reserve E,X,Y,x for set;
reserve A,B,C for Subset of E;

theorem
  E <> {} implies for B for x being Element of E st not x in B holds x in B`
proof
  assume
A1: E <> {};
  let B be Subset of E;
  let x be Element of E;
  assume
A2: not x in B;
  x in E by A1,Def1;
  hence thesis by A2,XBOOLE_0:def 5;
end;
