Jay Gatsby did not become wealthy until he was in his adult life. He grew up poor in the Midwestern United States. Then, he learned the workings of the upper class while working on a boat. After serving in World War I, he began to earn money and a place in high society. Gatsby earned his income through the illegal business of bootlegging. Despite being criminal, he was able to become rich quickly, which allowed him to move to New York to find Daisy. His story of rags to riches proves the theme of social class. It is what motivates characters since they are constantly trying to move higher in society.
The characters do not care about social class since they are already at the top of society. They know that their wealth and social connections will keep them in the upper class, so they don't have to worry about their status. Gatsby used his popularity in order to win back Daisy, not because he cared about being in the upper class.
[P1] Gatsby went from poor to rich. [P2] The characters care about being in the upper class. [P3] Therefore, the theme of The Great Gatsby is social class.