reserve A,B,C for Ordinal,
  X,X1,Y,Y1,Z for set,a,b,b1,b2,x,y,z for object,
  R for Relation,
  f,g,h for Function,
  k,m,n for Nat;
reserve M,N for Cardinal;
reserve S for Sequence;

theorem Th21:
  aleph A = aleph B implies A = B
proof
  assume
A1: aleph A = aleph B;
A2: now
    assume B in A;
    then aleph B in aleph A by Th20;
    hence contradiction by A1;
  end;
  now
    assume A in B;
    then aleph A in aleph B by Th20;
    hence contradiction by A1;
  end;
  hence thesis by A2,ORDINAL1:14;
end;
