Theorem undif3 4244

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.) (Proof shortened by JJ, 13-Jul-2021.)

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 4151	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
2		pm4.53 1001	. . . . 5 ⊢ (¬ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴) ↔ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴))
3		eldif 4048	. . . . 5 ⊢ (𝑥 ∈ (𝐶 ∖ 𝐴) ↔ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴))
4	2, 3	xchnxbir 336	. . . 4 ⊢ (¬ 𝑥 ∈ (𝐶 ∖ 𝐴) ↔ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴))
5	1, 4	anbi12i 640	. . 3 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ ¬ 𝑥 ∈ (𝐶 ∖ 𝐴)) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)))
6		eldif 4048	. . 3 ⊢ (𝑥 ∈ ((𝐴 ∪ 𝐵) ∖ (𝐶 ∖ 𝐴)) ↔ (𝑥 ∈ (𝐴 ∪ 𝐵) ∧ ¬ 𝑥 ∈ (𝐶 ∖ 𝐴)))
7		elun 4151	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ (𝐵 ∖ 𝐶)))
8		eldif 4048	. . . . 5 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶))
9	8	orbi2i 926	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ (𝐵 ∖ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)))
10		ordi 1023	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐶)))
11		orcom 884	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐶) ↔ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴))
12	11	anbi2i 635	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)))
13	10, 12	bitri 278	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)))
14	7, 9, 13	3bitri 300	. . 3 ⊢ (𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ (¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴)))
15	5, 6, 14	3bitr4ri 307	. 2 ⊢ (𝑥 ∈ (𝐴 ∪ (𝐵 ∖ 𝐶)) ↔ 𝑥 ∈ ((𝐴 ∪ 𝐵) ∖ (𝐶 ∖ 𝐴)))
16	15	eqriv 2757	1 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐶)) = ((𝐴 ∪ 𝐵) ∖ (𝐶 ∖ 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2144   ∖ cdif 4044   ∪ cun 4148

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-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-dif 4046  df-un 4150

This theorem is used by:  undifabs  4433  psdmullem  22493  llycmpkgen2  23876  hgt750lemb  35333  cantnfresb  44433