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\documentclass{article}
\usepackage{amsmath, amsthm}
\usepackage{
thm-listof,
thm-restate,
thm-autoref,
thm-kv,
}
\usepackage{hyperref}
\declaretheorem[unnumbered,
title={Zorn's Lemma}]{zl}
%\newtheorem*{zl}{Zorn's Lemma}
\declaretheorem[numberwithin=section]{theorem}
%\newtheorem{theorem}{Theorem}[section]
\declaretheorem[sibling=theorem]{lemma}
%\newtheorem{lemma}[theorem]{Lemma}
\declaretheorem[numberlike=lemma]{axiom}
%\newtheorem{axiom}[lemma]{Axiom}
\begin{document}
\section{Introduction}
In this dummy document, we will show important things. One very important
insight is
\begin{lemma}[Zorn]
If every chain in $X$ is bounded, $X$ has a maximal element.
(Here, $X$ is a set system.)
\end{lemma}
This lemma is so important that it's a fixed name:
\begin{restatable}{zl}{zornslemma}
If every chain in $X$ is bounded, $X$ has a maximal element.
(Here, $X$ is a set system.)
\end{restatable}
We will conclude in important theorem from this:
\begin{restatable}[Well-ordering]{theorem}{wohlordnung}\label{thm:order}
Every set is well-ordered.
\end{restatable}
%\show\wohlordnung
\section{Main}
Here, we will prove \wohlordnung which first appeared
as~\autoref{thm:order} on page~\pageref{thm:order} and is
actually equivalent to
\zornslemma
Another equivalent formulation is
\begin{axiom}[Axiom of Choice]
If you have a non-empty set, you can take an element out of it.
\end{axiom}
\section{Conclusion}
To remind you, these are the theorems that occur in this document,
ignoring Lemmas:
\ignoretheorems{lemma}
\listoftheorems
\end{document}
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