If $\psi(s,x)$ is a formula in which symbols $``s,x"$ occur free, in which the symbol $``u"$ doesn't occur free, then all closures of:

$$\forall S \ [ \forall s (s \subseteq S \exists! x (\psi(s,x))) \to \exists u \forall z (z \in u \leftrightarrow \exists x (\exists s \subseteq S (\psi(s,x)) \wedge z \in x))]$$

are axioms.

In other words this would be: $\forall S \exists u [u=\bigcup(\{F(x)| x \subseteq S\})]$ for a definable function $F$.

In English if we replace each subset of a set by a set after a definable function $F$; then the set union of all replacing sets exists.

Now this axiom with Extensionality, Singletons and the Empty set, would easily interpret: Pairing, Set Union, Power, Separation, and Replacement. Which are the main comprehension axioms of $ZFC$.

What is noticeable is that this axiom uses basically the subset operator and set union, which reminds one of Part-hood and Mereological fusions, which appear to support an underlying Mereological motivation for comprehension in $ZFC$.

Had there been known work on that particular axiom in relation to a Mereological foundation of Set Theory?


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