Special sum set

A finite set $A$ of positive integers such that nonempty disjoint sets contained in $A$ have different sums, and the larger subset has the larger sum when their sizes differ. For $a_1<\cdots <a_n$, the size condition is equivalent to $a_1+\cdots +a_{k+1}>a_{n-k+1}+\cdots +a_n$ for $1\le k\le \lfloor (n-1)/2\rfloor$: the left side is the smallest possible sum of $k+1$ elements and the right side the largest sum of $k$ elements.