Ordinal is Less than Successor

Theorem
Let $x \in \operatorname{On}$. $x$ is an ordinal and must also be a set.

Let $x^+$ denote the successor of $x$


 * $\displaystyle x \in x^+$


 * $\displaystyle x \subset x^+$

Proof

 * $\displaystyle x \in ( x \cup \{ x \} ) \land x \subset ( x \cup \{ x \} )$ so by applying the definition of a successor set,


 * $\displaystyle x \in x^+ \land x \subset x^+$