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

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Theorem

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


Proof

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

$\blacksquare$