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An algebraic way to prove the set identity is to show that $D=\overline{A}\cup \overline{B}$ is a complement of a set for $A\cap B$. By the distributive property,andA complement of a set is unique: if $D,E$ both complement $S$, then $D=D\cap (S\cup E)=D\cap E$, and similarly $E=D\cap E$. Consequently

Every complement of a set is taken in the same universal set $\mathcal{U}$. Apply De Morgan's laws for propositional logic to the membership propositions. For every $x\in \mathcal{U}$,Equality of membership for every element proves the set identities

An equality between sets expressed in terms of arbitrary input sets that holds for every allowed choice of those inputs. A double inclusion proof proves a set identity by proving both set inclusions. Membership equivalences give another proof: the expressions are equal if and only if each element belongs to either both or neither.