Definition: "A tautology is a propositional formula that is true under any truth assignment to each of the atomic propositions in the domain of propositional function."
Let $p$ be a simple (or atomic) proposition (e.g. "9 is a square root of 81").
I understand that a proposition may be either true or false (but not both true and false at the same time). That is, $p$ may be either true or false, exclusively. Under all possible truth assignments, $p$ is not always true. Therefore, from the definition of tautology, $p$ is not a tautology.
However, suppose I proved $p$ to be true. I am tempted to write $p \iff \top$, but this means "$p$ is a tautology". However, the previous paragraph's conclusions was that "$p$ is not a tautology".
What's going on here?
Using the notation in symbolic logic, how does one write that $p$ is indeed true?
Thanks in advance for your help.