Successor Set of Transitive Set is Transitive

Theorem
Let $S$ be a transitive set.

Then its successor set $S\,^{+} = S \cup \left\{{S}\right\}$ is also transitive.

Proof
Suppose that $x \in S\,^{+}$.

Then either $x \in S$ or $x = S$.

If $x \in S$, it follows by the transitivity of $S$ that $x \subseteq S$.

If $x = S$, then $x = S \subseteq S$ because a set is a subset of itself.

Since $S \subseteq S\,^{+}$, it follows by the transitivity of the subset relation that $x \subseteq S\,^{+}$.