Sandwich Principle for G-Towers/Corollary 1

Theorem
Let $M$ be a class.

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

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

Let $x, y \in M$ be arbitrary.

Then:
 * $x \subsetneqq y \implies \map g x \subseteq y$

Proof
From Lemma $2$ of $g$-Tower is Nest we have that:


 * $\forall x, y \in M: \map g x \subseteq y \lor y \subseteq x$

From the Sandwich Principle: Corollary 1:


 * $\forall x, y \in M: x \subseteq y \subseteq \map g x \implies x = y \lor y = \map g x$

That is, if:
 * $x \subsetneqq y$

then:
 * $\map g x \subseteq y$

Also see

 * Sandwich Principle