"Minimum" would indicate the elements of the set have a total ordering.

1. Let A be a set with a total order and be a finite set. If |B| = 1, then the minimum element is the only element in B. Assume for |B| = n contains a minimum element p. Then if an element j and , has a minimum element, since or , so either p or j is the minimum element. Since | | = n+1, the induction on the size of the set is complete.