A long-standing expectation is that large, dense, and cosmopolitan areas support socioeconomic mixing and exposure between diverse individuals. It has been difficult to assess this hypothesis because past approaches to measuring socioeconomic mixing have relied on static residential housing data rather than real-life exposures between people at work, in places of leisure, and in home neighborhoods. Here we develop a new measure of exposure segregation (ES) that captures the socioeconomic diversity of everyday encounters. Leveraging cell phone mobility data to represent 1.6 billion exposures among 9.6 million people in the United States, we measure exposure segregation across 382 Metropolitan Statistical Areas (MSAs) and 2829 counties. We discover that exposure segregation is 67% higher in the 10 largest Metropolitan Statistical Areas (MSAs) than in small MSAs with fewer than 100,000 residents. This means that, contrary to expectation, residents of large cosmopolitan areas have significantly less exposure to diverse individuals. Second, we find evidence that large cities offer a greater choice of differentiated spaces targeted to specific socioeconomic groups, a dynamic that accounts for this increase in everyday socioeconomic segregation. Third, we discover that this segregation-increasing effect is countered when a city's hubs (e.g. shopping malls) are positioned to bridge diverse neighborhoods and thus attract people of all socioeconomic statuses. Overall, our findings challenge a long-standing conjecture in human geography and urban design, and highlight how built environment can both prevent and facilitate exposure between diverse individuals.