Connected component of (A∩ B)ᶜ through a point of A∖ B

Let $X$ be a topological space and let $A, B \subseteq X$ be subsets, both assumed closed, whose union is all of $X$. Assume further that the difference $A \setminus B$ is preconnected, and let $e$ be a point of $X$ lying in $A \setminus B$, that is, $e \in A$ and $e \notin B$. The conclusion is an equality of subsets of $X$: the connected component of $e$ inside the subset $(A \cap B)^{c}$, in the sense of Mathlib's connectedComponentIn (the connected component of $e$ in the subspace $(A\cap B)^{c}$, pushed forward to $X$), equals $A \cap (A \cap B)^{c}$. Since $A$ is covered by $A \cap B$ and $A \setminus B$, this right-hand side is exactly $A \setminus B$; the statement is phrased with the intersection with the complement rather than with the set difference.

An elementary point-set fact: when $X$ is the union of two closed sets $A$ and $B$, the open set $(A \cap B)^{c}$ is the disjoint union of the relatively clopen pieces $A \setminus B$ and $B \setminus A$, so a preconnected piece containing $e$ is the whole connected component of $e$. It is used in the analysis of degenerations of a two-component fibre, where $A$ and $B$ are the two components and $(A\cap B)^{c}$ the locus away from their intersection: it is cited by ModularCurve.DRModelPackage.exists_twoLineDegeneration_of_not_smooth and ModularCurve.DRModelPackage.exists_twoLineDegeneration_of_not_smooth_iso_comp_eq.