Question about Markov chain

Question

We know that if $\{X_n\}$ is a Markov chain, then $X_{n+1}$ is independent with the past states $X_0,\ldots,X_{n-1}$ given current state $X_n$, that is $$P\{X_{n+1}=j|X_0=i_0,\ldots,X_n=i\}=P\{X_{n+1}=j|X_n=i\}$$ What if we give several possible current states, do we still have $$P\{X_{n+1}=j|X_0=i_0,\ldots,X_n=i\ \text{or}\ i'\}=P\{X_{n+1}=j|X_n=i\ \text{or}\ i'\}$$ Intuitively it is correct, but I wonder how to prove it using the original definition.

Answer 1

This is incorrect. For a counterexample, assume that the state space has size $2$ then, if $i\ne i'$, $P(X_2=j\mid X_0=x_0,X_1\in\{i,i'\})=P(X_2=j\mid X_0=x_0)$ and $P(X_2=j\mid X_1\in\{i,i'\})=P(X_2=j)$ but, in general, $X_0$ and $X_2$ are not independent.