Theorem difun2 3338

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 3233	. 2 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐵))
2		difid 3328	. . 3 ⊢ (𝐵 ∖ 𝐵) = ∅
3	2	uneq2i 3133	. 2 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐵)) = ((𝐴 ∖ 𝐵) ∪ ∅)
4		un0 3294	. 2 ⊢ ((𝐴 ∖ 𝐵) ∪ ∅) = (𝐴 ∖ 𝐵)
5	1, 3, 4	3eqtri 2107	1 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐵) = (𝐴 ∖ 𝐵)

Syntax hints:   = wceq 1285   ∖ cdif 2979   ∪ cun 2980  ∅c0 3267

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268

This theorem is referenced by:  uneqdifeqim  3344  difprsn1  3544  orddif  4318  fisseneq  6474  dfn2  8420