Consider following proof of Axiom of Choice $\Rightarrow$ Zorn's Lemma:
Suppose that there is a poset $(P,\le)$ in which every chain has an upper bound, yet $P$ has no maximal element. We shall prove, using AC, that in that case for every well-ordered set $L$ there is an embedding of $L$ into $P$ (since this would contradict Hartogs' Lemma, stating that for any set $X$ there is a well-order $L_X$ such that there is no injection $L_X\to X$).
Since in $P$ every chain has an upper bound, $P$ is nonempty: let $p_0\in P$. By AC [Q, below] there is a function $R: \mathcal{P}(P)\to P$ such that $$ R(C)=\begin{cases} \text{an upperbound for C} & \text{if } C\text{ is a chain in } P\\ p_0 & \text{otherwise}\end{cases}.$$
Q: We apply AC here, but we need a surjective function $P\to \mathcal{P}(P)$ to do so, right? Cantor, however, states that such a surjection does not exist. How is AC applied here then? I suspect it might have to do with following fact:
AC $\iff$ for every family of sets $\{X_i\mid i\in I\}$ such that $X_i$ is nonempty for each $i\in I$, there is a function $f:I\to \bigcup_{i\in I} X_i$ such that $f(i)\in X_i$ for each $i\in I$.
Thanks.
Why would you need a surjection $P\to\mathcal{P}(P)$? You simply split $\mathcal{P}(P)$ into two: $\mathcal{C}$ are the subsets that are nonempty chains and $\mathcal{Y}$ those that aren't. For $C\in\mathcal{C}$ you can consider $U(C)$, the set of upper bounds of $C$. Then $\mathcal{C}'=\{U(C):C\in\mathcal{C}\}$ is a set of nonempty subsets of $P$ and there is a choice function $g\colon\mathcal{C}'\to P$.
For $C\in\mathcal{C}$, define $R(C)=g(U(C))$. For $C\in\mathcal{Y}$ define $R(C)=p_0$.