We already know that the class of ordinals (denoted by $\operatorname{Ord}$) is too big to be a set. I wonder if there is a sub class of $\operatorname{Ord}$ that is not a set.
My question: Is there a proper class $X$ such that $X\subsetneq \operatorname{Ord}$?
Thank you so much for your dedicated help!
The class of all non-empty ordinals is a subclass of $\operatorname{Ord}$ different from $\operatorname{Ord}$. Of course, it isn't a set. In general, a subclass $X$ of $\operatorname{Ord}$ is not a set if and only if for all ordinals $\alpha$ there is some $\beta\in X$ such that $\alpha\in \beta$.