Theorem undif4

Description: Distribute union over difference. (Contributed by NM, 17-May-1998) (Proof shortened by Andrew Salmon, 26-Jun-2011)

	Ref	Expression
Assertion	undif4	⊢ ( ( 𝐴 ∩ 𝐶 ) = ∅ → ( 𝐴 ∪ ( 𝐵 ∖ 𝐶 ) ) = ( ( 𝐴 ∪ 𝐵 ) ∖ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		pm2.621	⊢ ( ( 𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶 ) → ( ( 𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐶 ) → ¬ 𝑥 ∈ 𝐶 ) )
2		olc	⊢ ( ¬ 𝑥 ∈ 𝐶 → ( 𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐶 ) )
3	1 2	impbid1	⊢ ( ( 𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶 ) → ( ( 𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐶 ) ↔ ¬ 𝑥 ∈ 𝐶 ) )
4	3	anbi2d	⊢ ( ( 𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶 ) → ( ( ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐶 ) ) ↔ ( ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) ∧ ¬ 𝑥 ∈ 𝐶 ) ) )
5		eldif	⊢ ( 𝑥 ∈ ( 𝐵 ∖ 𝐶 ) ↔ ( 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶 ) )
6	5	orbi2i	⊢ ( ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ ( 𝐵 ∖ 𝐶 ) ) ↔ ( 𝑥 ∈ 𝐴 ∨ ( 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶 ) ) )
7		ordi	⊢ ( ( 𝑥 ∈ 𝐴 ∨ ( 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶 ) ) ↔ ( ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐶 ) ) )
8	6 7	bitri	⊢ ( ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ ( 𝐵 ∖ 𝐶 ) ) ↔ ( ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∨ ¬ 𝑥 ∈ 𝐶 ) ) )
9		elun	⊢ ( 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) )
10	9	anbi1i	⊢ ( ( 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ∧ ¬ 𝑥 ∈ 𝐶 ) ↔ ( ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) ∧ ¬ 𝑥 ∈ 𝐶 ) )
11	4 8 10	3bitr4g	⊢ ( ( 𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶 ) → ( ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ ( 𝐵 ∖ 𝐶 ) ) ↔ ( 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ∧ ¬ 𝑥 ∈ 𝐶 ) ) )
12		elun	⊢ ( 𝑥 ∈ ( 𝐴 ∪ ( 𝐵 ∖ 𝐶 ) ) ↔ ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ ( 𝐵 ∖ 𝐶 ) ) )
13		eldif	⊢ ( 𝑥 ∈ ( ( 𝐴 ∪ 𝐵 ) ∖ 𝐶 ) ↔ ( 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ∧ ¬ 𝑥 ∈ 𝐶 ) )
14	11 12 13	3bitr4g	⊢ ( ( 𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶 ) → ( 𝑥 ∈ ( 𝐴 ∪ ( 𝐵 ∖ 𝐶 ) ) ↔ 𝑥 ∈ ( ( 𝐴 ∪ 𝐵 ) ∖ 𝐶 ) ) )
15	14	alimi	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶 ) → ∀ 𝑥 ( 𝑥 ∈ ( 𝐴 ∪ ( 𝐵 ∖ 𝐶 ) ) ↔ 𝑥 ∈ ( ( 𝐴 ∪ 𝐵 ) ∖ 𝐶 ) ) )
16		disj1	⊢ ( ( 𝐴 ∩ 𝐶 ) = ∅ ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐶 ) )
17		dfcleq	⊢ ( ( 𝐴 ∪ ( 𝐵 ∖ 𝐶 ) ) = ( ( 𝐴 ∪ 𝐵 ) ∖ 𝐶 ) ↔ ∀ 𝑥 ( 𝑥 ∈ ( 𝐴 ∪ ( 𝐵 ∖ 𝐶 ) ) ↔ 𝑥 ∈ ( ( 𝐴 ∪ 𝐵 ) ∖ 𝐶 ) ) )
18	15 16 17	3imtr4i	⊢ ( ( 𝐴 ∩ 𝐶 ) = ∅ → ( 𝐴 ∪ ( 𝐵 ∖ 𝐶 ) ) = ( ( 𝐴 ∪ 𝐵 ) ∖ 𝐶 ) )