Tukey's lemma TukeysLemma 2002-12-09 02:40:16 2002-12-09 02:56:00 Theorem % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Each nonempty family of finite character has a maximal element. Here, by a maximal element we \PMlinkescapetext{mean} a maximal element with respect to the inclusion ordering: $A\leq B$ iff $A\subseteq B$. This lemma is equivalent to the axiom of choice.