Transfinite Induction/Schema 2/Proof 2

Proof
Define the class:
 * $A := \left\{{x \in \operatorname{On}: \phi \left({x}\right) = \mathrm T}\right\}$.

Then $\phi \left({x}\right) = \mathrm T$ is equivalent to the statement:
 * that $x \in A$

The three conditions in the hypothesis become:


 * $(1a): \quad \varnothing \in A$
 * $(2a): \quad x \in A \implies x^+ \in A$
 * $(3a): \quad \left({\forall x < y: x \in A}\right) \implies y \in A$

These are precisely the conditions for the class $A$ in the second principle of transfinite induction.

Therefore, $\operatorname{On} \subseteq A$.

Thus, $\phi \left({x}\right)$ holds for all $x \in \operatorname{On}$.