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Theorem unabs 3719
 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 3698 . 2 (𝐴𝐵) ⊆ 𝐴
2 ssequn2 3652 . 2 ((𝐴𝐵) ⊆ 𝐴 ↔ (𝐴 ∪ (𝐴𝐵)) = 𝐴)
31, 2mpbi 218 1 (𝐴 ∪ (𝐴𝐵)) = 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1474   ∪ cun 3442   ∩ cin 3443   ⊆ wss 3444 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494 This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-v 3079  df-un 3449  df-in 3451  df-ss 3458 This theorem is referenced by:  volun  23023
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