For a collections of ordinals $C$, I think $C$ is an ordinal $\iff$ $C = pred(a, ON, \in)$ for some ordinal $a$. Is this correct?
For the $\implies$, just take $a = C$, and for $\Longleftarrow$ the set is transitive as if $x \in C$, $\forall b<x, b \in C$ so $x \subset C$ . It is well ordered because it is a collections of ordinal. Thus $C$ is an ordinal.
With this I think if C is a collection of ordinal, and an ordinal itself. Then $\sup C = C$ iff C is the limit ordinal or $\emptyset$.
For $\Longleftarrow $, Suppose $\sup C \neq C$, then, $\exists \alpha \in C$ s.t $\sup C = \alpha$. Thus C is a successor ordinal, which is a contraction For $\implies $, Suppose not if $C=S(b)$ for some b, then $b \in C$ and $b = \sup C$, which result in a contradiction
I did not find these results anywhere. They are easy to proof but seems to be quite useful, so I want to validate if these are correct or not.