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Metamath Proof Explorer

Theorem eqvreldisj3

Description: The elements of the quotient set of an equivalence relation are disjoint (cf. qsdisj2 ). (Contributed by Mario Carneiro, 10-Dec-2016) (Revised by Peter Mazsa, 20-Jun-2019) (Revised by Peter Mazsa, 19-Sep-2021)

		Ref	Expression
	Assertion	eqvreldisj3	$⊢ EqvRel R \to Disj ({E^{-1} ↾}_{(A / R)})$

Proof

Step	Hyp	Ref	Expression
1		eqvreldisj2	$⊢ EqvRel R \to ElDisj (A / R)$
2		df-eldisj	$⊢ ElDisj (A / R) \leftrightarrow Disj ({E^{-1} ↾}_{(A / R)})$
3	1 2	sylib	$⊢ EqvRel R \to Disj ({E^{-1} ↾}_{(A / R)})$