Subset is Element of Power Set
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Theorem
Let $x$ be a set.
Let $\powerset x$ denote the power set of $x$.
Then:
- $y \in \powerset x \iff y \subseteq x$
Proof
By definition of power set, $\powerset x$ is the set of subsets of $x$.
Hence the result, by definition of subset and power set.
$\blacksquare$
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 6$ The power axiom: Remarks $(2)$