The statement is indeed a first order statement in standard Set Theory. But it is no wonder you are a bit confused.

Most people think of sets and elements as different things. However, in standard set theory this is not the case.

In standard set theory, everything is a set (there are no "ur-elements", elements that are not sets). The objects of set theory are sets themselves. The primitive relation $\in$ is a relation between sets, not between ur-elements and sets. So, for example, the Axiom of the Power Set in ZFC states that $$\forall x\exists y(\forall z(z\in y \leftrightarrow z\subseteq x))$$ which is a first order statement, because all the things being quantified over are objects in the theory (namely, "sets").

So you need to forget the notion that "elements" are things in sets and sets are things that contain elements. In ZF, everything is a set.

It's a little hard to see the distinction between first and second order statements in ZF precisely because "sets" are the objects, and moreover, given any set, there is a set that contains all the subsets (the power set). In a way, the Axioms are set up precisely to allow you to talk about collections of sets without having to go to second order logic.

To get to second order you need to start talking about "proper classes" or "properties". In ZF, for instance, that's why Comprehension is not a single axiom, but an entire infinite family of axioms. Comprehension essentially says that if $x$ is a set and $P$ is a property, then $\{y\mid y\in x\wedge P(y)\}$ is a set. But trying to quantify over all propositions would be a second order statement. Instead, you have an "Axiom Schema" which says that for each property $P$, you have an axiom that says $$\forall x\exists y\Bigl(z\in y\leftrightarrow\bigl(z\in x\wedge P(z)\bigr)\Bigr).$$ If you try quantifying over "all $P$", then you get a second order statement in ZF.

The statement is indeed a first order statement in standard Set Theory. But it is no wonder you are a bit confused.

Most people think of sets and elements as different things. However, in standard set theory this is not the case.

In standard set theory, everything is a set (there are no "ur-elements", elements that are not sets). The objects of set theory are sets themselves. The primitive relation $\in$ is a relation between sets, not between ur-elements and sets. So, for example, the Axiom of the Power Set in ZFC states that $$\forall x\exists y(\forall z(z\in y \leftrightarrow z\subseteq x))$$ which is a first order statement, because all the things being quantified over are objects in the theory (namely, "sets").

So you need to forget the notion that "elements" are things in sets and sets are things that contain elements. In ZF, everything is a set.

It's a little hard to see the distinction between first and second order statements in ZF precisely because "sets" are the objects, and moreover, given any set, there is a set that contains all the subsets (the power set). In a way, the Axioms are set up precisely to allow you to talk about collections of sets without having to go to second order logic.

In ZF, to get to second order you need to start talking about "proper classes" or "properties". For instance, that's why Comprehension is not a single axiom, but an entire infinite family of axioms. Comprehension essentially says that for every property $P$ and every object $x$ of the theory (i.e., every set $x$), $\{y\mid y\in x\wedge P(y)\}$ is a set. But trying to quantify over all propositions would be a second order statement. Instead, you have an "Axiom Schema" which says that for each property $P$, you have an axiom that says $$\forall x\exists y\Bigl(z\in y\leftrightarrow\bigl(z\in x\wedge P(z)\bigr)\Bigr).$$ If you try quantifying over "all $P$", then you get a second order statement in ZF.

The statement is indeed a first order statement in standard Set Theory. But it is no wonder you are a bit confused.

Most people think of sets and elements as different things. However, in standard set theory this is not the case.

In standard set theory, everything is a set (there are no "ur-elements", elements that are not sets). The objects of set theory are sets themselves. The primitive relation $\in$ is a relation between sets, not between ur-elements and sets. So, for example, the Axiom of the Power Set in ZFC states that $$\forall x\exists y(\forall z(z\in y \leftrightarrow z\subseteq x))$$ which is a first order statement, because all the things being quantified over are objects in the theory (namely, "sets").

So you need to forget the notion that "elements" are things in sets and sets are things that contain elements. In ZF, everything is a set.

It's a little hard to see the distinction between first and second order statements in ZF precisely because "sets" are the objects, and moreover, given any set, there is a set that contains all the subsets (the power set). In a way, the Axioms are set up precisely to allow you to talk about collections of sets without having to go to second order logic.

The statement is indeed a first order statement in standard Set Theory. But it is no wonder you are a bit confused.

Most people think of sets and elements as different things. However, in standard set theory this is not the case.

In standard set theory, everything is a set (there are no "ur-elements", elements that are not sets). The objects of set theory are sets themselves. The primitive relation $\in$ is a relation between sets, not between ur-elements and sets. So, for example, the Axiom of the Power Set in ZFC states that $$\forall x\exists y(\forall z(z\in y \leftrightarrow z\subseteq x))$$ which is a first order statement, because all the things being quantified over are objects in the theory (namely, "sets").

So you need to forget the notion that "elements" are things in sets and sets are things that contain elements. In ZF, everything is a set.

It's a little hard to see the distinction between first and second order statements in ZF precisely because "sets" are the objects, and moreover, given any set, there is a set that contains all the subsets (the power set). In a way, the Axioms are set up precisely to allow you to talk about collections of sets without having to go to second order logic.

To get to second order you need to start talking about "proper classes" or "properties". In ZF, for instance, that's why Comprehension is not a single axiom, but an entire infinite family of axioms. Comprehension essentially says that if $x$ is a set and $P$ is a property, then $\{y\mid y\in x\wedge P(y)\}$ is a set. But trying to quantify over all propositions would be a second order statement. Instead, you have an "Axiom Schema" which says that for each property $P$, you have an axiom that says $$\forall x\exists y\Bigl(z\in y\leftrightarrow\bigl(z\in x\wedge P(z)\bigr)\Bigr).$$ If you try quantifying over "all $P$", then you get a second order statement in ZF.