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Metamath Proof Explorer

Theorem detidres

Description: The cosets by the restricted identity relation are in equivalence relation if and only if the restricted identity relation is disjoint. (Contributed by Peter Mazsa, 31-Dec-2021)

		Ref	Expression
	Assertion	detidres	⊢ ( Disj ( I ↾ 𝐴 ) ↔ EqvRel ≀ ( I ↾ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		disjALTVidres	⊢ Disj ( I ↾ 𝐴 )
2	1	detlem	⊢ ( Disj ( I ↾ 𝐴 ) ↔ EqvRel ≀ ( I ↾ 𝐴 ) )