Theorem inss1 3157

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 3126	. . 3 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
2	1	simplbi 259	. 2 |- ( x e. ( A i^i B ) -> x e. A )
3	2	ssriv 2949	1 |- ( A i^i B ) C_ A

Syntax hints:   e. wcel 1393   i^i cin 2916   C_ wss 2917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931

This theorem is referenced by:  inss2  3158  ssinss1  3165  unabs  3167  nssinpss  3169  inssddif  3178  inv1  3253  disjdif  3296  inundifss  3301  relin1  4455  resss  4635  resmpt3  4657  cnvcnvss  4775  funin  4970  funimass2  4977  fnresin1  5013  fnres  5015  fresin  5068  ssimaex  5234  fneqeql2  5276  isoini2  5458  ofrfval  5720  fnofval  5721  ofrval  5722  off  5724  ofres  5725  ofco  5729  smores  5907  smores2  5909  tfrlem5  5930  peano5nnnn  6966  peano5nni  7917  rexanuz  9587