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:
 * $\left({S \cap T}\right) \setminus S = \varnothing$
 * $\left({S \cap T}\right) \setminus T = \varnothing$

Proof
From Set Difference is Right Distributive over Set Intersection we have:
 * $\left({R \cap S}\right) \setminus T = \left({R \setminus T}\right) \cap \left({S \setminus T}\right)$

Hence: