Intersection of Unions with Complements is Subset of Union

Theorem
Let $R, S, T$ be sets.

Then:
 * $\left({R \cup S}\right) \cap \left({\overline R \cup T}\right) \subseteq S \cup T$

Proof
Let $x \in \left({R \cup S}\right) \cap \left({\overline R \cup T}\right)$.

Then by definition of set intersection, set union and set complement, we have:


 * $\left({x \in R \lor x \in S}\right) \land \left({x \notin R \lor x \in T}\right)$

From Conjunction of Disjunctions with Complements implies Disjunction‎ it follows that:
 * $x \in S \lor x \in T$

By definition of set union, it follows that:
 * $x \in S \cup T$

By definition of subset it follows that:
 * $\left({R \cup S}\right) \cap \left({\overline R \cup T}\right) \subseteq S \cup T$

Illustration by Venn Diagram
A Venn diagram illustrating this result is given below:


 * IntUnsCompsSubsetUnion.png

The red and purple field together mark $R \cup S$.

The blue and purple field together mark $\overline R \cup T$. Note that the outer rectangular area is also included in this, but has been left colorless for clarity.

The purple field marks $\left({R \cup S}\right) \cap \left({\overline R \cup T}\right)$, where the red and blue intersect with each other.