# Is $\subseteq$ technically a poset?

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?

• I would use the term quasi-poset, but that is personal. I never met that term. Commented Mar 20, 2020 at 11:00

• What if I want to talk about the subset relation in the category $\textbf{Rel}$? Am I allowed to talk about the subset relation as a(transitive,antisymmetric, reflexive) morphism(relation) between sets? Commented Mar 20, 2020 at 11:01
• @Garmekain The collection of relations between two sets $A$ and $B$ is always a subset of the power set of $A\times B$. Commented Mar 20, 2020 at 11:09
Given a set $$X$$, it is true that $$(\mathcal{P}(X), \subseteq)$$ is a partially ordered set.