Let be a finite Boolean ring with . Prove that for some .
We proceed by induction on the cardinality of . If , then .
Now suppose that for some , every Boolean ring with identity of cardinality at most is isomorphic to for some . Let be a Boolean ring of cardinality . Now contains an element not equal to 1 or 0 which is idempotent (since is Boolean). By a previous theorem, . Note that if , then , and likewise if , then . Thus and are at most , and we have for some and . Thus , and the result holds by induction.