Theorem inss2 3194

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

Step	Hyp	Ref	Expression
1		incom 3165	. 2 |- ( B i^i A ) = ( A i^i B )
2		inss1 3193	. 2 |- ( B i^i A ) C_ B
3	1, 2	eqsstr3i 3031	1 |- ( A i^i B ) C_ B

Syntax hints:   i^i cin 2973   C_ wss 2974

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-in 2980  df-ss 2987

This theorem is referenced by:  difin0  3324  bnd2  3955  ordin  4148  relin2  4484  relres  4667  ssrnres  4793  cnvcnv  4803  funinsn  4979  funimaexg  5014  fnresin2  5045  ssimaex  5266  ffvresb  5360  ofrfval  5751  fnofval  5752  ofrval  5753  off  5755  ofres  5756  ofco  5760  offres  5793  tpostpos  5913  smores3  5942  tfrlem5  5963  tfrexlem  5983  erinxp  6246  unfiin  6444  ltrelpi  6576  peano5nnnn  7120  peano5nni  8109  rexanuz  10012  peano5set  10893