Element in Set iff Singleton in Powerset

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
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