Question: show that if is finite, then any rational preference relation generates a nonempty choice rule; that is, for any with

We know that if a preference relation is rational, then it is both transitive and complete. Then the choice structured generated satisfies the weak axiom. If we have all subsets of up to 3 elements as well, then then for all . Then it's just a matter of showing that is nonempty, which will in turn show that is nonempty. However, we don't know if we have all subsets of up to 3 elements or not, so the proof cannot be done this way.

Any tips? Thanks!