A clean way to separate sets of surreals

Let surreal numbers be defined by means of sign sequences. We give a proof that if $S<T$ are sets of surreals, then there is some surreal $w$ such that $S<w<T$. The classical proof is simplified by observing that, for every set $S$ of surreals, there exists a surreal $s$ such that, for every surreal $w$, we have $S<w$ if and only if the restriction of $w$ to the length of $s$ is $\ge s$. Hence $S<w<T$ if and only if $w$ satisfies the above condition, as well as its symmetrical version with respect to $T$. It is now enough to check that if $S<T$, then the two conditions are compatible.