How to prove an axiomatic system

All systems are ultimately axiomatic. That is the definition of a system. 

So how might a system of ethics or virtue be demonstrated to be good? 

If we play this game and all the players of the game over time are satisfied (emotionally, motivationally) and wish to continue playing. We know something is good if it continues to work for all the community over time. 

(There are not actually that many games we could all play and want to go on playing!)

In other words: by their fruits you will know them. It seems this is the way of life we are made for and are suited to. 

It may be only later that we can explain the system. The way of life must be lived to be understood. Faith seeks understanding. 

There are natural hierarchies: some things work and some things don't. We need leaders and goals. Observing life, we are able to identify the admirable. We can make stories about it, dramas. And we can distil principles from these stories: general principles based on custom.