By the ASoC, . Since , we have . One can show similarly that provided that .
So, suppose that for some set A. For this particular A, is a set by ASoC. Moreover, for any other set A' such that , the set exists by ASoC and is equal to as shown above. So, if at least one set A exists such that , we can introduce the notation . It means: choose some set A such that and form . All such sets for various A's are equal, so the result is well-defined.