Set Difference and Intersection are Disjoint

Theorem

Let $S$ and $T$ be sets.

Then:

$S \setminus T$ and $S \cap T$ are disjoint

where $S \setminus T$ denotes set difference and $S \cap T$ denotes set intersection.

Proof

From Set Difference Intersection with Second Set is Empty Set:

$\paren {S \setminus T} \cap T = \O$

and hence immediately from Intersection with Empty Set:

$\paren {S \setminus T} \cap \paren {S \cap T} = \O$

So $S \setminus T$ and $S \cap T$ are disjoint.

$\blacksquare$