How much of Zorn's lemma can be saved if we assume only ZF+DC without full choice? More precisely: assume we have a partially ordered (inductive) set which is of size continuum. Then can we apply Zorn's lemma in this case assuming only dependent choice?
It is a theorem of Wolk  that for every infinite $\kappa$ the axiom $\sf DC_\kappa$ is equivalent to the following statement:
If $(P,\leq)$ is a partially ordered set in which every well-ordered chain has order type $<\kappa$ and has an upper-bound, then $P$ contains a maximal element.
In the case of $\kappa=\omega$ this means that every partial order in which every well-ordered chain is finite and bounded has a maximal element.
Note that there is no limitation on the cardinality of $P$. Only on its well-ordered chains.
- Elliot S. Wolk, On the principle of dependent choices and some forms of Zorn’s lemma, Canad. Math. Bull. 26 (1983), no. 3, 365–367.