In regular ZF, AC, WO and Zorn's Lemma are equivalent, but every proof I know (of the implication AC>WO and AC> Zorn) uses the axiom of choice on the powerset of X (where X is the Set which is to be wellordered). My question is, whether or not there is a proof of this equivalence that doesn't use the axiom of choice or whether there is a Model of ZFPowerset in which AC holds but the wellordering principle (or Zorns Lemma) fails.
This is a classic theorem of Zarach, that it is consistent that ${\sf ZF}^$ holds with the Axiom of Choice, but not every set can be wellordered.
Zarach, Andrzej, Unions of ${\sf ZF}^$models which are themselves ${\sf ZF}^$ models, Logic colloquium ’80, Eur. Summer Meet., Prague 1980, Stud. Logic Found. Math. 108, 315342 (1982). ZBL0524.03039.

1$\begingroup$ In the quoted paper (p.338), Zarach credits Zbigniew Szczepaniak with first having demonstrated in 1979 that in the ZF^ context, the axiom of choice does not imply that every set can be wellordered; and he acknowledges how Szcepaniak's work relates to his. $\endgroup$ – Ali Enayat Apr 26 at 19:24