The proofs I know of Zorn's lemma give the following refinement:

Let $(X,<)$ be a partially ordered set such that every well-ordered subset of $X$ has an upperbound. Then $X$ has a maximal element.

In fact, Zorn's lemma is sometimes stated as such. In comparison, the usual statement
asks that *every totally ordered subset of $X$ has an upper bound.*

Does this refinement have some application?

EDIT (after comments). Your comments make me realize that I did not think enough of my question, which is what the voters-to-close probably guessed. To help future readers, let me sum up the comments.

If $(X,<)$ satisfies the hypothesis of the refined Zorn lemma (RZL), then (modulo AC), it satisfies the hypothesis of classical Zorn lemma as well, since any totally ordered set contains a cofinal ordered subset. (Comment of Ramiro de la Vega).

ZF+RZL implies ZF+AC (Noah S).

In Cohen's first model of ZF, there exists a set which satisfies the hypothesis of RZL but not that of ZL (Asaf Karagila).

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