Theorem disjuniel

Description: A set of elements B of a disjoint set A is disjoint with another element of that set. (Contributed by Thierry Arnoux, 24-May-2020)

	Ref	Expression
Hypotheses	disjuniel.1	⊢ ( 𝜑 → Disj 𝑥 ∈ 𝐴 𝑥 )
	disjuniel.2	⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 )
	disjuniel.3	⊢ ( 𝜑 → 𝐶 ∈ ( 𝐴 ∖ 𝐵 ) )
Assertion	disjuniel	⊢ ( 𝜑 → ( ∪ 𝐵 ∩ 𝐶 ) = ∅ )

Proof

Step	Hyp	Ref	Expression
1		disjuniel.1	⊢ ( 𝜑 → Disj 𝑥 ∈ 𝐴 𝑥 )
2		disjuniel.2	⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 )
3		disjuniel.3	⊢ ( 𝜑 → 𝐶 ∈ ( 𝐴 ∖ 𝐵 ) )
4		uniiun	⊢ ∪ 𝐵 = ∪ 𝑥 ∈ 𝐵 𝑥
5	4	ineq1i	⊢ ( ∪ 𝐵 ∩ 𝐶 ) = ( ∪ 𝑥 ∈ 𝐵 𝑥 ∩ 𝐶 )
6		id	⊢ ( 𝑥 = 𝐶 → 𝑥 = 𝐶 )
7	1 6 2 3	disjiunel	⊢ ( 𝜑 → ( ∪ 𝑥 ∈ 𝐵 𝑥 ∩ 𝐶 ) = ∅ )
8	5 7	syl5eq	⊢ ( 𝜑 → ( ∪ 𝐵 ∩ 𝐶 ) = ∅ )