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We show that if 
$(M,{ \in _1},{ \in _2})$ satisfies the first-order Zermelo–Fraenkel axioms of set theory when the membership relation is 
${ \in _1}$ and also when the membership relation is 
${ \in _2}$, and in both cases the formulas are allowed to contain both 
${ \in _1}$ and 
${ \in _2}$, then 
$\left( {M, \in _1 } \right) \cong \left( {M, \in _2 } \right)$, and the isomorphism is definable in 
$(M,{ \in _1},{ \in _2})$. This extends Zermelo’s 1930 theorem in [6].
