[adrianratnapala]

It's not the factorial either, that would include composites in the product too.

[schoen]

Euclid didn't use the factorial in his original proof, but it still produces a logically correct argument and it has fewer syllables. Metri causa. :-)

[JadeNB]

Using the factorial is better, I think, even if ahistorical; it avoids the slight unpleasantness of having to show that every non-unit integer is divisible by a prime.