Sandwich Principle

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 $y \subseteq x$.

Then:
 * $\forall x, y \in A: x \subseteq y \subseteq \map g x \implies x = y \lor y = \map g x$

That is, there is no element $y$ of $A$ such that:
 * $x \subset y \subset \map g x$

where $\subset$ denotes a proper subset.

Also see

 * Sandwich Theorem or Sandwich Rule in the context of real analysis, also known as the Squeeze Theorem (preferred on )