reserve X, Y, Z for set, x, y, z for object;

theorem
 X c< Y implies ex x st x in Y & X c= Y \ {x}
proof
 assume
A1: X c< Y;
 then consider x such that
A2: x in Y and
A3: not x in X by Th6;
 take x;
 thus x in Y by A2;
 let y;
 assume
A4: y in X;
  then y <> x by A3;
  then
A5: not y in {x} by TARSKI:def 1;
  X c= Y by A1;
  then y in Y by A4;
 hence thesis by Def5,A5;
end;
