Can set containing not only all ordinal numbers exist?

Question

I've always seen the same proof of non-existence of set of all ordianal numbers.

It was based on a fact that such a set would fit into definition of ordinal number, and the set itself would be an ordinal number, which would not be included in the set.

Let $d_{'}$, be arbitrary mathematical object which is not an ordinal number.

Then let set $O$ to contain all ordinal numbers and the $d_{'}$ element.

Now the original proof has no problem with $O$, because it contains another element than ordinal number, thus can't be considered one.

Can set such as $O$, exist in for e.g. ZFC?

Answer 1

If $O$ exists then so does $$\{\,x\in O\mid x\text{ is an ordinal}\,\}, $$ which does not.