\begin{proposition}\label{phase5_nested_set_induction}
    Let $A$ be a set.
    Then for all $x\in A$ such that $x\neq A$ we have $x = x$.
\end{proposition}
\begin{proof}
    Show for all $x\in A$ such that $x\neq A$ we have $x = x$.
    \begin{subproof}[Proof by \in-induction]
        Assume $x\in A$ and $x\neq A$.
        Follows by assumption.
    \end{subproof}
    Follows by assumption.
\end{proof}