# Set Difference with Proper Subset is Proper Subset

## Theorem

Let $S$ be a set.

Let $T \subsetneq S$ be a proper subset of $S$.

Let $S \setminus T$ denote the set difference between $S$ and $T$.

Then:

$S \setminus T$ is a proper subset of $S$

## Proof

$S \setminus T \subseteq S$
$S \setminus T \ne \O$

By definition of proper subset:

$T \ne \O$
$S \cap T = T$

Hence:

$S \cap T \ne \O$
$S \setminus T \ne S$

It follows that $S \setminus T$ is a proper subset by definition.

$\blacksquare$