Let $X$ be a set of ordinals.


If $X$ has no largest element, then 
$$
\sup X \notin X \subseteq \sup X, 
$$
and $\sup X$ is the smallest ordinal $\alpha$ such that $X \subseteq \alpha$.


On the other hand, if $X$ has a largest element $\max X$, then 
$$
\max X = \sup X \in X \nsubseteq \sup X,
$$
and the smallest ordinal $\alpha$ such that $X \subseteq \alpha$ is $\sup X + 1$.


Is there any way to express "the smallest ordinal $\alpha$ such that $X \subseteq \alpha$" that works in both cases?


One possibility would be
$$
\sup \{ \beta + 1 : \beta \in X \},
$$
but I'm looking for something more concise or elegant than that.



How can we express the smallest ordinal $\alpha$ such that $X \subseteq \alpha$?
$\mathrm{rank}(X)$


