I am trying to prove that two definitions of a finite set are equivalent.(adsbygoogle = window.adsbygoogle || []).push({});

1.) A set ##A## is finite if and only if it is equipollent to a natural number ##n##. ( natural number as the set containing all the previous natural numbers including ##0## )

2.) A set ##A## is finite if and only if every collection of subsets has a minimal element with respect to the ##\subset## relation. ( this means there is at least one element of every collection that contains no others as subsets ).

I have figured out how to prove (1) ##\rightarrow## (2) but am struggling with (2) ##\rightarrow## (1).

Please don't post a full proof. Just a hint or basic sketch would be most appreciated.

Also, I'd like to prove it without using the axiom of choice. Induction on the natural numbers is okay ( that's how I proved (1) ##\rightarrow## (2) ).

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# A Somewhat difficult set theory proof

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