Art can't be just design. This would be a better interpretation. But design is part of art. You are using the terminology wrong way around. Have you ever seen someone using a broad term to justify specific term. I can also said Animal =/= Human. But Human is part of animal kingdom. And don't say design is not art DUDE. It is classed in the Faculty of Art in the university.
AND FYI
Arts includes languages, culture, performance, painting, DESIGN, Archeology, Architecture, Anthropology... etc.
And people also said Medicine and Biology should be part of art too. Do you know how and why Leonardo da Vinci can draw such beautiful painting cause he studies human bodies and animal bodies by dissecting them.
And the beauty of art is very objective. One may be beautiful for some but not for the others.