Sandwich Principle/Corollary 1

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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.

$\blacksquare$


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