Theorem undif3

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)

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

Proof

Step	Hyp	Ref	Expression
1		elun	⊢ ( 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) )
2		pm4.53	⊢ ( ¬ ( 𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴 ) ↔ ( ¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴 ) )
3		eldif	⊢ ( 𝑥 ∈ ( 𝐶 ∖ 𝐴 ) ↔ ( 𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴 ) )
4	2 3	xchnxbir	⊢ ( ¬ 𝑥 ∈ ( 𝐶 ∖ 𝐴 ) ↔ ( ¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴 ) )
5	1 4	anbi12i	⊢ ( ( 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ∧ ¬ 𝑥 ∈ ( 𝐶 ∖ 𝐴 ) ) ↔ ( ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) ∧ ( ¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴 ) ) )
6		eldif	⊢ ( 𝑥 ∈ ( ( 𝐴 ∪ 𝐵 ) ∖ ( 𝐶 ∖ 𝐴 ) ) ↔ ( 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ∧ ¬ 𝑥 ∈ ( 𝐶 ∖ 𝐴 ) ) )
7		elun	⊢ ( 𝑥 ∈ ( 𝐴 ∪ ( 𝐵 ∖ 𝐶 ) ) ↔ ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ ( 𝐵 ∖ 𝐶 ) ) )
8		eldif	⊢ ( 𝑥 ∈ ( 𝐵 ∖ 𝐶 ) ↔ ( 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶 ) )
9	8	orbi2i	⊢ ( ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ ( 𝐵 ∖ 𝐶 ) ) ↔ ( 𝑥 ∈ 𝐴 ∨ ( 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶 ) ) )
10		ordi	⊢ ( ( 𝑥 ∈ 𝐴 ∨ ( 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶 ) ) ↔ ( ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐶 ) ) )
11		orcom	⊢ ( ( 𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐶 ) ↔ ( ¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴 ) )
12	11	anbi2i	⊢ ( ( ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐶 ) ) ↔ ( ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) ∧ ( ¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴 ) ) )
13	10 12	bitri	⊢ ( ( 𝑥 ∈ 𝐴 ∨ ( 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶 ) ) ↔ ( ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) ∧ ( ¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴 ) ) )
14	7 9 13	3bitri	⊢ ( 𝑥 ∈ ( 𝐴 ∪ ( 𝐵 ∖ 𝐶 ) ) ↔ ( ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) ∧ ( ¬ 𝑥 ∈ 𝐶 ∨ 𝑥 ∈ 𝐴 ) ) )
15	5 6 14	3bitr4ri	⊢ ( 𝑥 ∈ ( 𝐴 ∪ ( 𝐵 ∖ 𝐶 ) ) ↔ 𝑥 ∈ ( ( 𝐴 ∪ 𝐵 ) ∖ ( 𝐶 ∖ 𝐴 ) ) )
16	15	eqriv	⊢ ( 𝐴 ∪ ( 𝐵 ∖ 𝐶 ) ) = ( ( 𝐴 ∪ 𝐵 ) ∖ ( 𝐶 ∖ 𝐴 ) )