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For a non-empty set $A$ and a partial ordering of $P(A)$ by the containment relation, given an order preserving $f: P(A)\to P(A)$, show that there exists a $B$ such that $f(B)=B$.

So far my thoughts are as follows: The minimal element in the group $P(A)$ ordered by containment has to be $[\emptyset]$, but I'm not sure where that gets me.

Also, can I say that $[\emptyset]$ is contained in $f([\emptyset])$?

I'm not sure how to coninute, any help would be appreciated!

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