Theorem undif3 3516

Description: An equality involving class union and class difference. The first equality of Exercise 13 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 17-Apr-2012.)

Assertion

Ref	Expression
undif3	⊢ (A ∪ (B ∖ C)) = ((A ∪ B) ∖ (C ∖ A))

Proof of Theorem undif3

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elun 3221	. . . 4 ⊢ (x ∈ (A ∪ B) ↔ (x ∈ A ∨ x ∈ B))
2		pm4.53 478	. . . . 5 ⊢ (¬ (x ∈ C ∧ ¬ x ∈ A) ↔ (¬ x ∈ C ∨ x ∈ A))
3		eldif 3222	. . . . 5 ⊢ (x ∈ (C ∖ A) ↔ (x ∈ C ∧ ¬ x ∈ A))
4	2, 3	xchnxbir 300	. . . 4 ⊢ (¬ x ∈ (C ∖ A) ↔ (¬ x ∈ C ∨ x ∈ A))
5	1, 4	anbi12i 678	. . 3 ⊢ ((x ∈ (A ∪ B) ∧ ¬ x ∈ (C ∖ A)) ↔ ((x ∈ A ∨ x ∈ B) ∧ (¬ x ∈ C ∨ x ∈ A)))
6		eldif 3222	. . 3 ⊢ (x ∈ ((A ∪ B) ∖ (C ∖ A)) ↔ (x ∈ (A ∪ B) ∧ ¬ x ∈ (C ∖ A)))
7		elun 3221	. . . 4 ⊢ (x ∈ (A ∪ (B ∖ C)) ↔ (x ∈ A ∨ x ∈ (B ∖ C)))
8		eldif 3222	. . . . 5 ⊢ (x ∈ (B ∖ C) ↔ (x ∈ B ∧ ¬ x ∈ C))
9	8	orbi2i 505	. . . 4 ⊢ ((x ∈ A ∨ x ∈ (B ∖ C)) ↔ (x ∈ A ∨ (x ∈ B ∧ ¬ x ∈ C)))
10		orc 374	. . . . . . 7 ⊢ (x ∈ A → (x ∈ A ∨ x ∈ B))
11		olc 373	. . . . . . 7 ⊢ (x ∈ A → (¬ x ∈ C ∨ x ∈ A))
12	10, 11	jca 518	. . . . . 6 ⊢ (x ∈ A → ((x ∈ A ∨ x ∈ B) ∧ (¬ x ∈ C ∨ x ∈ A)))
13		olc 373	. . . . . . 7 ⊢ (x ∈ B → (x ∈ A ∨ x ∈ B))
14		orc 374	. . . . . . 7 ⊢ (¬ x ∈ C → (¬ x ∈ C ∨ x ∈ A))
15	13, 14	anim12i 549	. . . . . 6 ⊢ ((x ∈ B ∧ ¬ x ∈ C) → ((x ∈ A ∨ x ∈ B) ∧ (¬ x ∈ C ∨ x ∈ A)))
16	12, 15	jaoi 368	. . . . 5 ⊢ ((x ∈ A ∨ (x ∈ B ∧ ¬ x ∈ C)) → ((x ∈ A ∨ x ∈ B) ∧ (¬ x ∈ C ∨ x ∈ A)))
17		simpl 443	. . . . . . 7 ⊢ ((x ∈ A ∧ ¬ x ∈ C) → x ∈ A)
18	17	orcd 381	. . . . . 6 ⊢ ((x ∈ A ∧ ¬ x ∈ C) → (x ∈ A ∨ (x ∈ B ∧ ¬ x ∈ C)))
19		olc 373	. . . . . 6 ⊢ ((x ∈ B ∧ ¬ x ∈ C) → (x ∈ A ∨ (x ∈ B ∧ ¬ x ∈ C)))
20		orc 374	. . . . . . 7 ⊢ (x ∈ A → (x ∈ A ∨ (x ∈ B ∧ ¬ x ∈ C)))
21	20	adantr 451	. . . . . 6 ⊢ ((x ∈ A ∧ x ∈ A) → (x ∈ A ∨ (x ∈ B ∧ ¬ x ∈ C)))
22	20	adantl 452	. . . . . 6 ⊢ ((x ∈ B ∧ x ∈ A) → (x ∈ A ∨ (x ∈ B ∧ ¬ x ∈ C)))
23	18, 19, 21, 22	ccase 912	. . . . 5 ⊢ (((x ∈ A ∨ x ∈ B) ∧ (¬ x ∈ C ∨ x ∈ A)) → (x ∈ A ∨ (x ∈ B ∧ ¬ x ∈ C)))
24	16, 23	impbii 180	. . . 4 ⊢ ((x ∈ A ∨ (x ∈ B ∧ ¬ x ∈ C)) ↔ ((x ∈ A ∨ x ∈ B) ∧ (¬ x ∈ C ∨ x ∈ A)))
25	7, 9, 24	3bitri 262	. . 3 ⊢ (x ∈ (A ∪ (B ∖ C)) ↔ ((x ∈ A ∨ x ∈ B) ∧ (¬ x ∈ C ∨ x ∈ A)))
26	5, 6, 25	3bitr4ri 269	. 2 ⊢ (x ∈ (A ∪ (B ∖ C)) ↔ x ∈ ((A ∪ B) ∖ (C ∖ A)))
27	26	eqriv 2350	1 ⊢ (A ∪ (B ∖ C)) = ((A ∪ B) ∖ (C ∖ A))

Syntax hints:  ¬ wn 3   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∖ cdif 3207   ∪ cun 3208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216

This theorem is referenced by:  undifabs  3628