Transfinite Induction/Schema 2/Proof 2

Proof
Define the class:
 * $A := \set {x \in \operatorname{On}: \map \phi x = \T}$.

Then $\map \phi x = \T$ is equivalent to the statement:
 * that $x \in A$

The three conditions in the hypothesis become:


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

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

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

Thus, $\map \phi x$ holds for all $x \in \operatorname{On}$.