Empty Set is Element of Power Set

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Theorem

The empty set is an element of all power sets:

$\forall S: \O \in \powerset S$


Proof

\(\displaystyle \forall S: \O\) \(\subseteq\) \(\displaystyle S\) Empty Set is Subset of All Sets
\(\displaystyle \leadsto \ \ \) \(\displaystyle \forall S: \O\) \(\in\) \(\displaystyle \powerset S\) Definition of Power Set

$\blacksquare$


Also see


Sources