 reserve X, Y for set, A for Ordinal;

theorem
  succRel(omega) = the set of all [n,n+1] where n is Nat
proof
  set S = the set of all [n,n+1] where n is Nat;
  now
    let z be object;
    hereby
      assume A0: z in succRel(omega);
      then consider x,y being object such that
        A0a: z = [x,y] by RELAT_1:def 1;
      reconsider a=x,b=y as set by TARSKI:1;
      [a,b] in succRel(omega) by A0, A0a;
      then A1: a in omega & b in omega & b = succ a by Def1;
      then reconsider a,b as Nat;
      b = succ Segm a by A1, ORDINAL1:def 17
        .= Segm(a+1) by NAT_1:38
        .= a + 1 by ORDINAL1:def 17;
      hence z in S by A0a;
    end;
    assume z in S;
    then consider n being Nat such that
      A2: z = [n,n+1];
    A4: n+1 = Segm(n+1) by ORDINAL1:def 17
      .= succ Segm n by NAT_1:38
      .= succ n by ORDINAL1:def 17;
    n in omega & n+1 in omega by ORDINAL1:def 12;
    hence z in succRel(omega) by A2, A4, Def1;
  end;
  hence thesis by TARSKI:2;
end;
