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 Th9:
  a in A implies a is Ordinal
proof
  assume
A1: a in A;
   reconsider a as set by TARSKI:1;
A2: a c= A by Def2,A1;
  now
    let y;
    assume
A3: y in a;
    then
A4: y c= A by A2,Def2;
    assume not y c= a;
    then consider b be object such that
A5: b in y & not b in a;
    b in y \ a by A5,XBOOLE_0:def 5;
    then consider z such that
A6: z in y \ a and
    not ex c being object st c in y \ a & c in z by TARSKI:3;
    z in y by A6;
    then z in a or z = a or a in z by A1,A4,Def3;
    hence contradiction by A3,A6,XBOOLE_0:def 5,XREGULAR:7;
  end;
  then
A7: a is epsilon-transitive;
  for y,z st y in a & z in a holds y in z or y = z or z in y by A2,Def3;
  then a is epsilon-connected;
  hence thesis by A7;
end;
