Proof

Let a premise $p(\beta)$ be given which can be proven by the principle of transfinite induction to be true for all ordinal numbers $\beta\supset\alpha,$ given some base case ordinal number $\alpha.$
Assume, the set $N$ of all ordinal numbers $\gamma\supseteq\alpha,$ for which the premise $p(\gamma)$ is false is not empty, $N\neq\emptyset.$
Since ordinal numbers are well-ordered by definition, $N$ contains a minimum $\gamma_0.$
But then $p(\alpha)$ was true for all $\alpha\subset \gamma_0,$ which is a contradiction to the minimality of $\gamma_0.$
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