Theorem undifabs 3628

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 3516	. 2 ⊢ (A ∪ (A ∖ B)) = ((A ∪ A) ∖ (B ∖ A))
2		unidm 3408	. . 3 ⊢ (A ∪ A) = A
3	2	difeq1i 3382	. 2 ⊢ ((A ∪ A) ∖ (B ∖ A)) = (A ∖ (B ∖ A))
4		difdif 3393	. 2 ⊢ (A ∖ (B ∖ A)) = A
5	1, 3, 4	3eqtri 2377	1 ⊢ (A ∪ (A ∖ B)) = A

This proof depends on syntax axioms:   = wceq 1642   ∖ cdif 3207   ∪ cun 3208

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216

This theorem is used by:  dfif5  3675