Lemma 100.4.16. Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$. Assume the residual gerbe $\mathcal{Z}_ x$ of $\mathcal{X}$ at $x$ exists. If $\mathcal{X}$ is DM, resp. quasi-DM, resp. separated, resp. quasi-separated, then so is $\mathcal{Z}_ x$.

**Proof.**
This is true because $\mathcal{Z}_ x \to \mathcal{X}$ is a monomorphism hence DM and separated by Lemma 100.4.15. Apply Lemma 100.4.11 to conclude.
$\square$

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