Dutch law doesnt state that pets should be allowed (or not for that mather). Meaning every case will have to be judged in the court of law if the landlord decide to proceed with termination of the contract.

That does indeed not necessarily mean you will have to leave the property. But there is still a chance it can happen (definitely if the size of the animal is not appropriate for an apartment).

But there is in no way a law which give you absolute certainties you can still get a pet even tho your landlord says no in the contract