If we consider two subcommittees, they have exactly one leader in common, therefore, together they have exactly 5 members. Hence there are at least 5 people in the committee. Denote them by $A = \{1, 2, 3, 4, 5\}$. However it's impossible to choose leaders for another subcommittee out of them. Therefore, the committee must have at least 6 members. Here's an example of four subcommittees and their leaders:
$$A = \{1, 2, 3, 4, 5, 6\} \rightarrow \{1, 2, 3\}; \{3, 4, 5\}; \{1, 5, 6\}; \{2, 4, 6\}.$$