Theorem undifabs 4437

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 4248	. 2 ⊢ (𝐴 ∪ (𝐴 ∖ 𝐵)) = ((𝐴 ∪ 𝐴) ∖ (𝐵 ∖ 𝐴))
2		unidm 4159	. . 3 ⊢ (𝐴 ∪ 𝐴) = 𝐴
3	2	difeq1i 4062	. 2 ⊢ ((𝐴 ∪ 𝐴) ∖ (𝐵 ∖ 𝐴)) = (𝐴 ∖ (𝐵 ∖ 𝐴))
4		difdif 4074	. 2 ⊢ (𝐴 ∖ (𝐵 ∖ 𝐴)) = 𝐴
5	1, 3, 4	3eqtri 2790	1 ⊢ (𝐴 ∪ (𝐴 ∖ 𝐵)) = 𝐴

This proof depends on syntax axioms:   = wceq 1570   ∖ cdif 4048   ∪ cun 4152

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-dif 4050  df-un 4154

This theorem is used by:  dfif5  4502  indifundif  33219  dfsucmap3  39661