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 Th19:
  for X st for a st a in X holds a is Ordinal
   holds union X is epsilon-transitive epsilon-connected
proof
  let X such that
A1: for a st a in X holds a is Ordinal;
  thus union X is epsilon-transitive
  proof
    let x;
    assume x in union X;
    then consider Y such that
A2: x in Y and
A3: Y in X by TARSKI:def 4;
    Y is Ordinal by A1,A3;
    then
A4: x c= Y by A2,Def2;
    let a be object;
    assume a in x;
    hence thesis by A3,A4,TARSKI:def 4;
  end;
  let x,y;
  assume that
A5: x in union X and
A6: y in union X;
  consider Z such that
A7: y in Z and
A8: Z in X by A6,TARSKI:def 4;
  Z is Ordinal by A1,A8;
  then
A9: y is Ordinal by A7,Th9;
  consider Y such that
A10: x in Y and
A11: Y in X by A5,TARSKI:def 4;
  Y is Ordinal by A1,A11;
  then x is Ordinal by A10,Th9;
  hence thesis by A9,Th10;
end;
