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

Theorem

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

Proof

\(\ds x\)

\(\in\)

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

\(\ds \leadstoandfrom \ \ \)

\(\ds x\)

\(\in\)

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

Definition of Set Intersection and Definition of Set Union

\(\ds \leadstoandfrom \ \ \)

\(\ds x\)

\(\in\)

\(\ds S\)

Disjunction Absorbs Conjunction

$\blacksquare$