Theorem eqvrelqseqdisj5

Description: Lemma for the Partition-Equivalence Theorem pet2 . (Contributed by Peter Mazsa, 15-Jul-2020) (Revised by Peter Mazsa, 22-Sep-2021)

	Ref	Expression
Assertion	eqvrelqseqdisj5	$$⊢\left(EqvRelR\wedge B/R=A\right)\to DisjS⋉\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)$$

Proof

Step	Hyp	Ref	Expression
1		eqvrelqseqdisj3	$$⊢\left(EqvRelR\wedge B/R=A\right)\to Disj\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)$$
2		disjimxrn	$$⊢Disj\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)\to DisjS⋉\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)$$
3	1 2	syl	$$⊢\left(EqvRelR\wedge B/R=A\right)\to DisjS⋉\left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)$$