I don't think you can prove it except by testing all the numbers that are not in the list.
There has been lots of research into prime numbers, and if one thing stands out, it is that you can not simply derive one prime number from another. So giving someone a list of primes does not actually say anything about numbers that are not in the list.
Fortunately for you, if the numbers that are not in the list are thought to be not prime, this can be tested relatively quickly. All you have to do is find one divisor, which is usually done very quickly for non-primes.
