Incidence equivalence in a set family ($x\sim y⟺\forall B\in \mathcal{F}:(x\in B⟺y\in B)$)

Two ground elements are incidence-equivalent when they belong to exactly the same members of a set family. Their incidence columns are identical. For the set basis problem, deleting one of two such elements preserves the answer. A basis before deletion can be restricted to the smaller ground set. Conversely, if $x$ remains and $y$ was deleted, lift each reduced basis set $A$ to $A\cup \{y\}$ when $x\in A$, and leave it unchanged otherwise. In every union of lifted basis sets, $y$ appears exactly when $x$ appears, which restores precisely the original input sets. Repeated deletion thus leaves one representative of every incidence class.