Theorem inss2 3152

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 3123	. 2 ⊢ (B ∩ A) = (A ∩ B)
2		inss1 3151	. 2 ⊢ (B ∩ A) ⊆ B
3	1, 2	eqsstr3i 2970	1 ⊢ (A ∩ B) ⊆ B

Syntax hints:   ∩ cin 2910   ⊆ wss 2911

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925

This theorem is referenced by:  difin0  3291  bnd2  3917  ordin  4088  relin2  4399  relres  4582  ssrnres  4706  cnvcnv  4716  funimaexg  4926  fnresin2  4957  ssimaex  5177  ffvresb  5271  ofrfval  5662  fnofval  5663  ofrval  5664  off  5666  ofres  5667  ofco  5671  offres  5704  tpostpos  5820  smores3  5849  tfrlem5  5871  tfrexlem  5889  erinxp  6116  ltrelpi  6308  peano5nni  7698  peano5set  9399  peano5setOLD  9400