reserve W,X,Y,Z for set,
  f,g for Function,
  a,x,y,z for set;
reserve u,v for Element of Tarski-Class(X),
  A,B,C for Ordinal,
  L for Sequence;

theorem Th31:
  A <> {} & A is limit_ordinal implies
  for x holds x in Rank A iff ex B st B in A & x in Rank B
proof
  assume
A1: A <> {} & A is limit_ordinal;
  consider L such that
A2: dom L = A & for B st B in A holds L.B = F(B) from ORDINAL2:sch 2;
A3: Rank A = union rng L by A1,A2,Lm2;
  let x;
  thus x in Rank A implies ex B st B in A & x in Rank B
  proof
    assume x in Rank A;
    then consider Y such that
A4: x in Y and
A5: Y in rng L by A3,TARSKI:def 4;
    consider y being object such that
A6: y in dom L and
A7: Y = L.y by A5,FUNCT_1:def 3;
    reconsider y as Ordinal by A6;
    take y;
    thus thesis by A2,A4,A6,A7;
  end;
  given B such that
A8: B in A and
A9: x in Rank B;
 L.B = Rank B by A2,A8;
then  Rank B in rng L by A2,A8,FUNCT_1:def 3;
  hence thesis by A3,A9,TARSKI:def 4;
end;
