Theorem difundi 3292

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 3044	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
2		eldif 3044	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶))
3	1, 2	anbi12i 453	. . 3 ⊢ ((𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶)))
4		elin 3223	. . 3 ⊢ (𝑥 ∈ ((𝐴 ∖ 𝐵) ∩ (𝐴 ∖ 𝐶)) ↔ (𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥 ∈ (𝐴 ∖ 𝐶)))
5		eldif 3044	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐵 ∪ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐵 ∪ 𝐶)))
6		elun 3181	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 ∪ 𝐶) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))
7	6	notbii 640	. . . . . . 7 ⊢ (¬ 𝑥 ∈ (𝐵 ∪ 𝐶) ↔ ¬ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶))
8	7	anbi2i 450	. . . . . 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 724	. . . . . 6 ⊢ (¬ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶) ↔ (¬ 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))
11	10	anbi2i 450	. . . . 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 562	. . . 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 2110	1 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = ((𝐴 ∖ 𝐵) ∩ (𝐴 ∖ 𝐶))

Syntax hints:  ¬ wn 3   ∧ wa 103   ∨ wo 680   = wceq 1312   ∈ wcel 1461   ∖ cdif 3032   ∪ cun 3033   ∩ cin 3034

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-dif 3037  df-un 3039  df-in 3041

This theorem is referenced by:  undm  3298  undifdc  6763  uncld  12119