In Elements of Set Theory by Enderton he defines the cardinal number of $A$ to be the least ordinal equinumerous to $A$.
Just before this definition is the Numeration Theorem which states that any set is equinumerous to some ordinal number. My issue is that no justification is given for the existence of a least ordinal number equinumerous to $A$, so I tried to justify it myself:
Let $\alpha$ be an ordinal equinumerous to $A$. Then the set $\{\beta\in\alpha^+:\beta \text{ is equinumerous to $A$}\}$ is a nonempty set of ordinals, and hence has a least element $l$. Then $l$ is the least ordinal equinumerous to $A$.
Does this justification seem OK?