Pokemon diamond and pearl online battleing???

When you battle someone over wi-fi like someone who lives kilometres away from you and you battle them and he has a ho-oh will ho-oh go into your pokedex or not becuase on the globe wi-fi people ask for ho-oh or mew but i havent seen a ho-oh or mew how do i see them if someone has it and i battle with them over wi-fi will it record it in my poke dex please help thanks!!!|||No It Wont But Im In Wi-Fi Right Now Ive Added You ill Invite Or You? talk much?