Theorem difindiss 3330

Description: Distributive law for class difference. In classical logic, for example, theorem 40 of [Suppes] p. 29, this is an equality instead of subset. (Contributed by Jim Kingdon, 26-Jul-2018.)

Assertion

Ref	Expression
difindiss	⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ 𝐶)) ⊆ (𝐴 ∖ (𝐵 ∩ 𝐶))

Proof of Theorem difindiss

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elun 3217	. . 3 ⊢ (𝑥 ∈ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ 𝐶)) ↔ (𝑥 ∈ (𝐴 ∖ 𝐵) ∨ 𝑥 ∈ (𝐴 ∖ 𝐶)))
2		eldif 3080	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
3		eldif 3080	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶))
4	2, 3	orbi12i 753	. . . . . 6 ⊢ ((𝑥 ∈ (𝐴 ∖ 𝐵) ∨ 𝑥 ∈ (𝐴 ∖ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∨ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶)))
5		andi 807	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ (¬ 𝑥 ∈ 𝐵 ∨ ¬ 𝑥 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∨ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶)))
6	4, 5	bitr4i 186	. . . . 5 ⊢ ((𝑥 ∈ (𝐴 ∖ 𝐵) ∨ 𝑥 ∈ (𝐴 ∖ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ (¬ 𝑥 ∈ 𝐵 ∨ ¬ 𝑥 ∈ 𝐶)))
7		pm3.14 742	. . . . . 6 ⊢ ((¬ 𝑥 ∈ 𝐵 ∨ ¬ 𝑥 ∈ 𝐶) → ¬ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))
8	7	anim2i 339	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ (¬ 𝑥 ∈ 𝐵 ∨ ¬ 𝑥 ∈ 𝐶)) → (𝑥 ∈ 𝐴 ∧ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)))
9	6, 8	sylbi 120	. . . 4 ⊢ ((𝑥 ∈ (𝐴 ∖ 𝐵) ∨ 𝑥 ∈ (𝐴 ∖ 𝐶)) → (𝑥 ∈ 𝐴 ∧ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)))
10		eldif 3080	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐵 ∩ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐵 ∩ 𝐶)))
11		elin 3259	. . . . . . 7 ⊢ (𝑥 ∈ (𝐵 ∩ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))
12	11	notbii 657	. . . . . 6 ⊢ (¬ 𝑥 ∈ (𝐵 ∩ 𝐶) ↔ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))
13	12	anbi2i 452	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐵 ∩ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)))
14	10, 13	bitr2i 184	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)) ↔ 𝑥 ∈ (𝐴 ∖ (𝐵 ∩ 𝐶)))
15	9, 14	sylib 121	. . 3 ⊢ ((𝑥 ∈ (𝐴 ∖ 𝐵) ∨ 𝑥 ∈ (𝐴 ∖ 𝐶)) → 𝑥 ∈ (𝐴 ∖ (𝐵 ∩ 𝐶)))
16	1, 15	sylbi 120	. 2 ⊢ (𝑥 ∈ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ 𝐶)) → 𝑥 ∈ (𝐴 ∖ (𝐵 ∩ 𝐶)))
17	16	ssriv 3101	1 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ 𝐶)) ⊆ (𝐴 ∖ (𝐵 ∩ 𝐶))

Syntax hints:  ¬ wn 3   ∧ wa 103   ∨ wo 697   ∈ wcel 1480   ∖ cdif 3068   ∪ cun 3069   ∩ cin 3070   ⊆ wss 3071

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 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084

This theorem is referenced by:  difdif2ss  3333  indmss  3335