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Theorem undifabs 4036
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 3880 . 2 (𝐴 ∪ (𝐴𝐵)) = ((𝐴𝐴) ∖ (𝐵𝐴))
2 unidm 3748 . . 3 (𝐴𝐴) = 𝐴
32difeq1i 3716 . 2 ((𝐴𝐴) ∖ (𝐵𝐴)) = (𝐴 ∖ (𝐵𝐴))
4 difdif 3728 . 2 (𝐴 ∖ (𝐵𝐴)) = 𝐴
51, 3, 43eqtri 2646 1 (𝐴 ∪ (𝐴𝐵)) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1481  cdif 3564  cun 3565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572
This theorem is referenced by:  dfif5  4093  indifundif  29328
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