Every Partially ordered set has a maximal independent subset I am working on this problem:-

Every Partially ordered set has a maximal independent subset.

Definition:Let $\langle E,\prec\rangle$ be a partially ordered set. A subset  $A\subset E$ is called independent set if for any two of its elements $a , b$ neither $a\prec b$ or $ b\prec a$.
My attempt:- I am trying to apply the  Teichmüller–Tukey lemma,but from my thinking I see  Teichmüller–Tukey lemma only give guarantee having the maximal subset. 
any help appreciated. 

This is a classical application of Zorn's lemma, but you can just as well use the Teichmüller–Tukey lemma instead.
HINT: Show that $\mathcal F=\{A\subseteq E\mid A\text{ is independent}\}$ has finite character. What can you conclude about a maximal element in $\cal F$?
