Transfinite Induction

Proposition 1 (Transfinite Induction). Let $C$ be a class of ordinals. Suppose:

1. \[0 \in C.\]
2. \[\forall \alpha \in C : \alpha + 1 \in C.\]

3. For every limit ordinal $\alpha$,

   \[\alpha \subseteq C \enspace\rimp\enspace \alpha \in C.\]

Then $C = \Ord$.