The Stacks project

Proof. Let $j : \mathcal{U} \to \mathcal{X}$ be an open immersion of algebraic stacks. Since $j$ is representable, it is DM by Lemma 100.4.3. On the other hand, if $X \to \mathcal{X}$ is a smooth and surjective morphism where $X$ is a scheme, then $U = \mathcal{U} \times _\mathcal {X} X$ is an open subscheme of $X$. Hence $U \to X$ is étale (Morphisms, Lemma 29.36.9) and we conclude that $j$ is étale from the definition. $\square$

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