Existence of sets

Question

Just wondering how you could prove whether or not the follow sets exist.

For each of the following predicates $\phi (x)$ prove whether the following sets exist {x : $\phi (x)$}

1: $\forall y(y\in x)$

2: $\forall y(x\in y)$

3: $\exists y(y\in x)$

4: $\exists y(x\in y)$

I have that 3 and 4 dont exist as they imply the existance of the set of all sets. Not sure about 1 and 2 i think they exist but are empty.

Answer 1

1. is the set of all all-sets. Since that no all-set exists (i.e., the class of all sets is proper), $\{x:\phi(x)\}$ is the empty set.
2. (after the question was edited) is the empty set because $x\in y$ cannot hold for all $y$ when it does not even hold for $y=\emptyset$ (nor for $y=x$)
3. is the class of all non-empty sets, which is a proper class as it would have to have itself as element if it were a set.
4. is the class of all sets (and so not a set) because we can always pick $y=\{x\}$.