Theorem difindiss 3485

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 3370	. . 3 ⊢ (𝑥 ∈ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ 𝐶)) ↔ (𝑥 ∈ (𝐴 ∖ 𝐵) ∨ 𝑥 ∈ (𝐴 ∖ 𝐶)))
2		eldif 3229	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
3		eldif 3229	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶))
4	2, 3	orbi12i 776	. . . . . 6 ⊢ ((𝑥 ∈ (𝐴 ∖ 𝐵) ∨ 𝑥 ∈ (𝐴 ∖ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∨ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶)))
5		andi 830	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ (¬ 𝑥 ∈ 𝐵 ∨ ¬ 𝑥 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∨ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶)))
6	4, 5	bitr4i 187	. . . . 5 ⊢ ((𝑥 ∈ (𝐴 ∖ 𝐵) ∨ 𝑥 ∈ (𝐴 ∖ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ (¬ 𝑥 ∈ 𝐵 ∨ ¬ 𝑥 ∈ 𝐶)))
7		pm3.14 765	. . . . . 6 ⊢ ((¬ 𝑥 ∈ 𝐵 ∨ ¬ 𝑥 ∈ 𝐶) → ¬ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))
8	7	anim2i 342	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ (¬ 𝑥 ∈ 𝐵 ∨ ¬ 𝑥 ∈ 𝐶)) → (𝑥 ∈ 𝐴 ∧ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)))
9	6, 8	sylbi 121	. . . 4 ⊢ ((𝑥 ∈ (𝐴 ∖ 𝐵) ∨ 𝑥 ∈ (𝐴 ∖ 𝐶)) → (𝑥 ∈ 𝐴 ∧ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)))
10		eldif 3229	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐵 ∩ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐵 ∩ 𝐶)))
11		elin 3412	. . . . . . 7 ⊢ (𝑥 ∈ (𝐵 ∩ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))
12	11	notbii 678	. . . . . 6 ⊢ (¬ 𝑥 ∈ (𝐵 ∩ 𝐶) ↔ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))
13	12	anbi2i 461	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐵 ∩ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)))
14	10, 13	bitr2i 185	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)) ↔ 𝑥 ∈ (𝐴 ∖ (𝐵 ∩ 𝐶)))
15	9, 14	sylib 122	. . 3 ⊢ ((𝑥 ∈ (𝐴 ∖ 𝐵) ∨ 𝑥 ∈ (𝐴 ∖ 𝐶)) → 𝑥 ∈ (𝐴 ∖ (𝐵 ∩ 𝐶)))
16	1, 15	sylbi 121	. 2 ⊢ (𝑥 ∈ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ 𝐶)) → 𝑥 ∈ (𝐴 ∖ (𝐵 ∩ 𝐶)))
17	16	ssriv 3252	1 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ 𝐶)) ⊆ (𝐴 ∖ (𝐵 ∩ 𝐶))

Syntax hints:  ¬ wn 3   ∧ wa 104   ∨ wo 720   ∈ wcel 2209   ∖ cdif 3217   ∪ cun 3218   ∩ cin 3219   ⊆ wss 3220

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233

This theorem is referenced by:  difdif2ss  3488  indmss  3490