Theorem difindiss 3431

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 3318	. . 3 ⊢ (𝑥 ∈ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ 𝐶)) ↔ (𝑥 ∈ (𝐴 ∖ 𝐵) ∨ 𝑥 ∈ (𝐴 ∖ 𝐶)))
2		eldif 3179	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
3		eldif 3179	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶))
4	2, 3	orbi12i 766	. . . . . 6 ⊢ ((𝑥 ∈ (𝐴 ∖ 𝐵) ∨ 𝑥 ∈ (𝐴 ∖ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∨ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐶)))
5		andi 820	. . . . . 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		pm3.14 755	. . . . . 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 3179	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐵 ∩ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐵 ∩ 𝐶)))
11		elin 3360	. . . . . . 7 ⊢ (𝑥 ∈ (𝐵 ∩ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))
12	11	notbii 670	. . . . . 6 ⊢ (¬ 𝑥 ∈ (𝐵 ∩ 𝐶) ↔ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))
13	12	anbi2i 457	. . . . 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 3201	1 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ 𝐶)) ⊆ (𝐴 ∖ (𝐵 ∩ 𝐶))

Syntax hints:  ¬ wn 3   ∧ wa 104   ∨ wo 710   ∈ wcel 2177   ∖ cdif 3167   ∪ cun 3168   ∩ cin 3169   ⊆ wss 3170

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183

This theorem is referenced by:  difdif2ss  3434  indmss  3436