I hope I interpreted this question correctly. Here is my proposed proof.
Consider 24 hours as being the pigeon holes. When 25 customers have arrived at least 2 will have arrived in the same hour. When 49 customers arrive at least 3 will arrive in the same hour. So for 53 customers there will be 5 hours in which at least 3 customers arrive: 19*2+5*3=53.
Each machine does no more than 7 washes in 24 hours. So each machine must take at least about 3 hours to complete a wash (and dry).
In 3 hours, a complete wash cycle, at least 6 customers will have arrived. They will each use an available machine. So 6 machines at least will be in use.
Over the course of the wash cycle at least 6 more customers arrive, but there are only 4 machines available. While the new customers are completing their wash cycles the previous customers will have completed their wash cycles and the machines will become available for more new customers. Until they arrive there will be at least 4 customers in the laundrette.