Proposition 4.7.1.20. The relation $\leq $ of Definition 4.7.1.19 determines a linear ordering on the collection of ordinals.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. The reflexivity of the relation $\leq $ follows from Example 4.7.1.12, and the transitivity follows from Remark 4.7.1.13. Let $\alpha $ and $\beta $ be ordinals, which we identify with the order types of well-ordered sets $(S, \leq )$ and $(T, \leq )$, respectively. Invoking Corollary 4.7.1.17, we deduce that $\alpha \leq \beta $ or $\beta \leq \alpha $. Moreover, if both conditions are satisfied, then Remark 4.7.1.18 shows that $\alpha = \beta $. $\square$