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The asserted set $x$ is just a set that contains all overlap sets between the set $a$ and any set, among its elements. Now in classical ZF all subsets of $a$ are overlaps with $a$, so all of them would be included in the weak power of $a$. Then by separation one can easily recover $P(a)$ axiom by separating on the weak power of $a$ using the property of being a subset of $a$.