In set theory (ZF) an ordinal is a transitive set of transitive sets. Thus (if exists) a set of all ordinals gives a contradiction therefore there is no set of all ordinals.
But what is wrong with the following: An ordinal is not only a transitive set of transitive sets, but also an ordinal is a limit ordinal or there is a 'greatest' ordinal belonging to that ordinal, but not both.
Thus, (if exists) a set of all ordinals then that set is not a limit-ordinal otherwise the successor of that limit-ordinal gives a contradiction, therefore if the set of all ordinals exists and is an ordinal then there is a 'greatest' ordinal belonging to that set of all ordinals, but that is impossible because there is no 'greatest' ordinal.
Therefore if exists a set of all ordinals then that set is not an ordinal.
Thank you for your attention, Doeko Homan
This proof is okay. It could use cleanup, but its mathematical essence is okay.
The usual, somewhat simpler proof that the collection of all ordinals is not an ordinal. Simply because then $\sf Ord\in Ord$, and this just contradicts the fact that $\in$ well-ordered the ordinals.
Really what you are proving (or should be proving) is that if $\alpha$ is an ordinal, then $\alpha+1$ exists, and that a transitive set of ordinals is an ordinal; and that if $\sf Ord$ is a set, then it is the largest ordinal.