Empty Set is Element of Power Set

Theorem

The empty set is an element of all power sets:

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

Proof

\(\ds \forall S: \, \)

\(\ds \O\)

\(\subseteq\)

\(\ds S\)

Empty Set is Subset of All Sets

\(\ds \leadsto \ \ \)

\(\ds \forall S: \, \)

\(\ds \O\)

\(\in\)

\(\ds \powerset S\)

Definition of Power Set

$\blacksquare$

Also see

• Set is Element of its Power Set

Sources

• 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 1.8$. Sets of sets: Example $25$
• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 2$: Sets and Subsets: Exercise $1 \ \text{(d)}$