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Theorem undifabs 4270
 Description: Absorption of difference by union. (Contributed by NM, 18-Aug-2013.)
Assertion
Ref Expression
undifabs (𝐴 ∪ (𝐴𝐵)) = 𝐴

Proof of Theorem undifabs
StepHypRef Expression
1 undif3 4120 . 2 (𝐴 ∪ (𝐴𝐵)) = ((𝐴𝐴) ∖ (𝐵𝐴))
2 unidm 3985 . . 3 (𝐴𝐴) = 𝐴
32difeq1i 3953 . 2 ((𝐴𝐴) ∖ (𝐵𝐴)) = (𝐴 ∖ (𝐵𝐴))
4 difdif 3965 . 2 (𝐴 ∖ (𝐵𝐴)) = 𝐴
51, 3, 43eqtri 2853 1 (𝐴 ∪ (𝐴𝐵)) = 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1656   ∖ cdif 3795   ∪ cun 3796 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803 This theorem is referenced by:  dfif5  4324  indifundif  29900
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