Theorem ssundif

Description: A condition equivalent to inclusion in the union of two classes. (Contributed by NM, 26-Mar-2007)

	Ref	Expression
Assertion	ssundif	⊢ ( 𝐴 ⊆ ( 𝐵 ∪ 𝐶 ) ↔ ( 𝐴 ∖ 𝐵 ) ⊆ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		pm5.6	⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵 ) → 𝑥 ∈ 𝐶 ) ↔ ( 𝑥 ∈ 𝐴 → ( 𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶 ) ) )
2		eldif	⊢ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵 ) )
3	2	imbi1i	⊢ ( ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → 𝑥 ∈ 𝐶 ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵 ) → 𝑥 ∈ 𝐶 ) )
4		elun	⊢ ( 𝑥 ∈ ( 𝐵 ∪ 𝐶 ) ↔ ( 𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶 ) )
5	4	imbi2i	⊢ ( ( 𝑥 ∈ 𝐴 → 𝑥 ∈ ( 𝐵 ∪ 𝐶 ) ) ↔ ( 𝑥 ∈ 𝐴 → ( 𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶 ) ) )
6	1 3 5	3bitr4ri	⊢ ( ( 𝑥 ∈ 𝐴 → 𝑥 ∈ ( 𝐵 ∪ 𝐶 ) ) ↔ ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → 𝑥 ∈ 𝐶 ) )
7	6	albii	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ∈ ( 𝐵 ∪ 𝐶 ) ) ↔ ∀ 𝑥 ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → 𝑥 ∈ 𝐶 ) )
8		dfss2	⊢ ( 𝐴 ⊆ ( 𝐵 ∪ 𝐶 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ∈ ( 𝐵 ∪ 𝐶 ) ) )
9		dfss2	⊢ ( ( 𝐴 ∖ 𝐵 ) ⊆ 𝐶 ↔ ∀ 𝑥 ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → 𝑥 ∈ 𝐶 ) )
10	7 8 9	3bitr4i	⊢ ( 𝐴 ⊆ ( 𝐵 ∪ 𝐶 ) ↔ ( 𝐴 ∖ 𝐵 ) ⊆ 𝐶 )