I don't know what are your "nine basic axioms of numbers".
Do you have the "least number principle"? I.e., if there is a natural number in A, then there is a LEAST natural number in A. If so:
Assume (1) and (2) in the second principle.
Use the first principle to show A=N.
From (1) in the second principe, we have 1 in A.
Suppose k in A. We need to show k+1 in A. So, from the second principle, we need to show that every n less than or equal to k is in A. But we have k in A. So we only need to show that every number less than k is in A. Toward a contradiction, suppose n is the LEAST number less than k that is not in A. So every number less than n is in A. But then, by (2) in the second principle, we have n in A, a contradiction.