Often in beginning set-theory courses, and in particular in Jech's book Set Theory, it is proved from scratch that given any two well-orderings, they are isomorphic or one is isomorphic to an initial segment of the other. This is the law of trichotomy for well-orderings, and while simple is not exactly trivial to prove. However, it is also proven in an apparently independent manner that every well-ordered set is isomorphic to an ordinal, and for any two ordinals $\alpha$ and $\beta$ either $\alpha \subseteq \beta$ or $\beta \subseteq \alpha$. It is then trivially seen from this fact that that either $\alpha = \beta$ or $\alpha$ is an initial segment of $\beta$ or vice versa. From these facts about ordinals, the law of trichotomy for well-orderings follows immediately.
My question is: is there any reason to prove from scratch the law of trichotomy? Is it secretly used somewhere to prove some of these basics facts about ordinals, and I simply missed it? Or can this proof just be omitted?