Theorem difun2 3630

Description: Absorption of union by difference. Theorem 36 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)

Assertion

Ref	Expression
difun2	⊢ ((A ∪ B) ∖ B) = (A ∖ B)

Proof of Theorem difun2

Step	Hyp	Ref	Expression
1		difundir 3509	. 2 ⊢ ((A ∪ B) ∖ B) = ((A ∖ B) ∪ (B ∖ B))
2		difid 3619	. . 3 ⊢ (B ∖ B) = ∅
3	2	uneq2i 3416	. 2 ⊢ ((A ∖ B) ∪ (B ∖ B)) = ((A ∖ B) ∪ ∅)
4		un0 3576	. 2 ⊢ ((A ∖ B) ∪ ∅) = (A ∖ B)
5	1, 3, 4	3eqtri 2377	1 ⊢ ((A ∪ B) ∖ B) = (A ∖ B)

Syntax hints:   = wceq 1642   ∖ cdif 3207   ∪ cun 3208  ∅c0 3551

This theorem was proved from 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 theorem 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-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552

This theorem is referenced by:  uneqdifeq  3639  difprsn1  3848  adj11  3890