$$It may EDIT: Originally I suggested that it might be interesting (in my opinion) to consider the a system of mereology with the implication above taken as an axiom. This could shed some light on However, in the nature comment below Andreas pointed to the fact that this entails linearity of difference between both understandings \sqsubseteq. The consequence is that the class of setsmodels of the theory which consists of poset axioms+separation+existence of mereological sums narrows down to one-element (up to isomorphism) class, the only model being degenerate one-element structure. 6 Minor corrections. I decided to add one more answer (instead of editing the previous one), since it is quite long. This will mainly address the OP question, Andreas Blass answer and Carl Mummert comment about defining sets as sets of atoms (points) in mereology. I hope it will shed some light on mereology and its relation to set theory. In mereology, as it is done in Lesniewskian tradition, it is assumed that part of relation (in symbols: \sqsubseteq) is a partial order (reflexive, antisymmetrical and transitive) and that it satisfies the separation condition (those familiar with forcing will find it very familiar):$$ \neg x\sqsubseteq y\longrightarrow\exists z(z\sqsubseteq x\wedge z\mathrel{\bot} y) $$where z\mathrel{\bot} y\iff\neg\exists u(u\sqsubseteq z\wedge u\sqsubseteq z) (z and y are incompatible, otherwise they are compatible). The crucial point is a definition of mereological sum (sometimes called fusion as well). The very idea of mereological sum is hidden in the following equivalence: an object x is a mereological sum of the group of S-es if and only if every S is part of x and every part of x is compatible with some S. Notice that it is a consequence of the definition that there cannot be a mereological set of an empty group of objects. Using sets and set theoretical notation we may define the sum of a set X as binary relation in the following way:$$ x\mathrel{\mathrm{Sum}} X\iff \forall y(y\in X\longrightarrow y\sqsubseteq x)\wedge\forall y(y\sqsubseteq x\longrightarrow\exists z(z\in X\wedge\neg z \mathrel{\bot} y). $$What is usually called classical mereology is a second order system which is obtain by adding the following axiom:$$ \forall X(X\neq\emptyset\longrightarrow\exists x(x\mathrel{\mathrm{Sum}} X). $$Building a first-order system is a little bit more painstaking. To simplify things a bit we may introduce some auxiliary notation:$$ x\mathrel{\mathbf{sum}_y}\varphi(y) $$as an abbreviation of the following formula:$$ \forall y(\varphi(y)\longrightarrow y\sqsubseteq x)\wedge\forall u(u\sqsubseteq x\longrightarrow\exists z(\varphi(z)\wedge \neg z\mathrel{\bot} u)). $$"x\mathrel{\mathbf{sum}_y}\varphi(y)" may be read as x is a mereological sum of all \varphi-ers. From this we can prove for example that: • \forall z(z\mathrel{\mathbf{sum}_y}\text‘z=y\text') • \forall z(z\mathrel{\mathbf{sum}_y}\text‘z\sqsubseteq y\text'). In this setting, mereological sum existence axiom schema can be expressed as:$$ \exists x\varphi(x)\longrightarrow\exists y(y\mathrel{\mathbf{sum}_x}\varphi(x)). $$Since the consequence of the axioms presented is that there can only be one mereological sum of \varphi-ers we can introduce notation (analogous to the set-theoretical abstraction operator):$$ \bigl[x\mid\varphi(x)\bigr], $$for those formulas, which are satisfied by at least one object. Now, important thing is that:$$ x=\bigl[x\bigr] $$so we cannot distinguish between any given object and its mereological singleton (so to say), which is the first problem to interpret ZF(C). Defining proper part as x\sqsubset y\iff x\sqsubseteq y\wedge x\neq y we may define mereological atoms (or points, if you prefer the name):$$ \mathrm{Atom}(x)\iff\neg\exists y(y\sqsubset x). $$Now, in case a_1,\ldots,a_n are atoms we can indeed treat \bigl[a_1,\ldots,a_n\bigr] as a counterpart of \{a_1,\ldots,a_n\} (and similarly in case of infinite collections), thus in this case the interpretation suggested by Carl Mummert and mentioned by Andreas Blass:$$ x\in y\iff\mathrm{Atom}(x)\wedge x\sqsubset y, $$works fine. But it does not work for example for:$$ \bigl[\bigl[a_1,\ldots,a_n\bigr],\bigl[b_1,\ldots,b_m\bigr]\bigr]=\bigl[a_1,\ldots,a_n, b_1,\ldots,b_m\bigr], $$since under the interpretation in question for every a_i:$$ a_i\in\bigl[\bigl[a_1,\ldots,a_n\bigr],\bigl[b_1,\ldots,b_m\bigr]\bigr]. $$Thus, as Andreas already pointed to it, there is no way to differentiate between sets of atoms and sets of sets of atoms and so on. Everything is reducible to a mereological set of atoms. (It is worth mentioning here as well that existence of atoms is independent from the axioms of the classical mereology.) To conclude this lengthy post, the crucial distinction between mereological sets and, so to say, standard ones is (I think) hidden in the following fact. The equivalence below is true about sets (with obvious restrictions, but assume that we limit our attention to a domain which is a set):$$ \varphi(x)\iff x\in\{z\mid\varphi(z)\}, $$while its mereological counterpart is usually not true. That is it is the case that:$$ \varphi(x)\longrightarrow x\sqsubseteq\bigl[z\mid\varphi(z)\bigr], $$but is NOT the case that:$$ x\sqsubseteq\bigl[z\mid\varphi(z)\bigr]\longrightarrow \varphi(x). $$It may be interesting (in my opinion) to consider the system of mereology with the implication above taken as an axiom. This could shed some light on the nature of difference between both understandings of sets. As Jeremy Shipley wrote above (in comments) part of is a decent interpretation of subsethood, but not membership. There are still some other points worth mentioning, but this post has already got out of control. 5 Corrected meaning of \bot. I decided to add one more answer (instead of editing the previous one), since it is quite long. This will mainly address the OP question, Andreas Blass answer and Carl Mummert comment about defining sets as sets of atoms (points) in mereology. I hope it will shed some light on mereology and its relation to set theory. In mereology, as it is done in Lesniewskian tradition, it is assumed that part of relation (in symbols: \sqsubseteq) is a partial order (reflexive, antisymmetrical and transitive) and that it satisfies the separation condition (those familiar with forcing will find it very familiar):$$ \neg x\sqsubseteq y\longrightarrow\exists z(z\sqsubseteq x\wedge \neg z\mathrel{\bot} y) $$where z\mathrel{\bot} y\iff\exists y\iff\neg\exists u(u\sqsubseteq z\wedge u\sqsubseteq z) (z and y are compatibleincompatible). The crucial point is a definition of mereological sum (sometimes called fusion as well). The very idea of mereological sum is hidden in the following equivalence: an object x is a mereological sum of the group of S-es if and only if every S is part of x and every part of x is compatible with some S. Notice that it is a consequence of the definition that there cannot be a mereological set of an empty group of objects. Using sets and set theoretical notation we may define the sum of a set X as binary relation in the following way:$$ x\mathrel{\mathrm{Sum}} X\iff \forall y(y\in X\longrightarrow y\sqsubseteq x)\wedge\forall y(y\sqsubseteq x\longrightarrow\exists z(z\in X\wedge z\mathrel{\bot} X\wedge\neg z \mathrel{\bot} y). $$What is usually called classical mereology is a second order system which is obtain by adding the following axiom:$$ \forall X(X\neq\emptyset\longrightarrow\exists x(x\mathrel{\mathrm{Sum}} X). $$Building a first-order system is a little bit more painstaking. To simplify things a bit we may introduce some auxiliary notation:$$ x\mathrel{\mathbf{sum}_y}\varphi(y) $$as an abbreviation of the following formula:$$ \forall y(\varphi(y)\longrightarrow y\sqsubseteq x)\wedge\forall u(u\sqsubseteq x\longrightarrow\exists z(\varphi(z)\wedge \neg z\mathrel{\bot} u)). $$"x\mathrel{\mathbf{sum}_y}\varphi(y)" may be read as x is a mereological sum of all \varphi-ers. From this we can prove for example that: • \forall z(z\mathrel{\mathbf{sum}_y}\text‘z=y\text') • \forall z(z\mathrel{\mathbf{sum}_y}\text‘z\sqsubseteq y\text'). In this setting, mereological sum existence axiom schema can be expressed as:$$ \exists x\varphi(x)\longrightarrow\exists y(y\mathrel{\mathbf{sum}_x}\varphi(x)). $$Since the consequence of the axioms presented is that there can only be one mereological sum of \varphi-ers we can introduce notation (analogous to the set-theoretical abstraction operator):$$ \bigl[x\mid\varphi(x)\bigr], $$for those formulas, which are satisfied by at least one object. Now, important thing is that:$$ x=\bigl[x\bigr] $$so we cannot distinguish between any given object and its mereological singleton (so to say), which is the first problem to interpret ZF(C). Defining proper part as x\sqsubset y\iff x\sqsubseteq y\wedge x\neq y we may define mereological atoms (or points, if you prefer the name):$$ \mathrm{Atom}(x)\iff\neg\exists y(y\sqsubset x). $$Now, in case a_1,\ldots,a_n are atoms we can indeed treat \bigl[a_1,\ldots,a_n\bigr] as a counterpart of \{a_1,\ldots,a_n\} (and similarly in case of infinite collections), thus in this case the interpretation suggested by Carl Mummert and mentioned by Andreas Blass:$$ x\in y\iff\mathrm{Atom}(x)\wedge x\sqsubset y, $$works fine. But it does not work for example for:$$ \bigl[\bigl[a_1,\ldots,a_n\bigr],\bigl[b_1,\ldots,b_m\bigr]\bigr]=\bigl[a_1,\ldots,a_n, b_1,\ldots,b_m\bigr], $$since under the interpretation in question for every a_i:$$ a_i\in\bigl[\bigl[a_1,\ldots,a_n\bigr],\bigl[b_1,\ldots,b_m\bigr]\bigr]. $$Thus, as Andreas already pointed to it, there is no way to differentiate between sets of atoms and sets of sets of atoms and so on. Everything is reducible to a mereological set of atoms. (It is worth mentioning here as well that existence of atoms is independent from the axioms of the classical mereology.) To conclude this lengthy post, the crucial distinction between mereological sets and, so to say, standard ones is (I think) hidden in the following fact. The equivalence below is true about sets (with obvious restrictions, but assume that we limit our attention to a domain which is a set):$$ \varphi(x)\iff x\in\{z\mid\varphi(z)\}, $$while its mereological counterpart is usually not true. That is it is the case that:$$ \varphi(x)\longrightarrow x\sqsubseteq\bigl[z\mid\varphi(z)\bigr], $$but is NOT the case that:$$ x\sqsubseteq\bigl[z\mid\varphi(z)\bigr]\longrightarrow \varphi(x).  It may be interesting (in my opinion) to consider the system of mereology with the implication above taken as an axiom. This could shed some light on the nature of difference between both understandings of sets.