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:
 * $\map g x \subseteq y$

Proof
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.