prove there is no city so that more than 2550 other cities have distance exactly four from it 
A problem and solution are shown below. My question is, why is it that in the solution, for each $y\in A_a,$ there exists some city $d_y$ in $D$ that can only be reached with four flights from $x$ while passing through $y$? From the definition of a substantial subset and the minimality of $A^*$, it seems that there exists $d\in D$ so that $d\not\in S_3(a)$ for any $a\in A_a$. But I can't see why this implies the claim above.


 A: The precise statement that follows from minimality: for each $y \in A_a$ (and in fact this would be true for each $y \in A^*$), there is some city $d_y \in D$ such that

*

*There is a way to get from $x$ to $d_y$ via some path $x - y - b_y - c_y - d_y$.

*There is no path $x - y' - b_y' - c_y' - d_y$ where $y' \in A^* \setminus \{y\}$.

This city $d_y$ is the city that justifies $y$'s inclusion in $A^*$; if for every $d_y \in D$ with a path $x - y - b_y - c_y - d_y$, there were also a path $x - y' - b_y' - c_y' - d_y$ with $y' \in A^* \setminus \{y\}$, then $A^* \setminus \{y\}$ would also have been substantial.
Note that we are not claiming that there is no length-$4$ path starting at $x$ and ending at $d_y$ that does not pass through $y$; there is simply no such path that goes through a vertex of $A^*$ other than $y$.
This is all we need to carry out the rest of the proof. When we show that all $2(m-1)$ cities $b_y, c_y$ with $y \in A_a$ are distinct, we are applying statement 2 to $y' = z$. When we show that $d(a,d_y) \ne 3$, we are applying statement 2 to $y' = a$.
