I am working with Zermelo-Fraenkel axioms. Specifically, I am allowed to assume the Axiom of Pair, Axiom Schema of Comprehension, Axiom of Union, and Axiom of Power Set, etc. (not yet allowed to use Axiom of Choice but that really should not be relevant to this question)
I'm given that there's a set $A$.
I know that by the Axiom of Union, the following collection is a set:
$$\bigcup A = \left\{x \mid \exists\, y \in A \text{ such that } x \in y\right\}$$
I also know how to prove that any set is a subset of the power set of the union of itself. For $A$, this would be $$A \subseteq \mathcal{P}\left(\bigcup A\right)$$
I need to show that the set of all sets whose unions are $A$ exists. This set should be a subset of the power set of $A$ but I'm not sure how to actually prove that it exists.
I want to contruct the power set of $A$ and use the Axiom Schema of Comprehension to show that this set $B$ exists, with
$$B = \left\{x \mid \bigcup x = A\right\}$$
Any help would be appreciated ... again, I feel like this is going to be a one-liner :/
\bigcup
gives you $\bigcup$, and\cup
gives you $\cup$. Similarly with\bigcap
and\cap
for intersection symbols. $\endgroup$ – Brian M. Scott Dec 17 '14 at 2:22