
theorem Th23:
  for X,Y being set holds X c= Y implies X in the_universe_of Y
proof
  let X,Y be set;
A1: Y c= the_transitive-closure_of Y by CLASSES1:52;
  Tarski-Class the_transitive-closure_of Y is_Tarski-Class_of
  the_transitive-closure_of Y by CLASSES1:def 4;
  then
A2: the_transitive-closure_of Y in Tarski-Class the_transitive-closure_of Y
  by CLASSES1:def 3;
  assume X c= Y;
  then X c= the_transitive-closure_of Y by A1;
  hence thesis by A2,CLASSES1:def 1;
end;
