With Axiom of Choice, if $\langle A,\prec\rangle$ is well ordered then A has no subset of order type $\omega^*$

Question

The question is raised when I look at this question which have been asked three years ago. I want to know if the other side of the question is still true or to be more specific :-

With Axiom of Choice, if $\langle A,\prec\rangle$ is well ordered then $A$ has no subset of order type $\omega^*$.

Answer 1

This can be shown even without choice: assume that $B\subseteq A$ has order type $\omega*$ when considered with $\prec$. Then $B$ has no minimal element, which says, by definition of well-order, that $A$ is not a well-order.