Major League Baseball Organizations

Spanish help please..??
Is this the appropriate way to translate this from English to Spanish?
Mariano Rivera is a professional baseball player in Major League Baseball for the New York Yankees.
Mariano Rivera es jugador de béisbol profesional en la Liga Nacional de Béisbol para los yanquis de Nueva York.
I don’t know if I should have translated the names Major League Baseball and New York Yankees since those are the names of American organizations. If my translation is wrong, how would I set up the sentence with using those American names, New York Yankees and Major League Baseball in a Spanish sentence?
Everything is virtually correct but one thing, after Mariano River you should add “UN” which makes it..”Mariano River IS……”
Good job
, take care.
7x Major League Baseball All-Star Dave Parker on Health