Theorem eqvrelqseqdisj4

Description: Lemma for petincnvepres2 . (Contributed by Peter Mazsa, 31-Dec-2021)

	Ref	Expression
Assertion	eqvrelqseqdisj4	$$⊢\left(EqvRelR\wedge B/R=A\right)\to Disj\left(S\cap \left({{\mathrm{E}}^{-1}\upharpoonright }_A\right)\right)$$

Proof

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