Theorem 5.4: (The Well-Definedness of Cardinality) Let S be a set. If there are natural numbers m and n such that #(S ) = n and #(S ) = m, then m = n.

Proof: Suppose not. Let S be a set with #(S ) = n and #(S ) = m where n and m are not equal. Then by, Theorem 4.6, the Trichotomy Law for natural numbers, one of them is smaller than the other. Say, after perhaps relabeling, that n < m. Then by Definition 4.6, n is a proper subset of m. Since #(S ) = n there is a one to one correspondence

and since #(S ) = m there is another one to one correspondence

Then by Theorem 3.7,

is a one to one correspondence, and by Theorem 3.4

sets up a one to one correspondence between m and a proper subset, which is impossible by Theorem 5.3.