It is well known that the class of cardinal numbers is well-ordered. But the proofs that we know are long ones, using Zorn's Theorem and the cumbersome theory of ordinal numbers. In this paper we give a very short proof of this theorem using both the Axiom of Choice and Zorn's Theorem. By the theorem of Bernstein-Cantor we know that the cardinal numbers form an order class. If we prove that every family of cardinal numbers has a first element it will follow that it is totally ordered (if not, a

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... set of two incomparable elements would not have a first element) and indeed, well-ordered. We shall use the notations and terminology of Bourbaki.