# How is the Markov Chain generated by the Metropolis-Hastings Algorithm Ergodic?

I understand that the Markov Chain generated by the Metropolis-Hastings algorithm satisfies the detailed balance condition, thus implying that the chain has a stationary distribution, $\pi(.)$, say.

However, I'm not sure how we can then apply the ergodic theorem to this Markov Chain? The ergodic theorem states that the chain must be positive recurrent, aperiodic and irreducible; but all we know about the Markov Chain generated by the MH algorithm is that it has $\pi(.)$ as its stationary distribution?