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Theorem undifabs 4269
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 4115 . 2 (𝐴 ∪ (𝐴𝐵)) = ((𝐴𝐴) ∖ (𝐵𝐴))
2 unidm 3979 . . 3 (𝐴𝐴) = 𝐴
32difeq1i 3947 . 2 ((𝐴𝐴) ∖ (𝐵𝐴)) = (𝐴 ∖ (𝐵𝐴))
4 difdif 3959 . 2 (𝐴 ∖ (𝐵𝐴)) = 𝐴
51, 3, 43eqtri 2806 1 (𝐴 ∪ (𝐴𝐵)) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1601  cdif 3789  cun 3790
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-rab 3099  df-v 3400  df-dif 3795  df-un 3797
This theorem is referenced by:  dfif5  4323  indifundif  29918
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