(finite case; not sure if this extends to infinite case)

Suppose not; then every person in the population has to be either a leader or a ruler (or neither, which doesn't help). In particular, a leader cannot rule another group, and a ruler cannot lead any group it is in.

Let person E be the ruler of the empty set, and let person F be the leader of the full population. Then someone other than E or F must rule the group (E), as E cannot be a leader, and F cannot be a ruler. Call the person in charge of this group "G"; G rules the group (E).

But now we can iterate: consider the group (E, G). It cannot be led by E, ruled by F, or led by G. Thus, this group must be ruled by some other person H. Now consider the group (E, G, H); it must be ruled by some other person I, and so on...

Eventually, you will run out of other people in the population, and will have no valid person to be in charge of the group - a contradiction!

(finite case; not sure if this extends to infinite case)

Suppose not; then every person in the population has to be either a leader or a ruler (or neither). In particular, a leader cannot rule another group, and a ruler cannot lead any group it is in.

Let person E be the ruler of the empty set, and let person F be the leader of the full population. Then someone other than E or F must rule the group (E), as E cannot be a leader, and F cannot be a ruler. Call the person in charge of this group "G"; G rules the group (E).

But now we can iterate: consider the group (E, G). It cannot be led by E, ruled by F, or led by G. Thus, this group must be ruled by some other person H. Now consider the group (E, G, H); it must be ruled by some other person I, and so on...

Eventually, you will run out of other people in the population, and will have no valid person to be in charge of the group - a contradiction!

(finite case; not sure if this extends to infinite case)

Suppose not; then every person in the population has to be either a leader or a ruler. In particular, a leader cannot rule another group, and a ruler cannot lead any group it is in.

Let person E be the ruler of the empty set, and let person F be the leader of the full population. Then someone other than E or F must rule the group (E), as E cannot be a leader, and F cannot be a ruler. Call the person in charge of this group "G"; G rules the group (E).

But now we can iterate: consider the group (E, G). It cannot be led by E, ruled by F, or led by G. Thus, this group must be ruled by some other person H. Now consider the group (E, G, H); it must be ruled by some other person I, and so on...

Eventually, you will run out of other people in the population, and will have no valid person to be in charge of the group - a contradiction!

(finite case; not sure if this extends to infinite case)

Suppose not; then every person in the population has to be either a leader or a ruler (or neither, which doesn't help). In particular, a leader cannot rule another group, and a ruler cannot lead any group it is in.

Let person E be the ruler of the empty set, and let person F be the leader of the full population. Then someone other than E or F must rule the group (E), as E cannot be a leader, and F cannot be a ruler. Call the person in charge of this group "G"; G rules the group (E).

But now we can iterate: consider the group (E, G). It cannot be led by E, ruled by F, or led by G. Thus, this group must be ruled by some other person H. Now consider the group (E, G, H); it must be ruled by some other person I, and so on...

Eventually, you will run out of other people in the population, and will have no valid person to be in charge of the group - a contradiction!