Ordinal is not Element of Itself/Proof 1

Proof
By Successor Set of Ordinal is Ordinal, the successor of $x$ is an ordinal.

That is, $x^+ = x \cup \left\{{x}\right\}$ is an ordinal.

By Set is Element of Successor, $x \in x^+$.

Since $x^+$ is an ordinal, it is strictly well-ordered by the epsilon relation.

Thus $x^+$ is certainly strictly ordered by the epsilon relation.

Since a strict ordering is antireflexive and $x \in x^+$, we conclude that $x \notin x$.