Theorem difundi 3459

Description: Distributive law for class difference. Theorem 39 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)

Assertion

Ref	Expression
difundi	⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = ((𝐴 ∖ 𝐵) ∩ (𝐴 ∖ 𝐶))

Proof of Theorem difundi

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldif 3209	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
2		eldif 3209	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶))
3	1, 2	anbi12i 460	. . 3 ⊢ ((𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶)))
4		elin 3390	. . 3 ⊢ (𝑥 ∈ ((𝐴 ∖ 𝐵) ∩ (𝐴 ∖ 𝐶)) ↔ (𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)))
5		eldif 3209	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐵 ∪ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐵 ∪ 𝐶)))
6		elun 3348	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 ∪ 𝐶) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))
7	6	notbii 674	. . . . . . 7 ⊢ (¬ 𝑥 ∈ (𝐵 ∪ 𝐶) ↔ ¬ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))
8	7	anbi2i 457	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐵 ∪ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶)))
9	5, 8	bitri 184	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐵 ∪ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶)))
10		ioran 759	. . . . . 6 ⊢ (¬ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶) ↔ (¬ 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))
11	10	anbi2i 457	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ (¬ 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
12	9, 11	bitri 184	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐵 ∪ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ (¬ 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
13		anandi 594	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ (¬ 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶)))
14	12, 13	bitri 184	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐵 ∪ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶)))
15	3, 4, 14	3bitr4ri 213	. 2 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐵 ∪ 𝐶)) ↔ 𝑥 ∈ ((𝐴 ∖ 𝐵) ∩ (𝐴 ∖ 𝐶)))
16	15	eqriv 2228	1 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = ((𝐴 ∖ 𝐵) ∩ (𝐴 ∖ 𝐶))

Syntax hints:  ¬ wn 3   ∧ wa 104   ∨ wo 715   = wceq 1397   ∈ wcel 2202   ∖ cdif 3197   ∪ cun 3198   ∩ cin 3199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202  df-un 3204  df-in 3206

This theorem is referenced by:  undm  3465  undifdc  7115  uncld  14836