Element in Set iff Singleton in Powerset
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Theorem
Let $S$ be a set.
Then:
- $x \in S \iff \set x \in \powerset S$
where $\powerset S$ denotes the power set of $S$.
Proof
| \(\ds x\) | \(\in\) | \(\ds S\) | ||||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds \set x\) | \(\subseteq\) | \(\ds S\) | Singleton of Element is Subset | ||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds \set x\) | \(\in\) | \(\ds \powerset S\) | Definition of Power Set |
$\blacksquare$
Sources
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 2$. Sets of sets