Lemma 100.50.3. Let $\mathcal{X}$ be an algebraic stack which is reduced and quasi-separated and whose associated topological space $|\mathcal{X}|$ is irreducible. Then $\mathcal{X}$ is integral.

Proof. If $\mathcal{X}$ is quasi-separated, then $\mathcal{X}$ is decent by Lemma 100.48.2. If $\mathcal{X}$ is quasi-separated, then $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is quasi-compact, hence $\mathcal{I}_\mathcal {X} \to \mathcal{X}$ is quasi-compact as the base change of $\Delta$ by $\Delta$, see Lemma 100.7.3. Thus we see that all the hypotheses of Definition 100.50.1 hold (and we also see that we may replace “quasi-separated” by “$\Delta _\mathcal {X}$ is quasi-compact”). $\square$

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