Theorem disjeq

Description: Equality theorem for disjoints. (Contributed by Peter Mazsa, 22-Sep-2021)

	Ref	Expression
Assertion	disjeq	⊢ ( 𝐴 = 𝐵 → ( Disj 𝐴 ↔ Disj 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		eqimss2	⊢ ( 𝐴 = 𝐵 → 𝐵 ⊆ 𝐴 )
2	1	disjssd	⊢ ( 𝐴 = 𝐵 → ( Disj 𝐴 → Disj 𝐵 ) )
3		eqimss	⊢ ( 𝐴 = 𝐵 → 𝐴 ⊆ 𝐵 )
4	3	disjssd	⊢ ( 𝐴 = 𝐵 → ( Disj 𝐵 → Disj 𝐴 ) )
5	2 4	impbid	⊢ ( 𝐴 = 𝐵 → ( Disj 𝐴 ↔ Disj 𝐵 ) )