Prime numbers are closed under subtraction

This is false. If we assumed the prime numbers were closed under subtraction, then the difference of any two primes would also get us a prime number.

Take the prime numbers 11 and 7. Their difference, 11 - 7, is 4, but 4 is composite, as it has 2 as a factor other than 1 and 4. All we need is this one counterexample to prove that prime numbers are not closed under subtraction.