Theorem undifabs 4441

Description: Absorption of difference by union. (Contributed by NM, 18-Aug-2013.)

Assertion

Ref	Expression
undifabs	⊢ (𝐴 ∪ (𝐴 ∖ 𝐵)) = 𝐴

Proof of Theorem undifabs

Step	Hyp	Ref	Expression
1		undif3 4261	. 2 ⊢ (𝐴 ∪ (𝐴 ∖ 𝐵)) = ((𝐴 ∪ 𝐴) ∖ (𝐵 ∖ 𝐴))
2		unidm 4119	. . 3 ⊢ (𝐴 ∪ 𝐴) = 𝐴
3	2	difeq1i 4085	. 2 ⊢ ((𝐴 ∪ 𝐴) ∖ (𝐵 ∖ 𝐴)) = (𝐴 ∖ (𝐵 ∖ 𝐴))
4		difdif 4097	. 2 ⊢ (𝐴 ∖ (𝐵 ∖ 𝐴)) = 𝐴
5	1, 3, 4	3eqtri 2796	1 ⊢ (𝐴 ∪ (𝐴 ∖ 𝐵)) = 𝐴

Syntax hints:   = wceq 1567   ∖ cdif 3910   ∪ cun 3911

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918

This theorem is referenced by:  dfif5  4506  indifundif  32807  dfsucmap3  38997