The well-ordering theorem

Let $P$ be the collection of pairs $(B, R)$ with $B \subseteq X$ and $R \subseteq B \times B$ a strict well-order of $B$; this is a set by Power Set and Separation applied to $\mathcal{P}(X) \times \mathcal{P}(X \times X)$, and it is nonempty because $(\emptyset, \emptyset) \in P$.

Order $P$ by end extension; this relation is reflexive, transitive, and antisymmetric, since mutual end extension forces the underlying sets and relations to coincide, so $(P, \preceq)$ is a poset.

The empty chain has an upper bound in $P$, namely $(\emptyset, \emptyset)$, so the hypothesis of Zorn's lemma is not vacuous at the bottom.

Let $\mathcal{C} \subseteq P$ be a nonempty chain and put $B^{*} = \bigcup \{B : (B, R) \in \mathcal{C}\}$ and $R^{*} = \bigcup \{R : (B, R) \in \mathcal{C}\}$.

$R^{*}$ is a strict total order on $B^{*}$: any two elements of $B^{*}$ lie in members of $\mathcal{C}$ one of which end extends the other, hence lie together in the larger member and are comparable there, and the comparisons made in different members agree, because along the chain each relation is the restriction of every larger one.

$R^{*}$ well-orders $B^{*}$: given $\emptyset \ne S \subseteq B^{*}$, choose a member $(B, R)$ of $\mathcal{C}$ meeting $S$ and let $\ell$ be the $R$-least element of $S \cap B$; if some $t \in S$ satisfied $t \mathbin{R^{*}} \ell$ then $t$ and $\ell$ would lie in a common member $(B', R')$ of $\mathcal{C}$, and either $(B', R') \preceq (B, R)$, giving $t \in B' \subseteq B$, or $(B, R) \preceq (B', R')$, in which case $B$ is an initial segment of $(B', R')$ and $t \mathbin{R'} \ell \in B$ again gives $t \in B$; either way $t \in S \cap B$ with $t \mathbin{R} \ell$, contradicting minimality of $\ell$.

Hence $(B^{*}, R^{*}) \in P$, and it end extends every member of $\mathcal{C}$: the inclusions and restrictions are immediate, and $B$ is an initial segment of $(B^{*}, R^{*})$ by the case analysis of step 5.1; so every nonempty chain has an upper bound.

Zorn's lemma applies to the nonempty poset $(P, \preceq)$, in which every chain has an upper bound, and yields a maximal element $(M, R_M)$; this is the one and only place the Axiom of Choice is spent, and it is spent inside Zorn's lemma.

$M = X$: otherwise there is $x \in X \setminus M$, and putting $M' = M \cup \{x\}$ with $R' = R_M \cup \{(m, x) : m \in M\}$ gives a well-order of $M'$, because a nonempty $S \subseteq M'$ has as least element the $R_M$-least element of $S \cap M$ when that is nonempty and $x$ otherwise, while $(M, R_M) \preceq (M', R')$ with $M \ne M'$ contradicts the maximality of $(M, R_M)$.

Therefore $R_M$ is a well-order of $M = X$, and $X$ can be well ordered.