Set is Element of its Power Set

Theorem

A set is an element of its power set:

$S \in \powerset S$


Proof

\(\ds \forall S: \, \) \(\ds S\) \(\subseteq\) \(\ds S\) Set is Subset of Itself
\(\ds \leadsto \ \ \) \(\ds \forall S: \, \) \(\ds S\) \(\in\) \(\ds \powerset S\) Definition of Power Set

$\blacksquare$


Also see


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 $1 \ \text{(a)}$
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