Transfinite Induction/Schema 2/Proof 1

Proof
It should be noted that for any two ordinals, $x \lt y \iff x \in y$.

Let $\map \phi x$ be a property that satisfies the above conditions.

$y$ be an ordinal such that $\neg \map \phi y$.

It is noted that $y \ne \O$.

Therefore $y$ must be either a successor ordinal or a limit ordinal.

If $y$ is a successor ordinal, then let $x$ be the ordinal such that $x^+ = y$.

By the Rule of Transposition, it is seen that $\neg \map \phi x$.

By Set is Element of Successor, it follows that $x \in y$, and so $x \lt y$.

If $y$ is a limit ordinal, then by the Rule of Transposition there exists an ordinal $x \lt y \iff x \in y$ such that $\neg \map \phi x$.

It has been shown that if $y$ is an ordinal such that $\neg \map \phi y$, then there exists an ordinal $x \in y$ such that $\neg \map \phi x$.

The rest follows from the proof of Schema 1.