Claim: Every chain in a partially ordered set is included in some maximal chain if Zorn's Lemma holds.
Proof: Let X be a poset, let P(X) be ordered by inclusion, and let S be the set of all chains in X. Since the empty set is in S, S is not empty. So, consider the chain C in S. Now, consider the set of chains C*={D in S: D[union]C is in S}; that is, C* is the set of all chains D included in X such that D[union]C is a chain included in X. Then, C* is a subset of P(P(X)), which, ordered by inclusion, has the maximum element, P(X). So, for all chains in C*, there is an upper bound, namely, P(X). So, by Zorn's Lemma, there is a maximal element in C*, call it M; that is, if there is a chain N in S such that M is a subset of N, then N=M. Since C is in C*, then, C is included in some maximal chain included X. qed.
Pretty sweet little proof, a lot simpler than my early trials, which involved, first, recursion and, second, a recursion-like action on the empty. Hope you enjoyed this as much as I enjoyed working on it. Truly the joy of why I do these things.
