Set is Subset of Power Set of Union
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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 $6.1. \ \text {(a)}$