Power Set of Empty Set

Theorem

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

Proof

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 $\O$.

$\blacksquare$

Sources

• 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 1$: The Language of Set Theory: Exercise $1.8$