Theorem difundi 3415

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 3166	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
2		eldif 3166	. . . 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 3346	. . 3 ⊢ (𝑥 ∈ ((𝐴 ∖ 𝐵) ∩ (𝐴 ∖ 𝐶)) ↔ (𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)))
5		eldif 3166	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐵 ∪ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐵 ∪ 𝐶)))
6		elun 3304	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 ∪ 𝐶) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))
7	6	notbii 669	. . . . . . 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 753	. . . . . 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 590	. . . 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 2193	1 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = ((𝐴 ∖ 𝐵) ∩ (𝐴 ∖ 𝐶))

Syntax hints:  ¬ wn 3   ∧ wa 104   ∨ wo 709   = wceq 1364   ∈ wcel 2167   ∖ cdif 3154   ∪ cun 3155   ∩ cin 3156

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-un 3161  df-in 3163

This theorem is referenced by:  undm  3421  undifdc  6985  uncld  14349