A colleague of mine suggested the following weakening of the axiom of choice:

If $\mathscr{F} := \{F_\alpha\}$ is a

well-orderedfamily of non-empty sets (i.e., there is a bijection between $\mathscr{F}$ and some von Neumann ordinal), then there is a choice function $f$ with domain $\mathscr F$ such that $f(F_\alpha) \in F_\alpha$ for all $\alpha$.

The intuition behind this axiom is that the well ordering allows us to imagine picking the representatives "one at a time," and at any stage there is always a clear candidate for the next $F_\alpha$ to consider.

**Question:** How does this axiom compare to other known weak forms of the axiom of choice?

Clearly it implies countable choice, but does it imply anything interesting that countable choice does not?