Set is Element of its Power Set
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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 $1 \ \text{(a)}$