Sandwich Principle for G-Towers/Corollary 2

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 such that $x \subseteq y$.

Then:

$\map g x \subseteq \map g y$

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 2:

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

$\blacksquare$

Also see

• Sandwich Principle

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 2$ Superinduction and double superinduction: Theorem $2.4 \ (3)$