Ordinal is Proper Subset of Successor

Theorem
Let $\alpha$ be an ordinal.

Then:
 * $\alpha \subsetneqq \alpha^+$

where $\alpha^+$ denotes the successor set of $\alpha$.

That is:
 * $\alpha$ is a proper subset of $\alpha^+$.

Proof
$\alpha = \alpha^+$.

By definition:
 * $\alpha^+ = \alpha \cup \set \alpha$

and so:
 * $\alpha \subseteq \alpha^+$

and:
 * $\alpha \in \alpha^+$

which leads to:
 * $\alpha \in \alpha$

But from Ordinal is not Element of Itself:
 * $\alpha \notin \alpha$

Hence by Proof by Contradiction:
 * $\alpha \ne \alpha^+$

But we have:
 * $\alpha \subseteq \alpha^+$

Hence:
 * $\alpha \subsetneqq \alpha^+$