 reserve X, Y for set, A for Ordinal;

theorem Th4:
  for B,C being Ordinal holds [B,C] in succRel(A) iff succ B in A & C = succ B
proof
  let B, C be Ordinal;
  hereby
    assume [B,C] in succRel(A);
    then C in A & C = succ B by Def1;
    hence succ B in A & C = succ B;
  end;
  assume A1: succ B in A & C = succ B;
  B in succ B by ORDINAL1:6;
  then B in A by A1, ORDINAL1:10;
  hence thesis by A1, Def1;
end;
