reserve X,Y,Z,X1,X2,X3,X4,X5,X6 for set, x,y for object;
reserve a,b,c for object, X,Y,Z,x,y,z for set;
reserve A,B,C,D for Ordinal;

theorem Th20:
  for X st for a st a in X holds a is Ordinal ex A st X c= A
proof
  let X;
  assume
A1: for a st a in X holds a is Ordinal;
  then
A2: union X is epsilon-transitive epsilon-connected by Th19;
  then reconsider A = succ(union X) as Ordinal;
  take A;
  let a be object;
  assume
A3: a in X;
  then reconsider B = a as Ordinal by A1;
  for b be object holds b in B implies b in union X by A3,TARSKI:def 4;
  then B c= union X;
  hence thesis by A2,Th18;
end;
