Is $\subseteq$ technically a poset?

Question

I know that for any two sets $A,B$, it holds that $A\subseteq B$ iff every element of $A$ is in $B$, intuitively. I also know that it is reflexive, antisymmetric and transitive.

But, is it technically a poset? In the wiki page of a poset says that the subset relation is defined on the power set of a set, but for that we need some universe set $U$, and then define the subset relation. But clearly, subset relation is defined for all sets, which is a proper class, not a set, and because there is no set of all sets, we cannot define a universe set, so I feel this definition is ill-formed.

So, what is the correct way of defining a subset "relation", If not a poset on the class of sets?

Answer 1

The answer is that you would restrict this to some universe that is a set if you want to get a poset. Otherwise the relation, while it exists, is a proper class and hence does not define a partially ordered set. In most subjects we are only concerned with sets contained in some larger set, hence the collection of all of the ones we care about forms a set. Otherwise we still have a relation that makes sense, and we can still analyze it, but as you said it does not define a poset.

Answer 2

You can still define order relations on a class that is not a set, but it is then not a partially ordered set.

See: https://en.wikipedia.org/wiki/Preordered_class

Given a set $X$, it is true that $(\mathcal{P}(X), \subseteq)$ is a partially ordered set.