[proofplan] We use the Axiom of Extensionality as a uniqueness principle. The hypothesis says exactly that $a$ and $b$ have the same elements. Extensionality then identifies the two sets. [/proofplan] [step:Match the hypothesis with the extensionality criterion] The Axiom of Extensionality states that two sets are equal whenever they have precisely the same elements: if $A$ and $B$ are sets and, for every set $x$, \begin{align*} x \in A \iff x \in B, \end{align*} then $A = B$. In the present theorem, $a$ and $b$ are sets, and the hypothesis gives, for every set $x$, \begin{align*} x \in a \iff x \in b. \end{align*} Thus the hypotheses of the Axiom of Extensionality are satisfied with $A := a$ and $B := b$. [/step] [step:Conclude equality of the two sets] Applying the Axiom of Extensionality to the sets $a$ and $b$ gives \begin{align*} a = b. \end{align*} This proves the claimed uniqueness statement. [/step]