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

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Theorem

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


Proof

\(\displaystyle x\) \(\in\) \(\displaystyle S \cap \paren {S \cup T}\)
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle S \land \paren {x \in S \lor x \in T}\) Definition of Set Intersection and Definition of Set Union
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle S\) Conjunction Absorbs Disjunction

$\blacksquare$