
theorem Th8:
  for X being set holds
  Tarski-Class the_transitive-closure_of {X} is Grothendieck of X
  proof
    let X be set;
    set R = the_transitive-closure_of {X};
    set T = Tarski-Class R;
A1: union T c= T by CLASSES1:48;
    X in {X} & {X} c= R by CLASSES1:52,TARSKI:def 1;
    then X in R & R in T by CLASSES1:2;
    then X in union T by TARSKI:def 4;
    hence thesis by A1,CLASSES3:def 4;
  end;
