Remark 2
Tag 06L1, permanent. Cite it as [Bourbaki, Tag 06L1].
Algebra, V, p. 45, § 7. Separable algebraic extensions
\* Let E be an algebraic and separable extension of K. If every algebraic and separable extension of K is isomorphic to a subextension of E, then E is a separable closure of K. For if E' is a separable closure of K, then each of the extensions E and E' is isomorphic to a subextension of the other; hence E and E' are isomorphic extensions of K (V, p. 52, Prop. 1, a)).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.