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Intersection**

Given a set \(X\), its intersection \(\cap X\) is the set which contains all elements contained in all sets in \(X\): \(\forall a:a\in \cap X\iff\forall x\in X:a\in x\), or equivalently \(\cap X=\{a|\forall x\in X:a\in x\}\). When \(X=\{x_1,...,x_n\}\) is finite, we typically write \(\cap X=x_1\cap\cdots \cap x_n\), and when \(X=\{x_i|i\in I\}\) is an indexed collection of sets we write \(\cap X=\cap_{i\in I}x_i\). The existence of the intersection of \(X\) is guaranteed by the axiom of replacement.