reserve k,m,n for Nat,
  a, b, c for object,
  x, y, X, Y, Z for set,
  D for non empty set;
reserve p, q, r, s, t, u, v for FinSequence;
reserve P, Q, R, P1, P2, Q1, Q2, R1, R2 for FinSequence-membered set;
reserve S, T for non empty FinSequence-membered set;
reserve A for Function of P, NAT;
reserve U, V, W for Subset of P*;
reserve k,l,m,n,i,j for Nat,
  a, b, c for object,
  x, y, z, X, Y, Z for set,
  D, D1, D2 for non empty set;
reserve p, q, r, s, t, u, v for FinSequence;
reserve P, Q, R for FinSequence-membered set;
reserve B, C for antichain;
reserve S, T for Polish-language;

theorem Th46:
  for B holds {} in B implies B = {{}}
proof
  let B;
  assume A1: {} in B;
  for a st a in B holds a = {}
  proof
    let a;
    assume A3: a in B;
    then reconsider a as FinSequence;
    {}^a in B by A3, FINSEQ_1:34;
    hence thesis by A1, Def16;
  end;
  then for a holds a in B iff a = {} by A1;
  hence thesis by TARSKI:def 1;
end;
