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## Homework Statement

We're working more or less with the standard ZF axioms.

Prove that [tex]A \subseteq \mathcal{P}(\bigcup A)[/tex] for any set A, whose elements are all sets. When are they equal?

## Homework Equations

Just the axioms

I) Extensionality

II) Emptyset and Pairset

III) Separation

IV) Powerset

V) Unionset

VI) Infinity

## The Attempt at a Solution

I can prove the first part pretty easily. We have that the union and power set of the union are sets directly from V and IV.

Briefly, suppose [tex]X \in A[/tex]. Then [tex]X \subseteq \bigcup A[/tex] by def. of union. So [tex]X \in \mathcal{P}(\bigcup A)[/tex].

The part I'm stuck on is when they're equal. I attempted to prove that [tex]\mathcal{P}(\bigcup A) \subseteq A[/tex] in order to get an idea of what condition A would need to meet.

My best guess is that they're only equal when A (at the very least) contains the emptyset and the singletons of every element in the union of all of A. Seems like a pretty circular definition, though, since I need to know what A is to know what the union of all of A is. Maybe I could say that A needs to be composed of the emptyset and the remaining elements must be singletons or else be the union of singletons already contained in A? I have thought about this a fair bit, and I'm pretty sure my condition ensures equality, so this isn't a random guess. Help would be appreciated.

Also, we have not really gotten into cardinality yet, but I do know that [tex]|S| \leq |\bigcup S|[/tex] if S is countable. But I also know that [tex]|\bigcup S| < |\mathcal{P}(\bigcup S)|[/tex]. So for our equality to hold, S must be uncountable...?

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