Each well-orderable set $X$ is equipotent to a unique initial ordinal.
Does my attempt look fine or contain logical flaws/gaps? Any suggestion is greatly appreciated. Thank you for your help!
Lemma: Every well-ordered set is isomorphic to a unique ordinal.
By Lemma and Axiom of Choice, $X$ is equipotent to some ordinal $\alpha$. Let $\alpha_0$ be the least ordinal equipotent to $X$. Then $\alpha_0$ is an initial ordinal. If not, $|\alpha_0|=|\beta|=|X|$ for some $\beta<\alpha_0$. This contradicts the minimality of $\alpha_0$.
Assume the contrary that $X$ is equipotent to initial ordinals $\alpha_0,\alpha_1$ such that $\alpha_0\neq\alpha_1$. WLOG, we assume $\alpha_0<\alpha_1$. Furthermore, $|\alpha_0|=|\alpha_1|$. This contradicts the fact that $\alpha_1$ is an initial ordinal.
Then the cardinal number of $X$, denoted by $|X|$, is defined as the unique initial ordinal equipotent to $X$.