\begin{proposition}\label{phase5_fixed_set_induction}
    For all sets $x$ we have $x=x$.
\end{proposition}
\begin{proof}
    Fix $x$.
    Show $x=x$.
    \begin{subproof}[Proof by \in-induction on $x$]
        Follows.
    \end{subproof}
    Follows by assumption.
\end{proof}