proof of principle of transfinite induction

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

The class $C=\{\alpha :\mathrm{\neg }\mathrm{\Phi }(\alpha )\}$ 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 , $\mathrm{\Phi }(\lambda )$ is true. Thus, by hypothesis, $\mathrm{\Phi }(\gamma )$ is true, contradiction.