Set Difference of Intersection with Set is Empty Set

Theorem
The set difference of the intersection of two sets with one of those sets is the empty set.

Let $S, T$ be sets.

Then:
 * $\paren {S \cap T} \setminus S = \O$
 * $\paren {S \cap T} \setminus T = \O$

Proof
From Set Difference is Right Distributive over Set Intersection:
 * $\paren {R \cap S} \setminus T = \paren {R \setminus T} \cap paren {S \setminus T}$

Hence: