be the representative of what is left of
. In this way we can go on until nothing is left. The successive representatives will form a well-ordered series containing all the members of
. (The above is, of course, only a hint of the general lines of the proof.) This proposition is called "Zermelo's theorem."
The multiplicative axiom is also equivalent to the assumption that of any two cardinals which are not equal, one must be the greater. If the axiom is false, there will be cardinals
and
such that