Remark 4.7.2.2. Let $S$ be a set, and let $A$ be the collection of all ordinals which arise as the order types of well-orderings on $S$. The collection $A$ is nonempty (Theorem 4.7.1.34), and therefore contains a smallest element (Corollary 4.7.1.24). It follows that the cardinality $|S|$ is well-defined.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$