Theorem inss2 3187

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 ∩ 𝐵) ⊆ 𝐵

Proof of Theorem inss2

Step	Hyp	Ref	Expression
1		incom 3158	. 2 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
2		inss1 3186	. 2 ⊢ (𝐵 ∩ 𝐴) ⊆ 𝐵
3	1, 2	eqsstr3i 3030	1 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵

Syntax hints:   ∩ cin 2972   ⊆ 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:  difin0  3317  bnd2  3947  ordin  4140  relin2  4474  relres  4657  ssrnres  4783  cnvcnv  4793  funimaexg  5003  fnresin2  5034  ssimaex  5255  ffvresb  5349  ofrfval  5740  fnofval  5741  ofrval  5742  off  5744  ofres  5745  ofco  5749  offres  5782  tpostpos  5902  smores3  5931  tfrlem5  5953  tfrexlem  5971  erinxp  6203  ltrelpi  6514  peano5nnnn  7058  peano5nni  8042  rexanuz  9874  peano5set  10735