TITLE: Does well-ordering of the proper class of cardinal numbers imply choice?
QUESTION [7 upvotes]: It is well-known (forgive the pun) that the axiom of choice (which states that the product of every non-empty family of non-empty sets is non-empty) implies that the proper class of all cardinal numbers is well-ordered, at least in the presence of the ZF axioms.
Does the converse hold? If not, is the statement that the proper class of all cardinal numbers is well-ordered consistent with axioms like determinacy, which, while interesting, contradict full-blown choice?

REPLY [5 votes]: To address the part of your question about determinacy, I will mention that there is a simple counterexample to trichotomy under $\mathsf{AD}$.  Namely, $\mathbb{R}$ does not inject into $\omega_1$ and $\omega_1$ does not inject into $\mathbb{R}$.
These statements both follow from the fact that under $\mathsf{AD}$ any well-ordered set of reals is countable.
For more information about the structure of cardinals under $\mathsf{AD}$ you may want to see the paper A trichotomy theorem in natural models of
$\mathsf{AD^+}$. (Note that the trichotomy here is unrelated to that which was disproved in the above paragraph.)