TITLE: What do the ZFC axioms look like in terms of subset?
QUESTION [9 upvotes]: I'm reasonably familiar with the ZFC axioms. I know they can be formalized in many different ways. However, I usually see them presented in terms of the elementary "propositional calculus primitives" $\rightarrow$, $\neg$, $\wedge$, $\vee$, $\leftrightarrow$; the "predicate calculus with equality" primitives $\forall$, $\exists$, $=$ ; and the single "set-theoretic primitive" $\in$. Then one usually proceeds to define other set operators like $\subseteq$ (and everything else) in terms of $\in$.
What happens if you instead take $\subseteq$ to be the primitive operator of set theory, and define $\in$ in terms of $\subseteq$?
Obviously we could rewrite any formalization of ZFC by replacing each occurrence $x \in y$ with $\{x\} \subseteq y$, but this would not be the most compact or elegant formulation. For example, it seems to me that Extensionality could best be cast as $x \subseteq y \wedge y \subseteq x \rightarrow x = y$.
Surely someone has studied this question and presented an elegant formalization of ZFC based on $\subseteq$. Where can I find it?

REPLY [6 votes]: As Noah so kindly mentioned, my work on set-theoretic mereology is exactly about this question. See my recent paper with Makoto Kikuchi:
J. D. Hamkins and M. Kikuchi, Set-theoretic mereology, Logic and Logical Philosophy, special issue “Mereology and beyond, part II”, pp. 1-24, 2016.  (click through to the arxiv for pdf)
Abstract. We consider a set-theoretic version of mereology based on the inclusion relation $\subseteq$ and analyze how well it might serve as a foundation of mathematics. After establishing the non-definability of $\in$ from $\subseteq$, we identify the natural axioms for $\subseteq$-based mereology, which constitute a finitely axiomatizable, complete, decidable theory. Ultimately, for these reasons, we conclude that this form of set-theoretic mereology cannot by itself serve as a foundation of mathematics. Meanwhile, augmented forms of set-theoretic mereology, such as that obtained by adding the singleton operator, are foundationally robust.
See more at the blog post, or take a look at the paper itself.