Theorem undif4 4419

Description: Distribute union over difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
undif4	⊢ ((𝐴 ∩ 𝐶) = ∅ → (𝐴 ∪ (𝐵 ∖ 𝐶)) = ((𝐴 ∪ 𝐵) ∖ 𝐶))

Proof of Theorem undif4

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pm2.621 912	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶) → ((𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐶) → ¬ 𝑥 ∈ 𝐶))
2		olc 882	. . . . . . 7 ⊢ (¬ 𝑥 ∈ 𝐶 → (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐶))
3	1, 2	impbid1 228	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶) → ((𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐶) ↔ ¬ 𝑥 ∈ 𝐶))
4	3	anbi2d 642	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶) → (((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ ¬ 𝑥 ∈ 𝐶)))
5		eldif 4048	. . . . . . 7 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))
6	5	orbi2i 926	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ (𝐵 ∖ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
7		ordi 1023	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐶)))
8	6, 7	bitri 278	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ (𝐵 ∖ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐶)))
9		elun 4151	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
10	9	anbi1i 636	. . . . 5 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ ¬ 𝑥 ∈ 𝐶) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ ¬ 𝑥 ∈ 𝐶))
11	4, 8, 10	3bitr4g 317	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶) → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ (𝐵 ∖ 𝐶)) ↔ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ ¬ 𝑥 ∈ 𝐶)))
12		elun 4151	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ (𝐵 ∖ 𝐶)))
13		eldif 4048	. . . 4 ⊢ (𝑥 ∈ ((𝐴 ∪ 𝐵) ∖ 𝐶) ↔ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ ¬ 𝑥 ∈ 𝐶))
14	11, 12, 13	3bitr4g 317	. . 3 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶) → (𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐶)) ↔ 𝑥 ∈ ((𝐴 ∪ 𝐵) ∖ 𝐶)))
15	14	alimi 1843	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶) → ∀𝑥(𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐶)) ↔ 𝑥 ∈ ((𝐴 ∪ 𝐵) ∖ 𝐶)))
16		disj1 4404	. 2 ⊢ ((𝐴 ∩ 𝐶) = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶))
17		dfcleq 2753	. 2 ⊢ ((𝐴 ∪ (𝐵 ∖ 𝐶)) = ((𝐴 ∪ 𝐵) ∖ 𝐶) ↔ ∀𝑥(𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐶)) ↔ 𝑥 ∈ ((𝐴 ∪ 𝐵) ∖ 𝐶)))
18	15, 16, 17	3imtr4i 295	1 ⊢ ((𝐴 ∩ 𝐶) = ∅ → (𝐴 ∪ (𝐵 ∖ 𝐶)) = ((𝐴 ∪ 𝐵) ∖ 𝐶))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861  ∀wal 1567   = wceq 1569   ∈ wcel 2144   ∖ cdif 4044   ∩ cin 4088   ∪ cun 4148  ∅c0 4277

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-v 3452  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278

This theorem is used by:  infdifsn  9665  difico  33425  lindsunlem  34306  caratheodorylem1  47672