Theorem difun2 1763

Description: Absorption of union by difference. Theorem 36 of [Suppes] p. 29.

Assertion

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

Proof of Theorem difun2

Step	Hyp	Ref	Expression
1		difundir 1682	. 2 ⊢ ((A ∪ B) ∖ B) = ((A ∖ B) ∪ (B ∖ B))
2		difid 1755	. . 3 ⊢ (B ∖ B) = ∅
3	2	uneq2i 1608	. 2 ⊢ ((A ∖ B) ∪ (B ∖ B)) = ((A ∖ B) ∪ ∅)
4		un0 1721	. 2 ⊢ ((A ∖ B) ∪ ∅) = (A ∖ B)
5	1, 3, 4	3eqtr 1123	1 ⊢ ((A ∪ B) ∖ B) = (A ∖ B)

Syntax hints:   = wceq 1091   ∖ cdif 1484   ∪ cun 1485  ∅c0 1707

This theorem is referenced by:  orddif 2326

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708

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