If we work in General Extensional Atomic Mereology [without bottom], so the primitives of the language are $P$ standing for "is a part of", and equality, now we add to it membership $\in$ relation symbol.

I observed that all rules of $\text{ZF}$ can be derived from the following Mereology-Set translation rule:

If $\phi$ is a formula in the pure language of mereology (only uses $P$ and $=$ as predicates) in which the symbol $x$ doesn't occur free, and such that all quantifiers in it are restricted to atoms (i.e. of the form $\forall x (\operatorname{atom}(x) \to\cdots)$, or $\exists x (\operatorname{atom}(x) \wedge\ldots)$, these denoted as $\forall ^* x (\cdots)$ and $\exists^* x(\ldots)$ respectively), and in which only symbols $y$, $p_1$, …, $p_n$ occur free, and if $\phi^{\in}$ is the formula obtained from $\phi$ by merely replacing each occurrence of the symbol $P$ by $\in$ and the symbols $\forall^*$, $\exists^*$ by symbols $\forall$, $\exists$ respectively, then:

$$[\forall^* p_1,\dotsc, \forall^* p_n \exists k (\operatorname{atom}(k) \wedge \forall y (\phi(y) \to y=k)] \to \forall p_1,\dotsc,\forall p_n \exists x \forall y (y \in x \leftrightarrow \phi^{\in})$$

is an axiom.

Now take the formulas: $$y=p_1,$$ $$\exists^* z \ (z \ P \ p_1 \wedge y\ P \ z),$$ $$\forall^* z \ (z \ P \ y \to z \ P \ p_1),$$ $$\exists^* x \ P \ p_1 [ \phi(x,y)], \quad\text{where}\quad\forall^* x \ P \ p_1 \ \exists^*! y (\phi(x,y)), $$ $$\forall^* I (p_1 \ P \ I \wedge \forall^* m \ P \ I (\exists^* n \ P \ I (\forall^*z \ P \ n \leftrightarrow z=m)) \to y \ P \ I) $$

All of the above formulas fulfill the hypothesis of the above schema and so the consequence would be axioms of Singletons, Set Union, Power, Replacement, and Infinity. Which are all the constructive axioms of $\text{ZF}$, except empty set, which can be simply added.

Of Note is that the known paradoxes of Cantor's, Russell's, Burali-Forti, Mirimanoff, and Lesniewski's, all are avoidable here.

I'm not really sure if that schema is consistent, but the idea is that it reflects a certain relationship between atomic Mereology and set theory, all axioms of set theory can be viewed as in some sense paralleling or rather mimicking the atomic part-hood relation of Mereology. So membership relation can be viewed as a kind of a binary relation between atoms and sets themselves are viewed as atoms.

Has such forms of mimicry between pieces of Mereology and Sets been considered before, especially along lines of setting some Mereological basis for deriving rules of set theory?