Prime numbers composed entirely of 1s

I came across the following irresistible and alluring statement on futilitycloset:

If a prime number is made up entirely of 1s (e.g., 11) then the number of its digits is also prime.

First, all numbers that consist of ones (1s) as the only used digit, for example 11, 111, 1111 and 11111111111111111111111111. This means a subset of all numbers possible. If any of those are a prime, the statement says that the number of digits making up that 1-number should also be prime. Or the sum, as they are just 1s.

This I had to test for myself. I do believe the statement, it just nerd sniped me enough to see for myself. So with the following code (this time using isprime() from sympy), I ran a test against the first 10 000 digits long 1-numbers. Yes, that took hours as ten thousand digits means huge numbers.

From the ten thousand tested 1-numbers, only five were prime numbers. The length or sum of those were indeed prime as well.

These five were the only ones (pun intended) I found under 10 000 digits:

11 and (2) is prime

1111111111111111111 and (19) is prime

11111111111111111111111 and (23) is prime

11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 and (317) is prime

11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 and (1031) is prime

So the summed/length of primes under the length of 10 000 digits of just ones adds up to the primes 2, 19, 23, 317 and 1031.