In this article we prove step by step the main theorem of Smullyan sets. This main theorem says that each element of a Smullyan set is simultaneously satisfiable in a semantic system of model sets and therefore also, for example, in a semantic system of truth sets as well as in a semantic system according to Henkin. After that, we prove the semantical compactness of a propositional logic system as an example of a utilisation of Smullyan sets and its main (...) theorem. (shrink)