\begin{axiom}\label{phase5_nested_failure_guard_1}
    For all $A, x$ we have if $x\in\pow{\cumul{A}}$, then
        $x\in\cumul{A}$.
\end{axiom}

\begin{axiom}\label{phase5_nested_failure_guard_2}
    For all $A, x$ we have if $x\in\pow{\pow{\cumul{A}}}$, then
        $x\in\cumul{A}$.
\end{axiom}

\begin{inductive}\label{phase5_nested_failure}
    Define $\phasefivenestedfailure{A}\subseteq\cumul{A}$ inductively as follows.
    \begin{enumerate}
        \item If $x\in\pow{\phasefivenestedfailure{A}}$, then
            $x\in\phasefivenestedfailure{A}$.
        \item If $x\in\pow{\pow{\phasefivenestedfailure{A}}}$, then
            $x\in\phasefivenestedfailure{A}$.
        \item If $y\in\pow{\phasefivenestedfailure{A}}$, then
            $y\in\phasefivenestedfailure{A}$.
    \end{enumerate}
\end{inductive}