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Metamath Proof Explorer

Theorem eqvreldisj4

Description: Intersection with the converse epsilon relation restricted to the quotient set of an equivalence relation is disjoint. (Contributed by Peter Mazsa, 30-May-2020) (Revised by Peter Mazsa, 31-Dec-2021)

		Ref	Expression
	Assertion	eqvreldisj4	$⊢ EqvRel R \to Disj (S \cap ({E^{-1} ↾}_{(B / R)}))$

Proof

Step	Hyp	Ref	Expression
1		eqvreldisj3	$⊢ EqvRel R \to Disj ({E^{-1} ↾}_{(B / R)})$
2		disjimin	$⊢ Disj ({E^{-1} ↾}_{(B / R)}) \to Disj (S \cap ({E^{-1} ↾}_{(B / R)}))$
3	1 2	syl	$⊢ EqvRel R \to Disj (S \cap ({E^{-1} ↾}_{(B / R)}))$