unbounded class of ordinals not a set

Question

A class $C$ of ordinals is unbounded just in case ∀α∈ORD (the class of all ordinals), there exists a β ∈ $C$ with α ∈ β.

How would I show that no unbounded class of ordinals is a set? Do I need to use the fact that every set has a "rank" (in context of the Von Neumann universe construction)?

Answer 1

Show that the union of a set of (von Neumann-)ordinals is an ordinal that is an upper bound, actually the least upper bound.

Answer 2

That certainly gives a short, simple proof, assuming that you’ve already proved basic facts about rank. Alternatively, you could note that if $C$ is a set, so is $\bigcup C$; now what is $\bigcup C$ in this case?