Transfinite Induction/Principle 2

Theorem
Let $A$ be a class satisfying the following conditions:


 * $\O \in A$
 * $\forall x \in A: x^+ \in A$
 * If $y$ is a limit ordinal, then $\paren {\forall x < y: x \in A} \implies y \in A$

where $x^+$ denotes the successor of $x$.

Then $\operatorname{On} \subseteq A$.

Proof
We shall prove this using the first principle of transfinite induction.

Assume $y$ is an ordinal such that $y \subseteq A$.

This implies $\forall x < y: x \in A$.

By definition, $y$ is a limit ordinal, $y = \O$, or $\exists x: y = x^+$.

If $y$ is a limit ordinal, then, since:


 * $\forall x < y: x \in A$

it follows that $y \in A$.

If $y = \O$, then $y \in A$.

Finally, assume $y = z^+$ for some ordinal $z$.

Since $z < z^+ = y$ and $\forall x < y: x \in A$, it follows that $z \in A$.

By hypothesis, we conclude $y = z^+ \in A$.

In any case, $y \in A$.

Therefore, for all ordinals $y$, if $y \subseteq A$, then $y \in A$.

By the first principle of transfinite induction, $\operatorname{On} \subseteq A$.