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?
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It is a theorem of Wolk  that for every infinite $\kappa$ the axiom $\sf DC_\kappa$ is equivalent to the following statement:
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.