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

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:

$$ $$ $$