Well ordering principle

Well ordering principle

For proofs by Well Ordering principle the general template is to consider
the negation of P(n). And then assume the set has a smallest element
according to WOP lets say m, and if we manage to prove that there is
another element that is less than m that also negates P(n) then we
contradict our assumption that m is the smallest element
My question is that should we be proving some sort of a base case as well
for the above mentioned template. As in proving that for some base case,
P(base) is true?