Logic as a matter-o-taste

It has been sometime since I have thought about these things. Here is a little argument about null sets that I want to present. Please find flaws.

If A and B are two sets such that 'x' belongs to A implies 'x' belongs to B, then A is a subset of B. The null set is a set that does not contain any element. Let us look at the following statement which is generally held to be true in Set Theory.

Statement S --> The null set is a subset of every set.

The argument used (or is hidden) to support this is as follows. In the first part of the definition of a subset, ("x belongs to A") since finding an x belonging to A, when A is the null set, is impossible - we say that every x belonging to the null set belongs to any other set.

In other words, we are saying that if at all there is an x belonging to the null set, it would belong to any other set. But since there is no such x, we can safely say that the null set is a subset of every other set (including itself).

On the contrary, equally logically, we can argue that the statement S is not a verifiable statement because the definition of subset needs an element x.

Thus, I claim that the accepting statement S as true is just a logical choice. It is not a 'primitive' concept which would be evidently true.

In more mathematical terms, if we make the choice of accepting that S is not verifiable, then the partial order relation "is a subset of" cannot be defined over any collection of sets that has the null set as one of its members.