Theorem disjss1

Description: A subset of a disjoint collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016)

	Ref	Expression
Assertion	disjss1	⊢ ( 𝐴 ⊆ 𝐵 → ( Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐴 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		ssel	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) )
2	1	anim1d	⊢ ( 𝐴 ⊆ 𝐵 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶 ) → ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) ) )
3	2	moimdv	⊢ ( 𝐴 ⊆ 𝐵 → ( ∃* 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) → ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶 ) ) )
4	3	alimdv	⊢ ( 𝐴 ⊆ 𝐵 → ( ∀ 𝑦 ∃* 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) → ∀ 𝑦 ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶 ) ) )
5		dfdisj2	⊢ ( Disj 𝑥 ∈ 𝐵 𝐶 ↔ ∀ 𝑦 ∃* 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶 ) )
6		dfdisj2	⊢ ( Disj 𝑥 ∈ 𝐴 𝐶 ↔ ∀ 𝑦 ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶 ) )
7	4 5 6	3imtr4g	⊢ ( 𝐴 ⊆ 𝐵 → ( Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐴 𝐶 ) )