Theorem disjeq1

Description: Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016)

	Ref	Expression
Assertion	disjeq1	⊢ ( 𝐴 = 𝐵 → ( Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		eqimss2	⊢ ( 𝐴 = 𝐵 → 𝐵 ⊆ 𝐴 )
2		disjss1	⊢ ( 𝐵 ⊆ 𝐴 → ( Disj 𝑥 ∈ 𝐴 𝐶 → Disj 𝑥 ∈ 𝐵 𝐶 ) )
3	1 2	syl	⊢ ( 𝐴 = 𝐵 → ( Disj 𝑥 ∈ 𝐴 𝐶 → Disj 𝑥 ∈ 𝐵 𝐶 ) )
4		eqimss	⊢ ( 𝐴 = 𝐵 → 𝐴 ⊆ 𝐵 )
5		disjss1	⊢ ( 𝐴 ⊆ 𝐵 → ( Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐴 𝐶 ) )
6	4 5	syl	⊢ ( 𝐴 = 𝐵 → ( Disj 𝑥 ∈ 𝐵 𝐶 → Disj 𝑥 ∈ 𝐴 𝐶 ) )
7	3 6	impbid	⊢ ( 𝐴 = 𝐵 → ( Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶 ) )