Subset is Element of Power Set

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 $(2)$