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Metamath Proof Explorer

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 df-ss ⊢ ( 𝐴 ⊆ ( 𝐵 ∪ 𝐶 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ∈ ( 𝐵 ∪ 𝐶 ) ) )
9 df-ss ⊢ ( ( 𝐴 ∖ 𝐵 ) ⊆ 𝐶 ↔ ∀ 𝑥 ( 𝑥 ∈ ( 𝐴 ∖ 𝐵 ) → 𝑥 ∈ 𝐶 ) )
10 7 8 9 3bitr4i ⊢ ( 𝐴 ⊆ ( 𝐵 ∪ 𝐶 ) ↔ ( 𝐴 ∖ 𝐵 ) ⊆ 𝐶 )