The following is an exercise in Just/Weese:
Prove that for every set $X$ there exists exactly one initial ordinal $\kappa$ such that $X \approx \kappa$.
An ordinal is called initial ordinal if it is not equipotent with any smaller ordinal.
Can you tell me if my proof is correct? Thank you.
Assume there are two initial ordinals, $\kappa \neq \kappa'$ with $\kappa \approx X \approx \kappa'$. Wlog let $\kappa < \kappa' $. Then this is a contradiction to $\kappa'$ being the smallest ordinal in bijection with $X$.
To show existence assume AC so that $X$ can be well-ordered. Let $o$ be the order type of $X$. Define $S = \{o' \mid o' \in \mathbf{ON}, o' \approx o \} \subseteq \mathbf{ON}$. Since $\mathbf{ON}$ is a well-order, $S$ has a minimal element $s_0$. This is the initial ordinal of $X$.
The exercise is rated difficult so I expect this proof to be flawed but I can't seem to spot the mistake.