\begin{definition}\label{phase5_separation_definition}
    $\phasefiveseparation{A} = \{ x \in A \mid x = x \}$.
\end{definition}

\begin{proposition}\label{phase5_separation_theorem}
    For all $A, x$ we have if $x \in \phasefiveseparation{A}$, then $x \in A$.
\end{proposition}
\begin{proof}
    Fix $A, x$.
    Assume $x \in \phasefiveseparation{A}$.
\end{proof}