Hello, ruprotein!
It is difficult to provide a proof if we don't know what axioms and theorems you are allowed.
I will assume you are familiar with these rules . . .
. . . . . definition of set subtraction
. . . . . one of DeMorgan's Laws.
. . . . . the intersection of and a set is the set .
Prove the following DeMorgan's law:
If are sets such that: then:
. .
The left side is: .
. . . . . . . . . . . . . from [1]
. . . . . . . . . . . . . from [2]
. . . . . . . . . . . . . from [3]
The right side is: .
. . . . . . . . . . . . . . from [1]
. . . . . . . . . . . . . . from [3]
Therefore, the two sides are equal . . . QED