% Mizar problem: t31_relset_2,relset_2,793,76 
fof(t31_relset_2,conjecture,(
    ! [A,B] : 
      ( ~ v1_xboole_0(B)
     => ! [C] : 
          ( m2_relset_1(C,A,B)
         => ! [D] : 
              ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(A)))
             => ! [E] : 
                  ( m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(B)))
                 => ( E = a_4_2_relset_2(A,B,C,D)
                   => k8_relset_2(A,B,k5_setfam_1(A,D),C) = k8_setfam_1(B,E) ) ) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,cc1_relat_1,cc1_relset_1,d10_setfam_1,d10_xboole_0,d1_setfam_1,d1_tarski,d3_tarski,d4_relset_2,d4_tarski,dt_k10_relset_1,dt_k1_setfam_1,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k3_pua2mss1,dt_k3_tarski,dt_k4_relset_2,dt_k5_relset_2,dt_k5_setfam_1,dt_k6_relset_2,dt_k6_setfam_1,dt_k7_relset_2,dt_k8_relset_2,dt_k8_setfam_1,dt_k9_relat_1,dt_m1_eqrel_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,existence_m1_eqrel_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,fc12_relat_1,fc1_subset_1,fc1_sysrel,fc2_subset_1,fc4_relat_1,fc4_subset_1,fraenkel_a_4_2_relset_2,rc1_relat_1,rc1_subset_1,rc2_partfun1,rc2_relat_1,rc2_subset_1,rc3_relat_1,redefinition_k10_relset_1,redefinition_k4_relset_2,redefinition_k5_setfam_1,redefinition_k6_relset_2,redefinition_k6_setfam_1,redefinition_k8_relset_2,redefinition_m2_relset_1,reflexivity_r1_tarski,t11_setfam_1,t1_subset,t23_relset_2,t24_relset_2,t25_relset_2,t29_relset_2,t2_subset,t2_tarski,t3_subset,t4_subset,t5_subset,t6_boole,t7_boole,t7_setfam_1,t8_boole,t91_orders_1]),
    [file(relset_2,t31_relset_2)]).

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

fof(cc1_relat_1,axiom,(
    ! [A] : 
      ( v1_xboole_0(A)
     => v1_relat_1(A) ) ),
    file(relat_1,cc1_relat_1),
    []).

fof(cc1_relset_1,axiom,(
    ! [A,B,C] : 
      ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
     => v1_relat_1(C) ) ),
    file(relset_1,cc1_relset_1),
    []).

fof(d10_setfam_1,axiom,(
    ! [A,B] : 
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ( ( B != k1_xboole_0
         => k8_setfam_1(A,B) = k6_setfam_1(A,B) )
        & ( B = k1_xboole_0
         => k8_setfam_1(A,B) = A ) ) ) ),
    file(setfam_1,d10_setfam_1),
    []).

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_setfam_1,axiom,(
    ! [A,B] : 
      ( ( A != k1_xboole_0
       => ( B = k1_setfam_1(A)
        <=> ! [C] : 
              ( r2_hidden(C,B)
            <=> ! [D] : 
                  ( r2_hidden(D,A)
                 => r2_hidden(C,D) ) ) ) )
      & ( A = k1_xboole_0
       => ( B = k1_setfam_1(A)
        <=> B = k1_xboole_0 ) ) ) ),
    file(setfam_1,d1_setfam_1),
    []).

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

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

fof(d4_relset_2,axiom,(
    ! [A,B,C] : 
      ( m1_subset_1(C,k1_zfmisc_1(A))
     => ! [D] : 
          ( m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(A,B)))
         => k7_relset_2(A,B,C,D) = k8_setfam_1(B,k4_relset_2(k1_zfmisc_1(A),B,k6_relset_2(B,A,D),k3_pua2mss1(C))) ) ) ),
    file(relset_2,d4_relset_2),
    []).

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(dt_k10_relset_1,axiom,(
    ! [A,B,C,D] : 
      ( m1_relset_1(C,A,B)
     => m1_subset_1(k10_relset_1(A,B,C,D),k1_zfmisc_1(B)) ) ),
    file(relset_1,k10_relset_1),
    []).

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_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k2_zfmisc_1),
    []).

fof(dt_k3_pua2mss1,axiom,(
    ! [A] : m1_eqrel_1(k3_pua2mss1(A),A) ),
    file(pua2mss1,k3_pua2mss1),
    []).

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

fof(dt_k4_relset_2,axiom,(
    ! [A,B,C,D] : 
      ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,k1_zfmisc_1(B))))
     => m1_subset_1(k4_relset_2(A,B,C,D),k1_zfmisc_1(k1_zfmisc_1(B))) ) ),
    file(relset_2,k4_relset_2),
    []).

fof(dt_k5_relset_2,axiom,(
    ! [A,B] : 
      ( v1_relat_1(B)
     => ( v1_relat_1(k5_relset_2(A,B))
        & v1_funct_1(k5_relset_2(A,B)) ) ) ),
    file(relset_2,k5_relset_2),
    []).

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_k6_relset_2,axiom,(
    ! [A,B,C] : 
      ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(B,A)))
     => ( v1_funct_1(k6_relset_2(A,B,C))
        & v1_funct_2(k6_relset_2(A,B,C),k1_zfmisc_1(B),k1_zfmisc_1(A))
        & m2_relset_1(k6_relset_2(A,B,C),k1_zfmisc_1(B),k1_zfmisc_1(A)) ) ) ),
    file(relset_2,k6_relset_2),
    []).

fof(dt_k6_setfam_1,axiom,(
    ! [A,B] : 
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => m1_subset_1(k6_setfam_1(A,B),k1_zfmisc_1(A)) ) ),
    file(setfam_1,k6_setfam_1),
    []).

fof(dt_k7_relset_2,axiom,(
    $true ),
    file(relset_2,k7_relset_2),
    []).

fof(dt_k8_relset_2,axiom,(
    ! [A,B,C,D] : 
      ( ( m1_subset_1(C,k1_zfmisc_1(A))
        & m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(A,B))) )
     => m1_subset_1(k8_relset_2(A,B,C,D),k1_zfmisc_1(B)) ) ),
    file(relset_2,k8_relset_2),
    []).

fof(dt_k8_setfam_1,axiom,(
    ! [A,B] : 
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => m1_subset_1(k8_setfam_1(A,B),k1_zfmisc_1(A)) ) ),
    file(setfam_1,k8_setfam_1),
    []).

fof(dt_k9_relat_1,axiom,(
    $true ),
    file(relat_1,k9_relat_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_relset_1,axiom,(
    $true ),
    file(relset_1,m1_relset_1),
    []).

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

fof(dt_m2_relset_1,axiom,(
    ! [A,B,C] : 
      ( m2_relset_1(C,A,B)
     => m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ),
    file(relset_1,m2_relset_1),
    []).

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

fof(existence_m1_relset_1,axiom,(
    ! [A,B] : 
    ? [C] : m1_relset_1(C,A,B) ),
    file(relset_1,m1_relset_1),
    []).

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

fof(existence_m2_relset_1,axiom,(
    ! [A,B] : 
    ? [C] : m2_relset_1(C,A,B) ),
    file(relset_1,m2_relset_1),
    []).

fof(fc12_relat_1,axiom,
    ( v1_xboole_0(k1_xboole_0)
    & v1_relat_1(k1_xboole_0)
    & v3_relat_1(k1_xboole_0) ),
    file(relat_1,fc12_relat_1),
    []).

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

fof(fc1_sysrel,axiom,(
    ! [A,B] : v1_relat_1(k2_zfmisc_1(A,B)) ),
    file(sysrel,fc1_sysrel),
    []).

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

fof(fc4_relat_1,axiom,
    ( v1_xboole_0(k1_xboole_0)
    & v1_relat_1(k1_xboole_0) ),
    file(relat_1,fc4_relat_1),
    []).

fof(fc4_subset_1,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B) )
     => ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ),
    file(subset_1,fc4_subset_1),
    []).

fof(fraenkel_a_4_2_relset_2,axiom,(
    ! [A,B,C,D,E] : 
      ( ( ~ v1_xboole_0(C)
        & m2_relset_1(D,B,C)
        & m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(B))) )
     => ( r2_hidden(A,a_4_2_relset_2(B,C,D,E))
      <=> ? [F] : 
            ( m1_subset_1(F,k1_zfmisc_1(B))
            & A = k8_relset_2(B,C,F,D)
            & r2_hidden(F,E) ) ) ) ),
    file(relset_2,a_4_2_relset_2),
    []).

fof(rc1_relat_1,axiom,(
    ? [A] : 
      ( v1_xboole_0(A)
      & v1_relat_1(A) ) ),
    file(relat_1,rc1_relat_1),
    []).

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(rc2_partfun1,axiom,(
    ! [A,B] : 
    ? [C] : 
      ( m1_relset_1(C,A,B)
      & v1_relat_1(C)
      & v1_funct_1(C) ) ),
    file(partfun1,rc2_partfun1),
    []).

fof(rc2_relat_1,axiom,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_relat_1(A) ) ),
    file(relat_1,rc2_relat_1),
    []).

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(rc3_relat_1,axiom,(
    ? [A] : 
      ( v1_relat_1(A)
      & v3_relat_1(A) ) ),
    file(relat_1,rc3_relat_1),
    []).

fof(redefinition_k10_relset_1,axiom,(
    ! [A,B,C,D] : 
      ( m1_relset_1(C,A,B)
     => k10_relset_1(A,B,C,D) = k9_relat_1(C,D) ) ),
    file(relset_1,k10_relset_1),
    []).

fof(redefinition_k4_relset_2,axiom,(
    ! [A,B,C,D] : 
      ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,k1_zfmisc_1(B))))
     => k4_relset_2(A,B,C,D) = k9_relat_1(C,D) ) ),
    file(relset_2,k4_relset_2),
    []).

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(redefinition_k6_relset_2,axiom,(
    ! [A,B,C] : 
      ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(B,A)))
     => k6_relset_2(A,B,C) = k5_relset_2(B,C) ) ),
    file(relset_2,k6_relset_2),
    []).

fof(redefinition_k6_setfam_1,axiom,(
    ! [A,B] : 
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => k6_setfam_1(A,B) = k1_setfam_1(B) ) ),
    file(setfam_1,k6_setfam_1),
    []).

fof(redefinition_k8_relset_2,axiom,(
    ! [A,B,C,D] : 
      ( ( m1_subset_1(C,k1_zfmisc_1(A))
        & m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(A,B))) )
     => k8_relset_2(A,B,C,D) = k7_relset_2(A,B,C,D) ) ),
    file(relset_2,k8_relset_2),
    []).

fof(redefinition_m2_relset_1,axiom,(
    ! [A,B,C] : 
      ( m2_relset_1(C,A,B)
    <=> m1_relset_1(C,A,B) ) ),
    file(relset_1,m2_relset_1),
    []).

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

fof(t11_setfam_1,axiom,(
    ! [A] : k1_setfam_1(k1_tarski(A)) = A ),
    file(setfam_1,t11_setfam_1),
    []).

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

fof(t23_relset_2,axiom,(
    ! [A,B,C] : 
      ( m1_subset_1(C,k1_zfmisc_1(A))
     => ! [D] : 
          ( m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(A,B)))
         => ( k4_relset_2(k1_zfmisc_1(A),B,k6_relset_2(B,A,D),k3_pua2mss1(C)) = k1_xboole_0
          <=> C = k1_xboole_0 ) ) ) ),
    file(relset_2,t23_relset_2),
    []).

fof(t24_relset_2,axiom,(
    ! [A,B,C,D] : 
      ( m1_subset_1(D,k1_zfmisc_1(A))
     => ! [E] : 
          ( m1_subset_1(E,k1_zfmisc_1(k2_zfmisc_1(A,B)))
         => ( r2_hidden(C,k8_relset_2(A,B,D,E))
           => ! [F] : 
                ( r2_hidden(F,D)
               => r2_hidden(C,k9_relat_1(E,k1_tarski(F))) ) ) ) ) ),
    file(relset_2,t24_relset_2),
    []).

fof(t25_relset_2,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B,C] : 
          ( m1_subset_1(C,k1_zfmisc_1(B))
         => ! [D] : 
              ( m1_subset_1(D,A)
             => ! [E] : 
                  ( m1_subset_1(E,k1_zfmisc_1(k2_zfmisc_1(B,A)))
                 => ( r2_hidden(D,k8_relset_2(B,A,C,E))
                  <=> ! [F] : 
                        ( r2_hidden(F,C)
                       => r2_hidden(D,k9_relat_1(E,k1_tarski(F))) ) ) ) ) ) ) ),
    file(relset_2,t25_relset_2),
    []).

fof(t29_relset_2,axiom,(
    ! [A,B,C] : 
      ( m1_subset_1(C,k1_zfmisc_1(A))
     => ! [D] : 
          ( m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(A,B)))
         => ( C = k1_xboole_0
           => k8_relset_2(A,B,C,D) = B ) ) ) ),
    file(relset_2,t29_relset_2),
    []).

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

fof(t2_tarski,axiom,(
    ! [A,B] : 
      ( ! [C] : 
          ( r2_hidden(C,A)
        <=> r2_hidden(C,B) )
     => A = B ) ),
    file(tarski,t2_tarski),
    []).

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

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_setfam_1,axiom,(
    ! [A,B] : 
      ( r1_tarski(A,B)
     => ( A = k1_xboole_0
        | r1_tarski(k1_setfam_1(B),k1_setfam_1(A)) ) ) ),
    file(setfam_1,t7_setfam_1),
    []).

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

fof(t91_orders_1,axiom,(
    ! [A] : 
      ( ~ ( ? [B] : 
              ( B != k1_xboole_0
              & r2_hidden(B,A) )
          & k3_tarski(A) = k1_xboole_0 )
      & ~ ( k3_tarski(A) != k1_xboole_0
          & ! [B] : ~ ( B != k1_xboole_0
              & r2_hidden(B,A) ) ) ) ),
    file(orders_1,t91_orders_1),
    []).