2	1	simplbi 268	. 2 |- ( x e. ( A i^i B ) -> x e. A )
3	2	ssriv 3003	1 |- ( A i^i B ) C_ A

Syntax hints:   e. wcel 1433   i^i cin 2972   C_ wss 2973

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979  df-ss 2986

This theorem is referenced by:  inss2  3187  ssinss1  3194  unabs  3196  inssddif  3205  inv1  3280  disjdif  3316  inundifss  3321  relin1  4473  resss  4653  resmpt3  4677  cnvcnvss  4795  funin  4990  funimass2  4997  fnresin1  5033  fnres  5035  fresin  5088  ssimaex  5255  fneqeql2  5297  isoini2  5478  ofrfval  5740  fnofval  5741  ofrval  5742  off  5744  ofres  5745  ofco  5749  smores  5930  smores2  5932  tfrlem5  5953  peano5nnnn  7058  peano5nni  8042  rexanuz  9874