G-Tower is Well-Ordered under Subset Relation/Empty Set

Theorem

Let $M$ be a class.

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

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

$\O$ is the smallest element of $M$.

Proof

Follows directly from $g$-Tower is Well-Ordered under Subset Relation.

$\blacksquare$

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: $\S 3$ The well ordering of $g$-towers: Theorem $3.3 \ (1)$