Ordinal is not Element of Itself
Let $x$ be an ordinal.
Then $x \notin x$.
That is, $x^+ = x \cup \set x$ is an ordinal.
By Set is Element of Successor, $x \in x^+$.
Because a strict ordering is antireflexive and $x \in x^+$, we conclude that $x \notin x$.
This result follows immediately from Set is Not Element of Itself.