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Theorem unabs 4081
 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 4053 . 2 (𝐴𝐵) ⊆ 𝐴
2 ssequn2 4009 . 2 ((𝐴𝐵) ⊆ 𝐴 ↔ (𝐴 ∪ (𝐴𝐵)) = 𝐴)
31, 2mpbi 222 1 (𝐴 ∪ (𝐴𝐵)) = 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1601   ∪ cun 3790   ∩ cin 3791   ⊆ wss 3792 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-v 3400  df-un 3797  df-in 3799  df-ss 3806 This theorem is referenced by:  volun  23753
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