Boolean cover by a generic quasi-closed subtopos (${\mathrm{sh}}_{q(\top )}(\mathcal{E}/\Omega )\to \mathcal{E}$)

In $\mathcal{E}/\Omega$, the generic truth mono $\top :1\hookrightarrow \Omega$ defines a quasi-closed Boolean subtopos. Its composite to $\mathcal{E}$ is surjective. If a predicate $p(x)$ pulled back from $\mathcal{E}$ becomes dense, then $((p(x)\Rightarrow u)\Rightarrow u)=1$ for the generic truth $u$. Substitution $u=p(x)$ yields $p(x)=1$, proving reflection of invertible monos and hence faithfulness.