Sandwich Principle/Corollary 1

Theorem

Let $A$ be a class.

Let $g: A \to A$ be a mapping on $A$ such that:

for all $x, y \in A$, either $\map g x \subseteq y$ or $y \subseteq x$.

Let:

$x \subset y$

where $\subset$ denotes a proper subset.

Then:

$\map g x \subseteq y$

Let $x \subset y$.

By hypothesis, either $\map g x \subseteq y$ or $y \subseteq x$.

But because $x \subset y$, it follows that $y \subseteq x$ cannot be the case.

Hence the result.

$\blacksquare$

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: Lemma $4.9 \ (2)$