Set Equality is Equivalence Relation

Theorem
Let $S$ be a set.

Set equality is an equivalence relation on the power set $\mathcal P \left({S}\right)$ of $S$.

Proof
Checking in turn each of the criteria for equivalence:

Reflexivity
Let $A \in \mathcal P \left({S}\right)$.

From Set Equals Itself:
 * $A = A$

So set equality has been shown to be reflexive on $\mathcal P \left({S}\right)$.

Symmetry
Let $A, B \in \mathcal P \left({S}\right)$.

Let $A = B$.

Then by definition of set equality:
 * $A \subseteq B$
 * $B \subseteq A$

from which it follows by definition of set equality that $B = A$.

So set equality has been shown to be symmetric on $\mathcal P \left({S}\right)$.

Transitivity
Let $A, B, C \in \mathcal P \left({S}\right)$.

Let $A = B$ and $B = C$.

Then by definition of set equality:


 * $(1): \quad A \subseteq B$
 * $(2): \quad B \subseteq C$
 * $(3): \quad C \subseteq B$
 * $(4): \quad B \subseteq A$

From $(1)$ and $(2)$ and Subset Relation is Transitive:
 * $A \subseteq C$

From $(3)$ and $(4)$ and Subset Relation is Transitive:
 * $C \subseteq A$

from which it follows by definition of set equality that $A = C$.

So set equality has been shown to be transitive on $\mathcal P \left({S}\right)$.

Set equality has been shown to be reflexive, symmetric and transitive on $\mathcal P \left({S}\right)$.

Hence by definition it is an equivalence relation on $\mathcal P \left({S}\right)$.