Zorn's lemma

Suppose $P$ has no maximal element.

Let $\mathcal{C}$ be the set of all chains of $P$, a subset of the power set of $P$, partially ordered by inclusion.

$(\mathcal{C},\subseteq )$ is a chain-complete poset: if $\mathcal{D}\subseteq \mathcal{C}$ is a chain under inclusion then $\bigcup \mathcal{D}$ is a chain of $P$, since any two of its elements lie in a common member of $\mathcal{D}$, and it is the least upper bound of $\mathcal{D}$ under inclusion; the empty chain has least upper bound $\bigcup \mathrm{\varnothing }=\mathrm{\varnothing }$, which is a chain.

For $C\in \mathcal{C}$ let $S_C$ be the set of strict upper bounds of $C$ in $P$.

Each $S_C$ is nonempty: $C$ has an upper bound $u$ in $P$ by hypothesis, taking any element of the nonempty $P$ when $C=\mathrm{\varnothing }$; by assumption $u$ is not maximal, so there is $v$ with $u<v$; then for every $c\in C$ transitivity gives $c\le v$ from $c\le u\le v$, and $c\mathrm{\ne }v$, since $c=v$ would give $v\le u$ and $u\le v$, hence $u=v$ by antisymmetry, contradicting $u<v$; so $c<v$ for every $c\in C$ and $v\in S_C$.

Apply the Axiom of Choice to the family $\mathcal{S}=\{S_C:C\in \mathcal{C}\}$, every member of which is nonempty, obtaining a choice function $g$ with $g(S)\in S$ for each $S\in \mathcal{S}$; composing with the map $C\mapsto S_C$, which is a function on $\mathcal{C}$, yields a selection $C\mapsto g(S_C)\in S_C$ defined for every chain $C$, and no injectivity of $C\mapsto S_C$ is needed, since two chains with the same set of strict upper bounds simply receive the same chosen element.

Define $f(C)=C\cup \{g(S_C)\}$ for $C\in \mathcal{C}$; this is again a chain, because $g(S_C)$ is a strict upper bound of $C$ and so is comparable to every element of $C$.

$f$ is progressive for inclusion, since $C\subseteq f(C)$ by construction.

By Bourbaki–Witt applied to the chain-complete $\mathcal{C}$ and the progressive $f$, there is $C_0\in \mathcal{C}$ with $f(C_0)=C_0$, that is $g(S_{C_0})\in C_0$.

But $g(S_{C_0})$ is a strict upper bound of $C_0$, so every element of $C_0$ is strictly below it, giving $g(S_{C_0})<g(S_{C_0})$, which is impossible because $<$ is irreflexive.