Axiom of Choice and one of its standard equivalent maximality principles

The source lists five familiar principles equivalent to the Axiom of Choice: Zorn’s Lemma, Tukey’s Lemma, the well-ordering principle, the Hausdorff maximality principle, and König’s Theorem.

1. State the Axiom of Choice and any two of those five principles.
2. Choose one of the five principles, call it \(P\), and prove either \[ \mathrm{AC}\Longrightarrow P \] or \[ P\Longrightarrow\mathrm{AC}. \]