
theorem
  for A be 2-element set, a, b be Element of A st a <> b holds
    IntPrefSpace (A,a,b) is non empty preference-like
  proof
    let A be 2-element set, a, b be Element of A;
    assume
Z1: a <> b;
    set X = IntPrefSpace (A,a,b);
a3: the ToleranceRel of X = {[a, a], [b, b]} by Def4
    .= id A by Lemma4,Z1
    .= id (the carrier of X) by Def4;
a4: the InternalRel of X = {[a, b], [b, a]} by Def4;
    the PrefRel of X = {}(A, A) & the ToleranceRel of X = {[a, a], [b, b]} &
      the InternalRel of X = {[a, b], [b, a]} by Def4; then
    (the PrefRel of X) /\ (the InternalRel of X) = {} &
      (the ToleranceRel of X) /\ (the InternalRel of X) = {} &
      (the PrefRel of X) /\ (the ToleranceRel of X) = {}
        by Z1,Lemma11,XBOOLE_0:def 7; then
A5: the PrefRel of X, the ToleranceRel of X, the InternalRel of X
      are_mutually_disjoint by XBOOLE_0:def 7;
C4: the PrefRel of X = {}(A, A) by Def4;
C6: the ToleranceRel of X = {[a, a], [b, b]} by Def4;
C1: the carrier of X = A by Def4;
C2: the InternalRel of X = {[a, b], [b, a]} by Def4;
D1: A = {a,b} by Z1,Lemma3;
    (the PrefRel of X) \/ (the PrefRel of X)~ \/
      (the ToleranceRel of X) \/
      (the InternalRel of X) = {}(A, A) \/ {}(A, A) \/ {[a, a], [b, b]} \/
        {[a, b], [b, a]} by C2, C4, C6
      .= {[a, a], [a, b], [b, a], [b, b]} by Lemma10
      .= nabla the carrier of X by C1,D1,ZFMISC_1:122;
    hence thesis by Def4,a3,a4,A5,Lemma9,Z1;
  end;
