Atlas Belongs to Unique Differentiable Structure

Theorem
Let $M$ be a locally Euclidean space of dimension $d$.

Let $\mathscr F$ be an atlas on $M$.

Then there exists a unique differentiable structure $\mathcal F$ on $M$ with $\mathscr F \in \mathcal F$.

Proof
Let $\mathcal F$ be the equivalence class of $\mathscr F$ under the equivalence relation of compatibility.

By definition we have $\mathscr F \in \mathcal F$.

By Relation Partitions Set iff Equivalence, $\mathcal F$ is an element of the partition of equivalence classes.

By definition, the elements of a partition are pairwise disjoint, so if $\mathcal G \ne \mathcal F$ is an element of the partition, we must have $\mathscr F \notin \mathcal G$.

Therefore $\mathscr F$ belongs to exactly one differentiable structure.