How to prove that for any nonempty set $A$, there exists a maximal partial order $\lambda$ with the axiom of choice?

Question

The ``maximal partial order'' means for any partial order $\alpha\in\mathscr{B}(A)$, $\lambda\subseteq\alpha$ implies $\lambda=\alpha$.

It is obvious if we apply the well-ordering theorem or Zorn's lemma. Since the equivalence among well-ordering theorem, Zorn's lemma and the axiom of choice, we can give a proof directly from the axiom of choice, I mean, not a pretended proof( to prove Zorn's lemma first and then use the lemma).

Could any one help me?

Answer 1

Apply Zorn's lemma to the collection $\mathcal{C}$ of partial orders in $A$. Here $\mathcal{C}$ is ordered by $\subseteq$.