G-Tower is Closed under Chain Unions

Theorem

Let $M$ be a class.

Let $g: M \to M$ be a progressing mapping on $M$.

Let $M$ be a $g$-tower.

Then $M$ is closed under chain unions.

Proof

By definition:

a $g$-tower is a class which is minimally superinductive under $g$

a class which is minimally superinductive under $g$ is superinductive under $g$

a superinductive class is closed under chain unions.

Hence the result.

$\blacksquare$