Successor Set of Transitive Set is Transitive
Let $S$ be a transitive set.
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 $x \subseteq S\,^+$.