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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. Weak Power as written above is simply:

$(\forall a)(\exists x)(\forall y)(a \cap y\in x) $

Now in classical ZF all subsets of $a$ are overlaps with $a$, i.e. $z \subseteq a \to z \cap a=z$, so all of them would be included in the weak power of $a$ (just substitute each subset $z$ of $a$ instead of $y$ in the above formula), Then by separation one can easily recover full $P(a)$ by separating on the weak power of $a$ using the property of being a subset of $a$.

The asserted set $x$ is just a set that contains all overlap sets between the set $a$ and any set, among its elements. Weak Power as written above is simply:

$(\forall a)(\exists x)(\forall y)(a \cap y\in x) $

Now in classical ZF all subsets of $a$ are overlaps with $a$, i.e. $z \subseteq a \to z \cap a=z$, so all of them would be included in the weak power of $a$ (just substitute each subset $z$ of $a$ instead of $y$ in the above formula), then by separation one can easily recover full $P(a)$ by separating on the weak power of $a$ using the property of being a subset of $a$.

The asserted set $x$ is just a set that contains all overlap sets between the set $a$ and any set, among its elements. Weak Power as written above is simply:

$(\forall a)(\exists x)(\forall y)(a \cap y\in x) $

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 $a$.

The asserted set $x$ is just a set that contains all overlap sets between the set $a$ and any set, among its elements. Weak Power as written above is simply:

$(\forall a)(\exists x)(\forall y)(a \cap y\in x) $

Now in classical ZF all subsets of $a$ are overlaps with $a$, i.e. $z \subseteq a \to z \cap a=z$, so all of them would be included in the weak power of $a$ (just substitute each subset $z$ of $a$ instead of $y$ in the above formula), Then by separation one can easily recover full $P(a)$ by separating on the weak power of $a$ using the property of being a subset of $a$.

The asserted set $x$ is just a set that contains all overlap sets between the set $a$ and any set, among its elements. Weak Power as written above is simply:

$(\forall a)(\exists x)(\forall y)(\exists z\in x) (z= a \cap y)$

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 $a$.

The asserted set $x$ is just a set that contains all overlap sets between the set $a$ and any set, among its elements. Weak Power as written above is simply:

$(\forall a)(\exists x)(\forall y)(a \cap y\in x) $

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 $a$.