Characteristics of Minimally Inductive Class under Progressing Mapping/Mapping Preserves Subsets

Theorem

Let $M$ be a class which is minimally inductive under a progressing mapping $g$.

Then for all $x, y \in M$:

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

Proof

From Minimally Inductive Class under Progressing Mapping induces Nest, we have that $M$ is a nest in which:

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

Thus corollary $2$ of the Sandwich Principle applies directly.

$\blacksquare$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 4$ A double induction principle and its applications: Theorem $4.10 \ (3)$