Cardinalities form Inequality implies Difference is Nonempty

Theorem
Let $X, Y$ be sets.

Let
 * $\card X < \card Y$

where $\card X$ denotes the cardinality of $X$.

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

Proof
that:
 * $Y \setminus X = \O$

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

Hence by Subset implies Cardinal Inequality:
 * $\card Y \le \card X$

This contradicts:
 * $\card X < \card Y$

Hence the result.