Set Difference is Right Distributive over Set Intersection/General Case

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Let $U$ be a collection of sets.

Let $T$ be a set.


$\ds \bigcap_{X \mathop \in U} \paren {X \setminus T} = \paren {\bigcap_{X \mathop \in U} X} \setminus T$

That is, the difference with an intersection equals the intersection of the differences.


\(\ds x\) \(\in\) \(\ds \bigcap_{X \mathop \in U} \paren {X \setminus T}\)
\(\ds \leadstoandfrom \ \ \) \(\ds \forall X \in U: \, \) \(\ds x\) \(\in\) \(\ds X \setminus T\) Definition of Set Intersection
\(\ds \leadstoandfrom \ \ \) \(\ds \forall X \in U: \, \) \(\ds x\) \(\in\) \(\ds X\) Definition of Set Difference
\(\, \ds \land \, \) \(\ds x\) \(\not \in\) \(\ds T\)
\(\ds \leadstoandfrom \ \ \) \(\ds x\) \(\in\) \(\ds \bigcap_{X \mathop \in U}\) Definition of Set Intersection
\(\, \ds \land \, \) \(\ds x\) \(\not \in\) \(\ds T\)
\(\ds \leadstoandfrom \ \ \) \(\ds x\) \(\in\) \(\ds \paren {\bigcap_{X \mathop \in U} X} \setminus T\) Definition of Set Difference