A family of sets F is called interesting if $$\exists n \in \omega \forall X (X \in F \iff \forall Y (Y \subseteq X \land |Y| \leq n \implies Y \in F)$$ I have the following lemma:
If F is an interesting family and $x\in F$, then there is a maximal $Y \in F$ (according to the $\subseteq$), such that $X \subseteq Y$.
I need to shouw that this lemma is equivalent to the axiom of choice.
My attempt:
I have noticed that this lemma has a tight connection with Teichmüller–Tukey lemma.(which is equivalent to the Axiom of Choice)
It's easy to see that every interesting Family is of finite character. So the given lemma is a bit "stronger" than the Teichmüller–Tukey lemma. But can I conclude that lemma and the AC are equivalent from that? Probably just the first direction (Lemma $\implies$ Axiom of Choice). But I can't figure the other direction. Any tips?
PS: I think the question was wrongfully closed: The given solution has nothing to do with the task.