Successor Set of Transitive Set is Transitive

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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 $x \subseteq S\,^+$.

$\blacksquare$