Prime Between nn and n!n!

Between 33 and 3!3!, there is 55, which a prime.

Between 44 and 4!4!, there is 55, which a prime.

Between 55 and 5!5!, there is 77, which is a prime.

True or False?

For every positive integer n3n \geq 3, there is always at least one prime pp with n<p<n!n<p<n!.

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