Suppose the integer a is a representative of a nonzero equivalence class in Z_p.
Then a is coprime to p and hence am+pn=1 for some integers m and n.
Projecting to equivalence classes we get [a][m]=1 (mod p). ([z] denotes the equivalence class in Z_p of the integer z.)
I.e. every element of U_p has an inverse.