Can the sheffer stroke do the work normally done with sets?

It seems to me that the sufficiency-necessity relation is effectively the same as the set-member relation.

Using a concrete examples to make the point:

• A rain drop ⊃ a bit of water
• A rain drop → a bit of water

So if all logical connectives can be built from the Sheffer stroke, and if the material conditional is a logical connective, and if it can perform the work of ⊃, then it would make sense that the Sheffer stroke could perform the work of the ⊃.

But I may have overlooked something. So, can the sheffer stroke do the work normally done with sets?

Thank you

Given a set $U$, the power set of $U$ is a Boolean algebra when we interpret $\land$ as $\cap$ and $\lor$ as $\cup$, negation is complement and implication is indeed $\subseteq$.
So yes, we can work with the Sheffer stroke, it would be exactly the same thing as it is in Boolean algebra, $$A\mid B=\{x\in U:x\notin A\cap B\}$$
So using the Sheffer stroke is possible, but it will inherently introduce a lot of clutter and problems because complement is not a well-defined notion in modern set theory. (whereas complement relative to a particular set $U$ is, but which $U$ are you going to use?)