Set Difference of Larger Set with Smaller is Not Empty

Theorem

Let $S$ and $T$ be finite sets.

Let $\card S > \card T$.

Then:

$S \setminus T \ne \O$

Proof

From Cardinality of Subset of Finite Set:

$S \nsubseteq T$

From the contrapositive statement of Set Difference with Superset is Empty Set:

$S \setminus T \ne \O$.

$\blacksquare$