Theorem detlem

Description: If a relation is disjoint, then it is equivalent to the equivalent cosets of the relation, inference version. (Contributed by Peter Mazsa, 30-Sep-2021)

	Ref	Expression
Hypothesis	detlem.1	⊢ Disj 𝑅
Assertion	detlem	⊢ ( Disj 𝑅 ↔ EqvRel ≀ 𝑅 )

Proof

Step	Hyp	Ref	Expression
1		detlem.1	⊢ Disj 𝑅
2		disjim	⊢ ( Disj 𝑅 → EqvRel ≀ 𝑅 )
3	1	a1i	⊢ ( EqvRel ≀ 𝑅 → Disj 𝑅 )
4	2 3	impbii	⊢ ( Disj 𝑅 ↔ EqvRel ≀ 𝑅 )