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	$$⊢EqvRelR\to Disj\left({{\mathrm{E}}^{-1}\upharpoonright }_{\left(A/R\right)}\right)$$

Proof

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