Cardinalities form Inequality implies Difference is Nonempty

Theorem
Let $X, Y$ be sets.

Let
 * $\left\vert{X}\right\vert < \left\vert{Y}\right\vert$

where $\left\vert{X}\right\vert$ denotes the cardinality of $X$.

Then:
 * $Y \setminus X \ne \varnothing$

Proof
Aiming for a contradiction suppose that
 * $Y \setminus X = \varnothing$

Then by Set Difference with Superset is Empty Set:
 * $Y \subseteq X$

Hence by Subset implies Cardinal Inequality:
 * $\left\vert{Y}\right\vert \leq \left\vert{X}\right\vert$

This contradicts:
 * $\left\vert{X}\right\vert < \left\vert{Y}\right\vert$

Hence the result.