A total order on finite subsets of an ordered set

Let $(A,<)$ be an ordered set. Then I define an ordering on the set of finite subsets of $A$ as follows. I say that $Xprec Y$ if and only if the smallest element of $Xtriangle Y$ (the symmetric difference) is contained in $X$. An easy, but not completely trivial, observation is that $prec$ actually defines a total order on the family of finite subsets of $A$. My question is whether this construction is well-known.