**Proof.**
The implication (1) $\Rightarrow $ (3) follows from Lemma 65.40.5. We will use Spaces, Lemma 63.5.8 without further mention in the rest of the proof. Recall that $\Delta _{X/Y}$ is a representable monomorphism which is locally of finite type, see Lemma 65.4.1. Since proper $\Rightarrow $ universally closed for morphisms of schemes we conclude that (3) implies (2). If $\Delta _{X/Y}$ is universally closed then Étale Morphisms, Lemma 41.7.2 implies that it is a closed immersion. Thus (2) $\Rightarrow $ (1) and we win.
$\square$

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