Theorem difundi 3323

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 3075	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
2		eldif 3075	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶))
3	1, 2	anbi12i 455	. . 3 ⊢ ((𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶)))
4		elin 3254	. . 3 ⊢ (𝑥 ∈ ((𝐴 ∖ 𝐵) ∩ (𝐴 ∖ 𝐶)) ↔ (𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)))
5		eldif 3075	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐵 ∪ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐵 ∪ 𝐶)))
6		elun 3212	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 ∪ 𝐶) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))
7	6	notbii 657	. . . . . . 7 ⊢ (¬ 𝑥 ∈ (𝐵 ∪ 𝐶) ↔ ¬ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))
8	7	anbi2i 452	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐵 ∪ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶)))
9	5, 8	bitri 183	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐵 ∪ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶)))
10		ioran 741	. . . . . 6 ⊢ (¬ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶) ↔ (¬ 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))
11	10	anbi2i 452	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ (¬ 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
12	9, 11	bitri 183	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐵 ∪ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ (¬ 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
13		anandi 579	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ (¬ 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶)))
14	12, 13	bitri 183	. . 3 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐵 ∪ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶)))
15	3, 4, 14	3bitr4ri 212	. 2 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐵 ∪ 𝐶)) ↔ 𝑥 ∈ ((𝐴 ∖ 𝐵) ∩ (𝐴 ∖ 𝐶)))
16	15	eqriv 2134	1 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = ((𝐴 ∖ 𝐵) ∩ (𝐴 ∖ 𝐶))

Syntax hints:  ¬ wn 3   ∧ wa 103   ∨ wo 697   = wceq 1331   ∈ wcel 1480   ∖ cdif 3063   ∪ cun 3064   ∩ cin 3065

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-dif 3068  df-un 3070  df-in 3072

This theorem is referenced by:  undm  3329  undifdc  6805  uncld  12271