Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  unabs Structured version   Visualization version   GIF version

Theorem unabs 4184
 Description: Absorption law for union. (Contributed by NM, 16-Apr-2006.)
Assertion
Ref Expression
unabs (𝐴 ∪ (𝐴𝐵)) = 𝐴

Proof of Theorem unabs
StepHypRef Expression
1 inss1 4158 . 2 (𝐴𝐵) ⊆ 𝐴
2 ssequn2 4113 . 2 ((𝐴𝐵) ⊆ 𝐴 ↔ (𝐴 ∪ (𝐴𝐵)) = 𝐴)
31, 2mpbi 233 1 (𝐴 ∪ (𝐴𝐵)) = 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∪ cun 3882   ∩ cin 3883   ⊆ wss 3884 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-un 3889  df-in 3891  df-ss 3901 This theorem is referenced by:  volun  24153
 Copyright terms: Public domain W3C validator