Power Set of Empty Set

From ProofWiki
Jump to navigation Jump to search


The power set of the empty set $\O$ is the set $\set \O$.


From Empty Set is Element of Power Set and Set is Element of its Power Set:

$\O \in \powerset \O$

From Empty Set is Subset of All Sets:

$S \subseteq \O \implies S = \O$

That is:

$S \in \powerset \O \implies S = \O$

Hence the only element of $\powerset \O$ is $\O$.