Intersection of Singleton

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Consider the set of sets $A$ such that $A$ consists of just one set $x$:

$A = \set x$

Then the intersection of $A$ is $x$:

$\bigcap A = x$


Let $A = \set x$.

Then from the definition:

$\bigcap \set x = \set {y: \forall z \in \set x: y \in z}$

from which it follows directly:

$\bigcap \set x = \set {y: y \in x}$

as $x$ is the only set in $\set x$.

That is:

$\bigcap A = x$


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