A prime number is a number that can only be completely divided by $1$ and itself.

Given the digits $1$, $2$, $3$, $4$, and $5$, we are tasked with forming five-digit prime numbers. We know that if the sum of the digits of a number is divisible by $3$, the number itself is also divisible by $3$.

The sum of the given digits is $1+2+3+4+5=15$, which is divisible by $3$.

Therefore, any number formed using these five digits will always be divisible by $3$, meaning it cannot be a prime number.

Thus, no five-digit prime number can be formed using these five digits without repetition.