The definition of supremum (from wikipedia) is as follows:
If $(P,\le)$ is a partially ordered set and $S$ is a subset of $P$ then $x\in P$ is an upper bound for $S$ if $y\le x$ for all $y\in S$. Further, upper bound $x$ is a supremum for $S$ if whenever $z$ is an upper bound for $S$ then $x\le z$.
Now consider the theorem:
If $\mathcal C$ is a collection of ordinal numbers then it possesses a supremum.
We take the union of all the ordinals and prove it is the supremum. But what is our set $P$ as per the above here? Also, the supremum of the $\omega=\{0,1,2\dots,\}$ is $\omega$ itself, right?