Successor Set of Transitive Set is Transitive

Theorem
Let $S$ be a transitive set.

Then its successor set $S^+ = S \cup \set S$ is also transitive.

Proof
Recall that $S$ is transitive :
 * $x \in S \implies x \subseteq S$

Hence:

Then:

Thus we have:
 * $x \in S \implies x \subseteq S$

Hence the result.