Definition:Set Intersection/Family of Sets/Two Sets

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Let $I = \set {\alpha, \beta}$ be an indexing set containing exactly two elements.

Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$.

From the definition of the intersection of $S_i$:

$\ds \bigcap_{i \mathop \in I} S_i := \set {x: \forall i \in I: x \in S_i}$

it follows that:

$\ds \bigcap \set {S_\alpha, S_\beta} := S_\alpha \cap S_\beta$