Search on Conformant Problem: solution for subset of a belief state

Question

I am having trouble understanding the following statement.

I have understood why in a sensorless/conformant problem, if there exists a solution (a sequence of actions) for a belief state $b$, then it is also a solution for any $b'$ that is a subset $b$.

Next claim is what is confusing me.

Then in a standard graph search, we can save time by pruning some branches from the search tree.

Example from the Sensorless Vacuum World problem in Russel Norvig.

When we have a successor $\{1,3,5,7\}$ generated during graph search, we do not add it to the frontier if we have already expanded the belief state $\{5, 7\}$ which is a subset.

I thought those cases are totally unrelated since each belief state may encode a different path from the start state. If $\{5, 7\}$ does give us a solution, we have no guarantee it will work for $\{1, 3, 5, 7\}$. It's also possible $\{1, 3, 5, 7\}$ can lead to a solution through a different path.

I must be missing something.. can someone please explain?