  We prove that for every Borel equivalence relation $E$, either $E$ is Borel
reducible to $\mathbb{E}\_0$, or the family of Borel equivalence relations
incompatible with $E$ has cofinal essential complexity. It follows that if $F$
is a Borel equivalence relation and $\cal F$ is a family of Borel equivalence
relations of non-cofinal essential complexity which together satisfy the
dichotomy that for every Borel equivalence relation $E$, either $E\in {\cal F}$
or $F$ is Borel reducible to $E$, then $\cal F$ consists solely of smooth
equivalence relations, thus the dichotomy is equivalent to a known theorem.
