 Notice also how we had to impose a universal quantification on the set of valuations. Without this explicit quantification, the formula would only hold for one specific model, which would not be an appropriate solution. If instead we are only interested in knowing whether there exists some outcome that is stable or, conversely, whether the core is empty, it is sufficient to require the formula above to be an axiom. This would amount to say that no outcome is stable, i.e., that the core is empty.