Can there be two possible worlds $w_1$ and $w_2$, an object $o$ (set or not) and a set $s$, such that $o\in s$ is true in $w_1$, $o\notin s$ is true in $w_2$ and $s$ is the same set in both worlds?
Another way to ask the question is whether the members of a set can differ between possible worlds or whether a set has all its members necessarily.
I came across an argument in some paper not really related to set theory that seemed to presuppose that sets have their members necessarily. But I was not aware of any such principle so far. I suppose it would already help me, if someone knows where such a principle is discussed and could point me towards that.