\begin{signature}\label{example_cons}
    $\cons{y}{X}$ is a set.
\end{signature}

\begin{signature}\label{example_pair}
    $(x,y)$ is a set.
\end{signature}

\begin{axiom}\label{cons}
    $x\in \cons{y}{X}$ iff $x = y$ or $x\in X$.
\end{axiom}

\begin{definition}\label{times}
    $A\times B = \{ (a,b) \mid a\in A, b\in B\}$.
\end{definition}

\begin{definition}\label{unit}
    $\unit = \{\emptyset\}$.
\end{definition}

\begin{proposition}\label{pair_emptyset_in_times_unit}
    $(\emptyset,\emptyset)\in \unit\times\unit$.
\end{proposition}


% Indirect definition of the Zermelo successor operation to test primitive replacement.
\begin{definition}\label{suc}
    $\suc{a} = \{ y \mid \exists x\in a. y = \{x\} \}$.
\end{definition}


% Dummy proposition to test unrolling of functional replacement in equations.
\begin{proposition}\label{times_replacement_test}
    $A\times B = \{ (a,b) \mid a\in A, b\in B\}$.
\end{proposition}
\begin{proof}
    Follows by \cref{times}.
\end{proof}
