The Stacks project

Proof. Let $E$ be constructible and stable under generalization. Let $Y \subset X$ be an irreducible closed subset with generic point $\xi \in Y$. If $E \cap Y$ is nonempty, then it contains $\xi$ (by stability under generalization) and hence is dense in $Y$, hence it contains a nonempty open of $Y$, see Lemma 5.16.3. Thus $E$ is open by Lemma 5.16.5. This proves (2). To prove (1) apply (2) to the complement of $E$ in $X$. $\square$