Absorption Laws (Set Theory)/Intersection with Union/Proof 2

Theorem

$S \cap \paren {S \cup T} = S$

Proof

\(\ds x\)

\(\in\)

\(\ds S \cap \paren {S \cup T}\)

\(\ds \leadstoandfrom \ \ \)

\(\ds x\)

\(\in\)

\(\ds S \land \paren {x \in S \lor x \in T}\)

Definition of Set Intersection and Definition of Set Union

\(\ds \leadstoandfrom \ \ \)

\(\ds x\)

\(\in\)

\(\ds S\)

Conjunction Absorbs Disjunction

$\blacksquare$