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Theorem undifabs 4078
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 3921 . 2 (𝐴 ∪ (𝐴𝐵)) = ((𝐴𝐴) ∖ (𝐵𝐴))
2 unidm 3789 . . 3 (𝐴𝐴) = 𝐴
32difeq1i 3757 . 2 ((𝐴𝐴) ∖ (𝐵𝐴)) = (𝐴 ∖ (𝐵𝐴))
4 difdif 3769 . 2 (𝐴 ∖ (𝐵𝐴)) = 𝐴
51, 3, 43eqtri 2677 1 (𝐴 ∪ (𝐴𝐵)) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  cdif 3604  cun 3605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612
This theorem is referenced by:  dfif5  4135  indifundif  29482
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