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\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}
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