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

Proof

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