The arithmetic of first uncountable ordinal number

Question

I think, I know the proof of 1+ω0 = ω0. (ω0 is countable ordinal s.t ω0=[N]). To prove this, I can define a function f: {-1,0,1,2,...}->{0,1,2,...} by f(x)=x+1. But if ω1 is first uncountable ordinal than how can I prove 1+ω1=ω1. Could anyone please give me an idea about this?