We can notice that the difference between the distances between the source`s` and two other vertices in the queue differs by at most one.

Especially, we know that $d[v] \le d[u] \le d[v] + 1$ for each $u \in Q$.

The reason for this is, that we only add vertices with equal distance or with distance plus one to the queue during each iteration.

Assuming there exists a $u$ in the queue with $d[u] - d[v] > 1$, then $u$ must have beeninsert in the queue via a different vertex $t$ with $d[t] \ge d[u] - 1 > d[v]$.

Assuming there exists a $u$ in the queue with $d[u] - d[v] > 1$, then $u$ must have beeninserted into the queue via a different vertex $t$ with $d[t] \ge d[u] - 1 > d[v]$.

However this is impossible, since Dijkstra's algorithm iterates over the vertices in increasing order.

This means, that the order of the queue looks like this: