Existence of transcendence bases

Apply Zorn's lemma to subsets of a field extension with algebraic independence, ordered by inclusion. The union of a chain is independent because any polynomial relation involves only finitely many distinct elements, all of which lie in one member of the chain. A maximal independent set $B$ is a transcendence basis: if any element were transcendental over $K(B)$, adjoining it would preserve independence, by clearing denominators, contradicting maximality. Tuple entries must be distinct in the set-based independence definition; otherwise the nonzero polynomial $X_1-X_2$ vanishes on a repeated pair and excludes every nonempty set.