Theorem inss2 3158

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 3129	. 2 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
2		inss1 3157	. 2 ⊢ (𝐵 ∩ 𝐴) ⊆ 𝐵
3	1, 2	eqsstr3i 2976	1 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵

Syntax hints:   ∩ cin 2916   ⊆ 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:  difin0  3297  bnd2  3926  ordin  4122  relin2  4456  relres  4639  ssrnres  4763  cnvcnv  4773  funimaexg  4983  fnresin2  5014  ssimaex  5234  ffvresb  5328  ofrfval  5720  fnofval  5721  ofrval  5722  off  5724  ofres  5725  ofco  5729  offres  5762  tpostpos  5879  smores3  5908  tfrlem5  5930  tfrexlem  5948  erinxp  6180  ltrelpi  6420  peano5nnnn  6964  peano5nni  7915  rexanuz  9561  peano5set  10038  peano5setOLD  10039