Equal Set Differences iff Equal Intersections/Proof 1

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Theorem

$R \setminus S = R \setminus T \iff R \cap S = R \cap T$


Proof

\(\displaystyle R \setminus S\) \(=\) \(\displaystyle R \setminus T\)
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle \set {x \in R: x \notin S}\) \(=\) \(\displaystyle \set {x \in R: x \notin T}\) Definition of Set Difference
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle \forall x \in R: \quad x \notin S\) \(\iff\) \(\displaystyle x \notin T\)
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle \forall x \in R: \quad x \in S\) \(\iff\) \(\displaystyle x \in T\)
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle \set {\paren {x \in R} \land \paren {x \in S} }\) \(=\) \(\displaystyle \set {\paren {x \in R} \land \paren {x \in T} }\)
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle R \cap S\) \(=\) \(\displaystyle R \cap T\) Definition of Set Intersection

$\blacksquare$