A new proof of Euclid's theorem by using set-theoretical concept

A prime number is a natural number greater than 1 that has no divisors other than 1 anditself. The fact that there are infinitely many prime numbers (Euclid's theorem) was proven byEuclid around 300 BC, and by 2022, about 200 different proof methods have beendiscovered. In Euclid's proof, a number obtained by adding 1 to the factorial of a prime isused. This paper will demonstrate the existence of new factors in natural numbers below acertain prime factorial, thereby proving that prime numbers are infinite using only set-theoretical concepts. In addition, this paper proves Euclid's theorem without assuming thatthere are a finite number of prime numbers.