% Mizar problem: t15_taxonom2,taxonom2,797,23 
fof(t15_taxonom2,conjecture,(
    ! [A,B] : 
      ( ( v2_abian(B,k1_zfmisc_1(k1_zfmisc_1(A)))
        & v5_taxonom2(B,A)
        & m1_taxonom2(B,A) )
     => ( v6_taxonom2(B,A)
       => ( r2_hidden(k1_xboole_0,B)
          | ! [C] : 
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => ! [D] : 
                  ( ( v4_taxonom2(D)
                    & m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(A))) )
                 => ( ( r2_hidden(C,D)
                      & r1_tarski(D,B)
                      & ! [E] : 
                          ( ( v4_taxonom2(E)
                            & m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(A))) )
                         => ( ( r2_hidden(C,E)
                              & r1_tarski(E,B)
                              & r1_tarski(k5_setfam_1(A,D),k5_setfam_1(A,E)) )
                           => k5_setfam_1(A,D) = k5_setfam_1(A,E) ) ) )
                   => m1_eqrel_1(D,A) ) ) ) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,cc1_eqrel_1,cc2_eqrel_1,commutativity_k2_xboole_0,d10_xboole_0,d1_tarski,d1_xboole_0,d3_tarski,d3_taxonom2,d4_tarski,d4_taxonom2,d5_taxonom2,d6_eqrel_1,d6_taxonom2,d7_taxonom2,d9_ordinal1,d9_xboole_0,dt_k1_setfam_1,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_xboole_0,dt_k3_tarski,dt_k5_setfam_1,dt_m1_eqrel_1,dt_m1_subset_1,dt_m1_taxonom2,existence_m1_eqrel_1,existence_m1_subset_1,existence_m1_taxonom2,fc1_subset_1,fc1_xboole_0,fc2_subset_1,fc2_xboole_0,fc3_xboole_0,idempotence_k2_xboole_0,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,rc3_taxonom2,rc4_taxonom2,redefinition_k5_setfam_1,reflexivity_r1_tarski,reflexivity_r3_xboole_0,s1_xboole_0__e5_31_1_1_1__taxonom2,symmetry_r1_xboole_0,symmetry_r3_xboole_0,t12_taxonom2,t1_boole,t1_subset,t1_xboole_1,t1_zfmisc_1,t2_subset,t39_zfmisc_1,t3_subset,t3_xboole_0,t3_xboole_1,t4_abian,t4_setfam_1,t4_subset,t5_subset,t6_boole,t7_boole,t7_xboole_1,t8_boole,t8_xboole_1,t95_zfmisc_1]),
    [file(taxonom2,t15_taxonom2)]).

fof(antisymmetry_r2_hidden,axiom,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => ~ r2_hidden(B,A) ) ),
    file(hidden,r2_hidden),
    []).

fof(cc1_eqrel_1,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ~ v1_xboole_0(B) ) ) ),
    file(eqrel_1,cc1_eqrel_1),
    []).

fof(cc2_eqrel_1,axiom,(
    ! [A,B] : 
      ( m1_eqrel_1(B,A)
     => v1_setfam_1(B) ) ),
    file(eqrel_1,cc2_eqrel_1),
    []).

fof(commutativity_k2_xboole_0,axiom,(
    ! [A,B] : k2_xboole_0(A,B) = k2_xboole_0(B,A) ),
    file(xboole_0,k2_xboole_0),
    []).

fof(d10_xboole_0,axiom,(
    ! [A,B] : 
      ( A = B
    <=> ( r1_tarski(A,B)
        & r1_tarski(B,A) ) ) ),
    file(xboole_0,d10_xboole_0),
    []).

fof(d1_tarski,axiom,(
    ! [A,B] : 
      ( B = k1_tarski(A)
    <=> ! [C] : 
          ( r2_hidden(C,B)
        <=> C = A ) ) ),
    file(tarski,d1_tarski),
    []).

fof(d1_xboole_0,axiom,(
    ! [A] : 
      ( A = k1_xboole_0
    <=> ! [B] : ~ r2_hidden(B,A) ) ),
    file(xboole_0,d1_xboole_0),
    []).

fof(d3_tarski,axiom,(
    ! [A,B] : 
      ( r1_tarski(A,B)
    <=> ! [C] : 
          ( r2_hidden(C,A)
         => r2_hidden(C,B) ) ) ),
    file(tarski,d3_tarski),
    []).

fof(d3_taxonom2,axiom,(
    ! [A] : 
      ( v3_taxonom2(A)
    <=> ! [B,C] : ~ ( r2_hidden(B,A)
          & r2_hidden(C,A)
          & ~ r1_tarski(B,C)
          & ~ r1_tarski(C,B)
          & ~ r1_xboole_0(B,C) ) ) ),
    file(taxonom2,d3_taxonom2),
    []).

fof(d4_tarski,axiom,(
    ! [A,B] : 
      ( B = k3_tarski(A)
    <=> ! [C] : 
          ( r2_hidden(C,B)
        <=> ? [D] : 
              ( r2_hidden(C,D)
              & r2_hidden(D,A) ) ) ) ),
    file(tarski,d4_tarski),
    []).

fof(d4_taxonom2,axiom,(
    ! [A,B] : 
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ( m1_taxonom2(B,A)
      <=> v3_taxonom2(B) ) ) ),
    file(taxonom2,d4_taxonom2),
    []).

fof(d5_taxonom2,axiom,(
    ! [A] : 
      ( v4_taxonom2(A)
    <=> ! [B,C] : 
          ( ( r2_hidden(B,A)
            & r2_hidden(C,A) )
         => ( B = C
            | r1_xboole_0(B,C) ) ) ) ),
    file(taxonom2,d5_taxonom2),
    []).

fof(d6_eqrel_1,axiom,(
    ! [A,B] : 
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ( ( A != k1_xboole_0
         => ( m1_eqrel_1(B,A)
          <=> ( k5_setfam_1(A,B) = A
              & ! [C] : 
                  ( m1_subset_1(C,k1_zfmisc_1(A))
                 => ( r2_hidden(C,B)
                   => ( C != k1_xboole_0
                      & ! [D] : 
                          ( m1_subset_1(D,k1_zfmisc_1(A))
                         => ~ ( r2_hidden(D,B)
                              & C != D
                              & ~ r1_xboole_0(C,D) ) ) ) ) ) ) ) )
        & ( A = k1_xboole_0
         => ( m1_eqrel_1(B,A)
          <=> B = k1_xboole_0 ) ) ) ) ),
    file(eqrel_1,d6_eqrel_1),
    []).

fof(d6_taxonom2,axiom,(
    ! [A,B] : 
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ( v5_taxonom2(B,A)
      <=> ! [C] : 
            ( m1_subset_1(C,k1_zfmisc_1(A))
           => ( r2_hidden(C,B)
             => ! [D] : 
                  ( m1_subset_1(D,A)
                 => ~ ( ~ r2_hidden(D,C)
                      & ! [E] : 
                          ( m1_subset_1(E,k1_zfmisc_1(A))
                         => ~ ( r2_hidden(D,E)
                              & r2_hidden(E,B)
                              & r1_xboole_0(C,E) ) ) ) ) ) ) ) ) ),
    file(taxonom2,d6_taxonom2),
    []).

fof(d7_taxonom2,axiom,(
    ! [A,B] : 
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ( v6_taxonom2(B,A)
      <=> ! [C] : ~ ( C != k1_xboole_0
            & r1_tarski(C,B)
            & v6_ordinal1(C)
            & ! [D] : ~ ( r2_hidden(D,B)
                & r1_tarski(D,k1_setfam_1(C)) ) ) ) ) ),
    file(taxonom2,d7_taxonom2),
    []).

fof(d9_ordinal1,axiom,(
    ! [A] : 
      ( v6_ordinal1(A)
    <=> ! [B,C] : 
          ( ( r2_hidden(B,A)
            & r2_hidden(C,A) )
         => r3_xboole_0(B,C) ) ) ),
    file(ordinal1,d9_ordinal1),
    []).

fof(d9_xboole_0,axiom,(
    ! [A,B] : 
      ( r3_xboole_0(A,B)
    <=> ( r1_tarski(A,B)
        | r1_tarski(B,A) ) ) ),
    file(xboole_0,d9_xboole_0),
    []).

fof(dt_k1_setfam_1,axiom,(
    $true ),
    file(setfam_1,k1_setfam_1),
    []).

fof(dt_k1_tarski,axiom,(
    $true ),
    file(tarski,k1_tarski),
    []).

fof(dt_k1_xboole_0,axiom,(
    $true ),
    file(xboole_0,k1_xboole_0),
    []).

fof(dt_k1_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k1_zfmisc_1),
    []).

fof(dt_k2_xboole_0,axiom,(
    $true ),
    file(xboole_0,k2_xboole_0),
    []).

fof(dt_k3_tarski,axiom,(
    $true ),
    file(tarski,k3_tarski),
    []).

fof(dt_k5_setfam_1,axiom,(
    ! [A,B] : 
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => m1_subset_1(k5_setfam_1(A,B),k1_zfmisc_1(A)) ) ),
    file(setfam_1,k5_setfam_1),
    []).

fof(dt_m1_eqrel_1,axiom,(
    ! [A,B] : 
      ( m1_eqrel_1(B,A)
     => m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) ),
    file(eqrel_1,m1_eqrel_1),
    []).

fof(dt_m1_subset_1,axiom,(
    $true ),
    file(subset_1,m1_subset_1),
    []).

fof(dt_m1_taxonom2,axiom,(
    ! [A,B] : 
      ( m1_taxonom2(B,A)
     => m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) ),
    file(taxonom2,m1_taxonom2),
    []).

fof(existence_m1_eqrel_1,axiom,(
    ! [A] : 
    ? [B] : m1_eqrel_1(B,A) ),
    file(eqrel_1,m1_eqrel_1),
    []).

fof(existence_m1_subset_1,axiom,(
    ! [A] : 
    ? [B] : m1_subset_1(B,A) ),
    file(subset_1,m1_subset_1),
    []).

fof(existence_m1_taxonom2,axiom,(
    ! [A] : 
    ? [B] : m1_taxonom2(B,A) ),
    file(taxonom2,m1_taxonom2),
    []).

fof(fc1_subset_1,axiom,(
    ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ),
    file(subset_1,fc1_subset_1),
    []).

fof(fc1_xboole_0,axiom,(
    v1_xboole_0(k1_xboole_0) ),
    file(xboole_0,fc1_xboole_0),
    []).

fof(fc2_subset_1,axiom,(
    ! [A] : ~ v1_xboole_0(k1_tarski(A)) ),
    file(subset_1,fc2_subset_1),
    []).

fof(fc2_xboole_0,axiom,(
    ! [A,B] : 
      ( ~ v1_xboole_0(A)
     => ~ v1_xboole_0(k2_xboole_0(A,B)) ) ),
    file(xboole_0,fc2_xboole_0),
    []).

fof(fc3_xboole_0,axiom,(
    ! [A,B] : 
      ( ~ v1_xboole_0(A)
     => ~ v1_xboole_0(k2_xboole_0(B,A)) ) ),
    file(xboole_0,fc3_xboole_0),
    []).

fof(idempotence_k2_xboole_0,axiom,(
    ! [A,B] : k2_xboole_0(A,A) = A ),
    file(xboole_0,k2_xboole_0),
    []).

fof(rc1_subset_1,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ? [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
          & ~ v1_xboole_0(B) ) ) ),
    file(subset_1,rc1_subset_1),
    []).

fof(rc1_xboole_0,axiom,(
    ? [A] : v1_xboole_0(A) ),
    file(xboole_0,rc1_xboole_0),
    []).

fof(rc2_subset_1,axiom,(
    ! [A] : 
    ? [B] : 
      ( m1_subset_1(B,k1_zfmisc_1(A))
      & v1_xboole_0(B) ) ),
    file(subset_1,rc2_subset_1),
    []).

fof(rc2_xboole_0,axiom,(
    ? [A] : ~ v1_xboole_0(A) ),
    file(xboole_0,rc2_xboole_0),
    []).

fof(rc3_taxonom2,axiom,(
    ! [A] : 
    ? [B] : 
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
      & v4_taxonom2(B) ) ),
    file(taxonom2,rc3_taxonom2),
    []).

fof(rc4_taxonom2,axiom,(
    ! [A] : 
    ? [B] : 
      ( m1_taxonom2(B,A)
      & v2_abian(B,k1_zfmisc_1(k1_zfmisc_1(A)))
      & v5_taxonom2(B,A) ) ),
    file(taxonom2,rc4_taxonom2),
    []).

fof(redefinition_k5_setfam_1,axiom,(
    ! [A,B] : 
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => k5_setfam_1(A,B) = k3_tarski(B) ) ),
    file(setfam_1,k5_setfam_1),
    []).

fof(reflexivity_r1_tarski,axiom,(
    ! [A,B] : r1_tarski(A,A) ),
    file(tarski,r1_tarski),
    []).

fof(reflexivity_r3_xboole_0,axiom,(
    ! [A,B] : r3_xboole_0(A,A) ),
    file(xboole_0,r3_xboole_0),
    []).

fof(s1_xboole_0__e5_31_1_1_1__taxonom2,axiom,(
    ! [A,B,C] : 
      ( ( v2_abian(B,k1_zfmisc_1(k1_zfmisc_1(A)))
        & v5_taxonom2(B,A)
        & m1_taxonom2(B,A) )
     => ? [D] : 
        ! [E] : 
          ( r2_hidden(E,D)
        <=> ( r2_hidden(E,B)
            & r2_hidden(C,E) ) ) ) ),
    file(taxonom2,s1_xboole_0__e5_31_1_1_1__taxonom2),
    []).

fof(symmetry_r1_xboole_0,axiom,(
    ! [A,B] : 
      ( r1_xboole_0(A,B)
     => r1_xboole_0(B,A) ) ),
    file(xboole_0,r1_xboole_0),
    []).

fof(symmetry_r3_xboole_0,axiom,(
    ! [A,B] : 
      ( r3_xboole_0(A,B)
     => r3_xboole_0(B,A) ) ),
    file(xboole_0,r3_xboole_0),
    []).

fof(t12_taxonom2,axiom,(
    ! [A,B] : 
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] : 
          ( ( v4_taxonom2(C)
            & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
         => ( ! [D] : 
                ( r2_hidden(D,C)
               => ( r1_xboole_0(B,D)
                  & A != k1_xboole_0 ) )
           => ( v4_taxonom2(k2_xboole_0(C,k1_tarski(B)))
              & m1_subset_1(k2_xboole_0(C,k1_tarski(B)),k1_zfmisc_1(k1_zfmisc_1(A)))
              & ~ ( B != k1_xboole_0
                  & k3_tarski(k2_xboole_0(C,k1_tarski(B))) = k5_setfam_1(A,C) ) ) ) ) ) ),
    file(taxonom2,t12_taxonom2),
    []).

fof(t1_boole,axiom,(
    ! [A] : k2_xboole_0(A,k1_xboole_0) = A ),
    file(boole,t1_boole),
    []).

fof(t1_subset,axiom,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => m1_subset_1(A,B) ) ),
    file(subset,t1_subset),
    []).

fof(t1_xboole_1,axiom,(
    ! [A,B,C] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(B,C) )
     => r1_tarski(A,C) ) ),
    file(xboole_1,t1_xboole_1),
    []).

fof(t1_zfmisc_1,axiom,(
    k1_zfmisc_1(k1_xboole_0) = k1_tarski(k1_xboole_0) ),
    file(zfmisc_1,t1_zfmisc_1),
    []).

fof(t2_subset,axiom,(
    ! [A,B] : 
      ( m1_subset_1(A,B)
     => ( v1_xboole_0(B)
        | r2_hidden(A,B) ) ) ),
    file(subset,t2_subset),
    []).

fof(t39_zfmisc_1,axiom,(
    ! [A,B] : 
      ( r1_tarski(A,k1_tarski(B))
    <=> ( A = k1_xboole_0
        | A = k1_tarski(B) ) ) ),
    file(zfmisc_1,t39_zfmisc_1),
    []).

fof(t3_subset,axiom,(
    ! [A,B] : 
      ( m1_subset_1(A,k1_zfmisc_1(B))
    <=> r1_tarski(A,B) ) ),
    file(subset,t3_subset),
    []).

fof(t3_xboole_0,axiom,(
    ! [A,B] : 
      ( ~ ( ~ r1_xboole_0(A,B)
          & ! [C] : ~ ( r2_hidden(C,A)
              & r2_hidden(C,B) ) )
      & ~ ( ? [C] : 
              ( r2_hidden(C,A)
              & r2_hidden(C,B) )
          & r1_xboole_0(A,B) ) ) ),
    file(xboole_0,t3_xboole_0),
    []).

fof(t3_xboole_1,axiom,(
    ! [A] : 
      ( r1_tarski(A,k1_xboole_0)
     => A = k1_xboole_0 ) ),
    file(xboole_1,t3_xboole_1),
    []).

fof(t4_abian,axiom,(
    ! [A,B] : 
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ( v2_abian(B,k1_zfmisc_1(k1_zfmisc_1(A)))
      <=> k5_setfam_1(A,B) = A ) ) ),
    file(abian,t4_abian),
    []).

fof(t4_setfam_1,axiom,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => r1_tarski(k1_setfam_1(B),A) ) ),
    file(setfam_1,t4_setfam_1),
    []).

fof(t4_subset,axiom,(
    ! [A,B,C] : 
      ( ( r2_hidden(A,B)
        & m1_subset_1(B,k1_zfmisc_1(C)) )
     => m1_subset_1(A,C) ) ),
    file(subset,t4_subset),
    []).

fof(t5_subset,axiom,(
    ! [A,B,C] : ~ ( r2_hidden(A,B)
      & m1_subset_1(B,k1_zfmisc_1(C))
      & v1_xboole_0(C) ) ),
    file(subset,t5_subset),
    []).

fof(t6_boole,axiom,(
    ! [A] : 
      ( v1_xboole_0(A)
     => A = k1_xboole_0 ) ),
    file(boole,t6_boole),
    []).

fof(t7_boole,axiom,(
    ! [A,B] : ~ ( r2_hidden(A,B)
      & v1_xboole_0(B) ) ),
    file(boole,t7_boole),
    []).

fof(t7_xboole_1,axiom,(
    ! [A,B] : r1_tarski(A,k2_xboole_0(A,B)) ),
    file(xboole_1,t7_xboole_1),
    []).

fof(t8_boole,axiom,(
    ! [A,B] : ~ ( v1_xboole_0(A)
      & A != B
      & v1_xboole_0(B) ) ),
    file(boole,t8_boole),
    []).

fof(t8_xboole_1,axiom,(
    ! [A,B,C] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(C,B) )
     => r1_tarski(k2_xboole_0(A,C),B) ) ),
    file(xboole_1,t8_xboole_1),
    []).

fof(t95_zfmisc_1,axiom,(
    ! [A,B] : 
      ( r1_tarski(A,B)
     => r1_tarski(k3_tarski(A),k3_tarski(B)) ) ),
    file(zfmisc_1,t95_zfmisc_1),
    []).