# 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

Aiming for a contradiction, suppose that:

$Y \setminus X = \O$
$Y \subseteq X$

Hence by Subset implies Cardinal Inequality:

$\card Y \le \card X$

$\card X < \card Y$
$\blacksquare$