Set is Subset of Power Set of Union

Theorem

Let $x$ be a set of sets.

Let $\bigcup x$ denote the union of $x$.

Let $\powerset {\bigcup x}$ denote the power set of $\bigcup x$.


Then:

$x \subseteq \powerset {\bigcup x}$


Proof

Let $z \in x$.

By Element of Class is Subset of Union of Class:

$z \subseteq \bigcup x$

By definition of power set:

$z \in \powerset {\bigcup x}$

The result follows by definition of subset.

$\blacksquare$


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: Exercise $6.1. \ \text {(a)}$
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