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 Th14:
  x is ordinal implies union x is epsilon-transitive epsilon-connected
proof
  assume x is ordinal;
  then reconsider A = x as Ordinal;
  thus y in union x implies y c= union x
  proof
    assume y in union x;
    then consider z such that
A1: y in z and
A2: z in x by TARSKI:def 4;
    z in A by A2;
    then reconsider z as Ordinal by Th9;
    z c= A by A2,Def2;
    hence thesis by A1,ZFMISC_1:74;
  end;
  let y,z;
  assume that
A3: y in union x and
A4: z in union x;
  consider X such that
A5: y in X and
A6: X in x by A3,TARSKI:def 4;
A7: X in A by A6;
  consider Y such that
A8: z in Y and
A9: Y in x by A4,TARSKI:def 4;
  reconsider X,Y as Ordinal by A9,A7,Th9;
  z in Y by A8;
  then
A10: z is Ordinal by Th9;
  y in X by A5;
  then y is Ordinal by Th9;
  hence thesis by A10,Th10;
end;
