Theorem undif4 3259

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		pm2.621 653	. . . . . . 7 ⊢ ((x ∈ A → ¬ x ∈ 𝐶) → ((x ∈ A ∨ ¬ x ∈ 𝐶) → ¬ x ∈ 𝐶))
2		olc 619	. . . . . . 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 440	. . . . 5 ⊢ ((x ∈ A → ¬ x ∈ 𝐶) → (((x ∈ A ∨ x ∈ B) ∧ (x ∈ A ∨ ¬ x ∈ 𝐶)) ↔ ((x ∈ A ∨ x ∈ B) ∧ ¬ x ∈ 𝐶)))
5		eldif 2902	. . . . . . 7 ⊢ (x ∈ (B ∖ 𝐶) ↔ (x ∈ B ∧ ¬ x ∈ 𝐶))
6	5	orbi2i 666	. . . . . 6 ⊢ ((x ∈ A ∨ x ∈ (B ∖ 𝐶)) ↔ (x ∈ A ∨ (x ∈ B ∧ ¬ x ∈ 𝐶)))
7		ordi 717	. . . . . 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 3059	. . . . . 6 ⊢ (x ∈ (A ∪ B) ↔ (x ∈ A ∨ x ∈ B))
10	9	anbi1i 434	. . . . 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 3059	. . . 4 ⊢ (x ∈ (A ∪ (B ∖ 𝐶)) ↔ (x ∈ A ∨ x ∈ (B ∖ 𝐶)))
13		eldif 2902	. . . 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 1324	. 2 ⊢ (∀x(x ∈ A → ¬ x ∈ 𝐶) → ∀x(x ∈ (A ∪ (B ∖ 𝐶)) ↔ x ∈ ((A ∪ B) ∖ 𝐶)))
16		disj1 3245	. 2 ⊢ ((A ∩ 𝐶) = ∅ ↔ ∀x(x ∈ A → ¬ x ∈ 𝐶))
17		dfcleq 2016	. 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 616  ∀wal 1226   = wceq 1228   ∈ wcel 1374   ∖ cdif 2889   ∪ cun 2890   ∩ cin 2891  ∅c0 3199

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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-v 2535  df-dif 2895  df-un 2897  df-in 2899  df-nul 3200

This theorem is referenced by: (None)