Sandwich Principle/Linguistic Note

Linguistic Note on Sandwich Principle
The name Sandwich Principle for this lemma appears to have been coined by and  in their

As they explain, when $x \subset y \subset z$, we may consider $y$ to be sandwiched between $x$ and $z$.

Thus the Sandwich Principle tells us that, given the conditions constraining $x$, $y$ and $\map g x$, there can be no such $y$ sandwiched between $x$ and $\map g x$.