reserve A,B,a,b,c,d,e,f,g,h for set;

theorem Th4:
  for X be trivial set, R be Relation of X st R is non empty holds
  ex x be object st R = {[x,x]}
proof
  let X be trivial set;
  let R be Relation of X;
  assume R is non empty;
  then consider x be object such that
A1: x in R;
  consider y,z be object such that
A2: x = [y,z] and
A3: y in X and
A4: z in X by A1,RELSET_1:2;
  consider a be object such that
A5: X = {a} by A3,ZFMISC_1:131;
A6: y = a & z = a by A3,A4,A5,TARSKI:def 1;
  R = {[a,a]}
  proof
    thus R c= {[a,a]}
    proof
      let r be object;
      assume r in R;
      then consider y,z be object such that
A7:   r = [y,z] and
A8:   y in X & z in X by RELSET_1:2;
      y = a & z = a by A5,A8,TARSKI:def 1;
      hence thesis by A7,TARSKI:def 1;
    end;
    let z be object;
    assume z in {[a,a]};
    hence thesis by A1,A2,A6,TARSKI:def 1;
  end;
  hence thesis;
end;
