Then it is a theorem, as it is tautologically implied by certain axioms imposed on preference relations or choice correspondences, such as completeness, transit

Then it is a theorem, as it is tautologically implied by certain axioms imposed on preference relations or choice correspondences, such as completeness, transitivity and a notion of continuity. Since the whole theory won't fit in one wikipedia article, I suggest you read the first chapters or either a) Mas-Collel, Whinston and Green or b) Kreps, Microeconomic Foundations I (not the old undergrad book from him) This would provide you with the theory to understand this result in a more mathematical correct way.