Proof

Please note that there is no set of all sets. Therefore, our universal set $U$ is some given set. We want to show that the equality of sets "$=$" fulfills all the defining properties of an equivalence relation for every subset of $A\subseteq U:$

For any subset $A\subseteq U$ we have $A=A$ by the axiom of extensionality. Thus "$=$" is reflexive.
If $A,B\subset U$, then, by the same axiom, from $A=B$ it follows $B=A.$ Thus, "$=$" is symmetric.
If $A,B,C\subset U$, then, by the same axiom, from $A=B$, and $B=C$, it follows $A=C$. Thus, "$=$" is "transitive.
∎

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