\begin{proposition}\label{phase5_nested_guard}
    For all $A, x$ we have if $x\in\pow{\cumul{A}}$, then
        $A\in\cumul{A}$.
\end{proposition}
\begin{proof}
    Fix $A, x$.
    Assume $x\in\pow{\cumul{A}}$.
\end{proof}

\begin{axiom}\label{phase5_nested_unsafe_support}
    For all $A$ we have $A=A$.
\end{axiom}

\begin{inductive}\label{phase5_nested}
    Define $\phasefivenested{A}\subseteq\cumul{A}$ inductively as follows.
    \begin{enumerate}
        \item If $x\in\pow{\phasefivenested{A}}$, then
            $A\in\phasefivenested{A}$.
    \end{enumerate}
\end{inductive}