Equivalence Relation on Power Set induced by Intersection with Subset/Equivalence Class of Empty Set

Theorem
Let $A, T$ be sets such that $A \subseteq T$.

Let $S = \powerset T$ denote the power set of $T$.

Let $\alpha$ denote the relation defined on $S$ by:
 * $\forall X, Y \in S: X \mathrel \alpha Y \iff X \cap A = Y \cap A$

We have that $\alpha$ is an equivalence relation.

The equivalence class of $\O$ in $S$ with respect to $\alpha$ is given by:


 * $\eqclass \O \alpha = \powerset {T \setminus A}$

Proof
That $\alpha$ is an equivalence relation is proved in Equivalence Relation on Power Set induced by Intersection with Subset.

We have that:
 * $\eqclass \O \alpha = \set {X \in S: X \cap A = \O \cap A = \O}$

Thus:

Then: