Stationary distribution in relation to irreducible and pos. recurrence

Question

I have the following statement that I can't reconcile with a theorem I studied during class:

The statement, which is supposed to be false is:

Let $X_n$ be a Markovchain with finite state space and a stationary distribution $\pi$. Then \begin{equation} \pi \text{ is unique stationary distribution} \iff X_n \text{ is irreducible.} \end{equation}

I thought this is a simple application of our following theorem:

$$\text{An irreducible Markovchain is positive recurrent} \iff X_n \text{ has unique stationary distribution } \pi.$$

My "proof" for the false statement would go like this:

"$\Rightarrow$" $\pi$ is unique stationary distribution, thus using theorem, $X_n$ is irreducible.

"$\Leftarrow$" $X_n$ is irreducible and has finite state space, thus is positive recurrent. Again using the theorem, we get $\pi$ is unique.

So where did this "proof" go wrong?

Thank you for your input!

Answer 1

Thanks to C.C, I understand now where my confusion lies, I interpreted the theorem wrongly.

The theorem states:

Let $X_n$ be an irreducible Markovchain: Then $X_n$ is positive recurrent $\iff X_n$ has unique stationary distribution $\pi$. Thus for the direction "Has unique stationary distribution" $\Rightarrow$ "Irreducible", I cannot make the conclusion that it is irreducible, because irreducible is a prerequisite of the theorem. Rather, I could only conclude positive recurrence (if $X_n$ was given to be irreducible).