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## Homework Statement

Given that [itex]F[/itex] is a family of sets, that [itex] \bigcup F[/itex] is the union of the sets members of the family [itex]F[/itex], that [itex]A[/itex] is a set, assume that

**(1)**[itex] \hspace{1cm} \forall F (\bigcup F = A \rightarrow A \in F) [/itex]

then prove that

**(G)**[itex] \hspace{1cm} \exists x (A= \left\{ x \right\} ) [/itex]

## Homework Equations

Given that [itex]P \rightarrow Q[/itex] is the same as [itex]\neg P \vee Q[/itex], we can rephrase the assumption as

**(2)**[itex] \hspace{1cm} \forall F (\bigcup F \neq A \vee A \in F) [/itex]

## The Attempt at a Solution

That's the biggest problem: I have no idea how to get the solution.

Just to remember, I am fighting with the software

*Proof Designer*: I love and hate it. I love it when it tells me I got the proof, I hate it when it doesn't... so right now I am hating it (or maybe I am hating myself!).

Btw, a nice part of the program is that it gives you the possibility to proceed in the proof even if you have no idea what - in this specific case - [itex]F[/itex] and [itex]x[/itex] are... which is exactly my situation! The program does it in order to mimic the way in which proofs actually work. You go around, you assume, and then you define... and everything goes right. Usually, but not this time.

First of all, just a confirmation of my general thought about the theorem: for whatever [itex]F[/itex] you pick up, you can always find an [itex]A[/itex] that has some characteristics (which have to be really specific).

So, basically we can vary [itex]F[/itex] and we will always find a corresponding [itex]x[/itex] that respect

**(1)**or

**(2)**.

__Right?__

About the solution. I focused a lot on

**(2)**, and then I tried to work out a proof by cases. It didn't really work, maybe cause I chose the wrong values for [itex]F[/itex] and [itex]x[/itex]. In particular I put [itex]A= \left\{ ∅ \right\} [/itex] and then [itex]F= \left\{ \left\{ ∅ \right\} \right\} [/itex]. With these choices, the second case is easily covered, while the second simply cannot be solved. This is related to the fact that the assumption is now [itex] \bigcup F \neq A [/itex], a negative statment that is not easy manageable (at least for me).

Now, the problem I found is that, even if I choose a different value for [itex]F[/itex] (let's say [itex]P(A)[/itex], defined as the Power Set of A) and wait to find the value for [itex]x[/itex], still I go nowhere. For example, [itex]P(A)[/itex] has some nice properties and it can be used

**(1)**instead of

**(2)**, because it can be easily proven that [itex] \bigcup P(A) =A [/itex], giving the possibility to use MP to get [itex]A \in P(A) [/itex]. However, in this case I have no idea what [itex]x[/itex] should be. It's like there is a huge gap between what I am assuming and what I have to prove and I cannot fill it...

Sorry for the long post, but this problem is really absorbing me.

I would really appreciate any kind of feedback.

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