Theorem inss2 1658

Description: The intersection of two classes is a subset of one of them. Part of Exercise 12 of [TakeutiZaring] p. 18.

Assertion

Ref	Expression
inss2	⊢ (A ∩ B) ⊆ B

Proof of Theorem inss2

Step	Hyp	Ref	Expression
1		incom 1636	. 2 ⊢ (B ∩ A) = (A ∩ B)
2		inss1 1657	. 2 ⊢ (B ∩ A) ⊆ B
3	1, 2	eqsstr3 1531	1 ⊢ (A ∩ B) ⊆ B

Syntax hints:   ∩ cin 1486   ⊆ wss 1487

This theorem is referenced by:  ssin 1659  ordin 2228  onfr 2237  relres 2591  intasym 2627  intirr 2628  cnvcnv 2661  fnresin2 2736  bnd2 3549  ltrelpi 3811  chdmm1 5398  chm0 5411  ledi 5447  pjoml2 5495  pjoml4 5497  cmcmlem 5500  cmbr4 5510  pjssm 5572  pjclem1 5649  pjc 5654  mdbr3 5729  mdbr4 5730  dmdbr2 5733  ssmd2 5735  cvexchlem 5759  atcvat4 5775

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-in 1491  df-ss 1492

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