Set Theory/The axiom of choice

Exercises

1. Prove that Zorn's lemma is equivalent to Tukey's lemma which states that whenever $X$ is a set and $ℱ\subseteq 2^X$ has the property that $A\in ℱ$ if and only if $B\in ℱ$ for all finite sets $B\subseteq A$, then for all $Y\subseteq X$ there exists a maximal $Z\subseteq X$ among all sets in $ℱ$ that contain $Y$ (where we consider $ℱ$ to be ordered by inclusion).