Cardinality of Set Difference with Subset

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

Let $T \subseteq S$.

Then:
 * $\card {S \setminus T} = \card S - \card T$

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

Proof
From Set Difference with Superset is Empty Set:
 * $T \subseteq S \iff T \setminus S = \O$

From Set Difference and Intersection form Partition:
 * $S = \paren {S \setminus T} \cup T$

Thus from Cardinality of Set Union:
 * $\card S = \card T + \card {S \setminus T} - \card {T \cap \paren {S \setminus T} }$

But from Set Difference Intersection with Second Set is Empty Set:
 * $T \cap \paren {S \setminus T} = 0$

Hence the result.