reserve W for Universe,
  H for ZF-formula,
  x,y,z,X for set,
  k for Variable,
  f for Function of VAR,W,
  u,v for Element of W;
reserve F for Function,
  A,B,C for Ordinal,
  a,b,b1,b2,c for Ordinal of W,
  fi for Ordinal-Sequence,
  phi for Ordinal-Sequence of W,
  H for ZF-formula;
reserve psi for Ordinal-Sequence;

theorem Th12:
  On Rank A = A
proof
  thus On Rank A c= A
  proof
    let x be object;
    assume
A1: x in On Rank A;
    then reconsider B = x as Ordinal by ORDINAL1:def 9;
    x in Rank A by A1,ORDINAL1:def 9;
    then the_rank_of B in A by CLASSES1:66;
    hence thesis by CLASSES1:73;
  end;
  On A c= On Rank A by CLASSES1:38,ORDINAL2:9;
  hence thesis by ORDINAL2:8;
end;
