Proof. If $U \to X$ is étale and $U$ is a scheme, then $U$ is a locally Noetherian scheme, see Properties of Spaces, Section 64.7. A locally Noetherian scheme has a locally finite set of irreducible components (Divisors, Lemma 31.26.1). Thus we conclude that $X$ passes condition (2) of the lemma. $\square$
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