Theorem undif4 3278

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		pm2.621 665	. . . . . . 7 ⊢ ((x ∈ A → ¬ x ∈ 𝐶) → ((x ∈ A ∨ ¬ x ∈ 𝐶) → ¬ x ∈ 𝐶))
2		olc 631	. . . . . . 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 2921	. . . . . . 7 ⊢ (x ∈ (B ∖ 𝐶) ↔ (x ∈ B ∧ ¬ x ∈ 𝐶))
6	5	orbi2i 678	. . . . . 6 ⊢ ((x ∈ A ∨ x ∈ (B ∖ 𝐶)) ↔ (x ∈ A ∨ (x ∈ B ∧ ¬ x ∈ 𝐶)))
7		ordi 728	. . . . . 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 3078	. . . . . 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 3078	. . . 4 ⊢ (x ∈ (A ∪ (B ∖ 𝐶)) ↔ (x ∈ A ∨ x ∈ (B ∖ 𝐶)))
13		eldif 2921	. . . 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 1341	. 2 ⊢ (∀x(x ∈ A → ¬ x ∈ 𝐶) → ∀x(x ∈ (A ∪ (B ∖ 𝐶)) ↔ x ∈ ((A ∪ B) ∖ 𝐶)))
16		disj1 3264	. 2 ⊢ ((A ∩ 𝐶) = ∅ ↔ ∀x(x ∈ A → ¬ x ∈ 𝐶))
17		dfcleq 2031	. 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 628  ∀wal 1240   = wceq 1242   ∈ wcel 1390   ∖ cdif 2908   ∪ cun 2909   ∩ cin 2910  ∅c0 3218

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-nul 3219

This theorem is referenced by: (None)