Theorem difun2 3494

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 3380	. 2 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐵))
2		difid 3483	. . 3 ⊢ (𝐵 ∖ 𝐵) = ∅
3	2	uneq2i 3278	. 2 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐵)) = ((𝐴 ∖ 𝐵) ∪ ∅)
4		un0 3448	. 2 ⊢ ((𝐴 ∖ 𝐵) ∪ ∅) = (𝐴 ∖ 𝐵)
5	1, 3, 4	3eqtri 2195	1 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐵) = (𝐴 ∖ 𝐵)

Syntax hints:   = wceq 1348   ∖ cdif 3118   ∪ cun 3119  ∅c0 3414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415

This theorem is referenced by:  uneqdifeqim  3500  difprsn1  3719  orddif  4531  fisseneq  6909  dfn2  9148