= Solution
Statement (iii) can fail because a nontransitive ambient set can omit the elements that distinguish two of its members. Take
$$
y=\{\varnothing,\{\{\varnothing\}\}\}.
$$
Neither element of $y$ has an element that also lies in $y$: the second contains $\{\varnothing\}$, which is omitted. Their predecessor sets for membership restricted to $y$ are therefore both empty, although the two elements are distinct. Hence this restricted membership relation, and any isomorphic relation, is not extensional.
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