Equivalence of Definitions of Prime Number

Theorem
The following definitions of prime number are equivalent:

Definition 1 iff Definition 2
This is proved in Prime Number has 4 Integral Divisors:

Definition 1 iff Definition 3
This is proved in Tau of Prime Number:

Definition 1 iff Definition 4
From these two results:
 * $1$ Divides all Integers
 * Integer Divides Itself

it follows that if $p$ has exactly two positive integer divisors then those are $1$ and $p$.

By the same coin, if the only positive integer divisors of $p$ are $1$ and $p$, then $p$ has exactly two positive integer divisors.

Definition 4 iff Definition 5
Let the only two positive integer divisors of $p$ be $1$ and $p$.

Then the only divisor of $p$ strictly less than $p$ is $1$.

Conversely, let the only divisor of $p$ strictly less than $p$ be $1$

From Integer Divides Itself we also have that $p$ is a divisor of $p$.

From Integer Absolute Value not less than Divisors it follows that any positive integer greater than $p$ is not a divisor of $p$.

Thus the only positive integer divisors of $p$ are $1$ and $p$.