Element of Minimal Infinite Successor Set is Transitive Set

Theorem

Let $\omega$ be the minimal infinite successor set.

Let $n \in \omega$.

Then $x \in n \implies x \subseteq n$.

That is, every element of $n$ is also a subset of it.

In other words, each element of $\omega$ is a transitive set.

Proof

Let $S \subseteq$ be the set of all transitive elements of $\omega$.

That is:

$n \in S \iff n \in \omega \land \forall x \in n: x \subseteq n$

It is vacuously true that $0 \in S$, as there are no $x \in 0$.

Now suppose $n \in S$.

If $x \in n^+$ then either $x \in n$ or $x = n$.

In the first case:

$x \subseteq n$ as $n \in S$

and so:

$x \subseteq n^+$

In the second case, by definition of successor set:

$x \subseteq n^+$

By Principle of Mathematical Induction for Minimal Infinite Successor Set it follows that:

$S = \omega$

Hence the result.

$\blacksquare$