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 Th7:
  for x being epsilon-transitive set, A being Ordinal st x c< A holds x in A
proof
  let x be epsilon-transitive set, A be Ordinal;
  set a = the Element of A \ x;
  assume
A1: x c< A;
  then
A2: x c= A;
  A \ x <> {} by A1,XBOOLE_1:37,60;
  then a in A \ x;
  then consider y such that
A3: y in A \ x and
A4: not ex a being object st a in A \ x & a in y by TARSKI:3;
A5: not y in x by A3,XBOOLE_0:def 5;
  now
    let a be object;
    assume a in x;
    then consider z such that
A6: z = a and
A7: z in x;
    z c= x by A7,Def2;
    then not y in z by A3,XBOOLE_0:def 5;
    hence a in y by A2,A3,A5,A6,A7,Def3;
  end;
  then
A8: x c= y;
A9: y c= A by A3,Def2;
  now
    let a be object;
    assume
A10: a in y;
    then not a in A \ x by A4;
    hence a in x by A9,A10,XBOOLE_0:def 5;
  end;
  then
A11: y c= x;
  y in A by A3;
  hence thesis by A11,A8,XBOOLE_0:def 10;
end;
