Theorem disjin2

Description: If a collection is disjoint, so is the collection of the intersections with a given set. (Contributed by Thierry Arnoux, 21-Jun-2020)

	Ref	Expression
Assertion	disjin2	⊢ ( Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐵 ( 𝐴 ∩ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		elinel2	⊢ ( 𝑦 ∈ ( 𝐴 ∩ 𝐶 ) → 𝑦 ∈ 𝐶 )
2	1	rmoimi	⊢ ( ∃* 𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 → ∃* 𝑥 ∈ 𝐵 𝑦 ∈ ( 𝐴 ∩ 𝐶 ) )
3	2	alimi	⊢ ( ∀ 𝑦 ∃* 𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 → ∀ 𝑦 ∃* 𝑥 ∈ 𝐵 𝑦 ∈ ( 𝐴 ∩ 𝐶 ) )
4		df-disj	⊢ ( Disj 𝑥 ∈ 𝐵 𝐶 ↔ ∀ 𝑦 ∃* 𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 )
5		df-disj	⊢ ( Disj 𝑥 ∈ 𝐵 ( 𝐴 ∩ 𝐶 ) ↔ ∀ 𝑦 ∃* 𝑥 ∈ 𝐵 𝑦 ∈ ( 𝐴 ∩ 𝐶 ) )
6	3 4 5	3imtr4i	⊢ ( Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐵 ( 𝐴 ∩ 𝐶 ) )