That's all you need to travel. However the immigration in every country has the right to decline your right to travel and that is the one of the purposes of the immigration at airports and always has been. If you watch the customs programs on TV, you'll see in most cases further proof is required to travel/get in although it's not stated on the ticket, but it is stated by the immigration website if your care to read it, and is part of the conditions of travelling abroad. It has nothing to do with the visa or airline.