Union of Empty Set

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Theorem

Consider the set of sets $\mathbb S$ such that $\mathbb S$ is the empty set $\O$.


Then the union of $\mathbb S$ is $\O$:

$\mathbb S = \O \implies \displaystyle \bigcup \mathbb S = \O$


Proof

Let $\mathbb S = \O$.

Then from the definition:

$\displaystyle \bigcup \mathbb S = \set {x: \exists X \in \mathbb S: x \in X}$

from which it follows directly:

$\displaystyle \bigcup \mathbb S = \O$

as there are no sets in $\mathbb S$.

$\blacksquare$


Sources