Strict Lower Closure of G-Tower is Set of Elements which are Proper Subsets

Theorem

Let $M$ be a class.

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

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

Let $x \in M$.

Then the strict lower closure of $x$ is the set of all elements of $M$ that are proper subsets of $M$.

Proof

Follows directly from the definitions.

$\blacksquare$

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