To prove the transfinite induction theorem, we note that the class of ordinals is well-ordered by $\in$ . So suppose for some $\Phi$ , there are ordinals $\alpha$ such that $\Phi(\alpha)$ is not true. Suppose further that $\Phi$ satisfies the hypothesis, i.e. $\forall\alpha(\forall\beta<\alpha(\Phi(\beta))\Rightarrow\Phi(\alpha))$ . We will reach a contradiction.

The class $C=\{\alpha:\neg\Phi(\alpha)\}$ is not empty. Note that it may be a proper class, but this is not important. Let $\gamma=\min(C)$ be the $\in$ -minimal element of $C$ . Then by assumption, for every $\lambda<\gamma$ , $\Phi(\lambda)$ is true. Thus, by hypothesis, $\Phi(\gamma)$ is true, contradiction.