Theorem undif4 3281

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		pm2.621 666	. . . . . . 7 ⊢ ((x ∈ A → ¬ x ∈ 𝐶) → ((x ∈ A ∨ ¬ x ∈ 𝐶) → ¬ x ∈ 𝐶))
2		olc 632	. . . . . . 7 ⊢ (¬ x ∈ 𝐶 → (x ∈ A ∨ ¬ x ∈ 𝐶))
3	1, 2	impbid1 130	. . . . . 6 ⊢ ((x ∈ A → ¬ x ∈ 𝐶) → ((x ∈ A ∨ ¬ x ∈ 𝐶) ↔ ¬ x ∈ 𝐶))
4	3	anbi2d 437	. . . . 5 ⊢ ((x ∈ A → ¬ x ∈ 𝐶) → (((x ∈ A ∨ x ∈ B) ∧ (x ∈ A ∨ ¬ x ∈ 𝐶)) ↔ ((x ∈ A ∨ x ∈ B) ∧ ¬ x ∈ 𝐶)))
5		eldif 2924	. . . . . . 7 ⊢ (x ∈ (B ∖ 𝐶) ↔ (x ∈ B ∧ ¬ x ∈ 𝐶))
6	5	orbi2i 679	. . . . . 6 ⊢ ((x ∈ A ∨ x ∈ (B ∖ 𝐶)) ↔ (x ∈ A ∨ (x ∈ B ∧ ¬ x ∈ 𝐶)))
7		ordi 729	. . . . . 6 ⊢ ((x ∈ A ∨ (x ∈ B ∧ ¬ x ∈ 𝐶)) ↔ ((x ∈ A ∨ x ∈ B) ∧ (x ∈ A ∨ ¬ x ∈ 𝐶)))
8	6, 7	bitri 173	. . . . 5 ⊢ ((x ∈ A ∨ x ∈ (B ∖ 𝐶)) ↔ ((x ∈ A ∨ x ∈ B) ∧ (x ∈ A ∨ ¬ x ∈ 𝐶)))
9		elun 3081	. . . . . 6 ⊢ (x ∈ (A ∪ B) ↔ (x ∈ A ∨ x ∈ B))
10	9	anbi1i 431	. . . . 5 ⊢ ((x ∈ (A ∪ B) ∧ ¬ x ∈ 𝐶) ↔ ((x ∈ A ∨ x ∈ B) ∧ ¬ x ∈ 𝐶))
11	4, 8, 10	3bitr4g 212	. . . 4 ⊢ ((x ∈ A → ¬ x ∈ 𝐶) → ((x ∈ A ∨ x ∈ (B ∖ 𝐶)) ↔ (x ∈ (A ∪ B) ∧ ¬ x ∈ 𝐶)))
12		elun 3081	. . . 4 ⊢ (x ∈ (A ∪ (B ∖ 𝐶)) ↔ (x ∈ A ∨ x ∈ (B ∖ 𝐶)))
13		eldif 2924	. . . 4 ⊢ (x ∈ ((A ∪ B) ∖ 𝐶) ↔ (x ∈ (A ∪ B) ∧ ¬ x ∈ 𝐶))
14	11, 12, 13	3bitr4g 212	. . 3 ⊢ ((x ∈ A → ¬ x ∈ 𝐶) → (x ∈ (A ∪ (B ∖ 𝐶)) ↔ x ∈ ((A ∪ B) ∖ 𝐶)))
15	14	alimi 1344	. 2 ⊢ (∀x(x ∈ A → ¬ x ∈ 𝐶) → ∀x(x ∈ (A ∪ (B ∖ 𝐶)) ↔ x ∈ ((A ∪ B) ∖ 𝐶)))
16		disj1 3267	. 2 ⊢ ((A ∩ 𝐶) = ∅ ↔ ∀x(x ∈ A → ¬ x ∈ 𝐶))
17		dfcleq 2034	. 2 ⊢ ((A ∪ (B ∖ 𝐶)) = ((A ∪ B) ∖ 𝐶) ↔ ∀x(x ∈ (A ∪ (B ∖ 𝐶)) ↔ x ∈ ((A ∪ B) ∖ 𝐶)))
18	15, 16, 17	3imtr4i 190	1 ⊢ ((A ∩ 𝐶) = ∅ → (A ∪ (B ∖ 𝐶)) = ((A ∪ B) ∖ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629  ∀wal 1241   = wceq 1243   ∈ wcel 1393   ∖ cdif 2911   ∪ cun 2912   ∩ cin 2913  ∅c0 3221

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-v 2556  df-dif 2917  df-un 2919  df-in 2921  df-nul 3222

This theorem is referenced by: (None)