We want to prove the "Trichotomy of Ordinals":
Definiton: An ordinal is a transitive set with elements that are all transitive.
Definiton: $\alpha$ and $\beta$ ordinals are comparable if one of the following is true, $\alpha \in \beta$ or $\beta \in \alpha$ or $\alpha = \beta$.
During the proof we set:
$$K=\{\langle\alpha,\beta\rangle \mid \alpha \text{ and } \beta \text{ are not comparable }\}$$
as the "set" of the ordered pairs of ordinals that are not comparable, and we want to show that it is empty. In order to claim that there is a minimal element in $K$ under a certain relation between ordered pairs of ordinals, it must be a set. Is $K$ a set? Why?
Thanks!