Given infinitely many finite maximal chains in a poset $P$, I am trying to construct an infinite antichain. My initial approach was to construct my infinite antichain by taking the maximal element of each maximal chain and show that none of them are ordered to any other. But showing they are not ordered to each other relies on each maximal element being different which I am not at liberty to assume. So I was wondering if anyone can recommend a different approach or tell me if there's a way I could get around the fact that the maximal elements aren't all necessarily different?
Any help is appreciated, cheers.
Let F = { x : x in finite maximal chain of P }.
Let K be a maximal antichain of F.
For all x in K, exists finite maximal chain C$_x$.
Show F = $\cup${ C$_x$ : x in K }.
Assume K is finite. Thus F is finite.
There are finite many finite maximal cbains of F.
So there exists a finite maximal chain C of P
that is not a finite maximal chain of F.
However C subset F, a contradiction,