Atlas Belongs to Unique Differentiable Structure

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

Let $\mathcal A$ be an atlas on $M$.

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

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

By Compatibility of Atlases is Equivalence Relation, this is indeed an equivalence relation.

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

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.

Therefore if $\mathcal G \ne \mathcal F$ is an element of the partition, we must have:
 * $\mathcal A \notin \mathcal G$

Therefore $\mathcal A$ belongs to exactly one differentiable structure.