Minimal Infinite Successor Set is Limit Ordinal

Theorem
Let $\omega$ denote the minimal infinite successor set.

Then $\omega$ is a limit ordinal.

Proof
Let $K_I$ denote the class of all nonlimit ordinals.

$\omega$ is not a limit ordinal.

Every element of $\omega$ is a nonlimit ordinal.

So, if $\omega$ is also a nonlimit ordinal, $\omega + 1 \subseteq K_I$.

By the definition of $\omega$:
 * $\omega \in \omega$

which violates No Membership Loops and is thus contradictory.

It follows from Proof by Contradiction that $\omega$ is a limit ordinal.