% Mizar ND problem: t59_classes2,classes2,1058,40 
fof(dh_c1_53__classes2,definition,
    ( ( ( ~ v1_xboole_0(c1_53__classes2)
        & v1_classes2(c1_53__classes2) )
     => ! [A] : 
          ( ( ~ v1_xboole_0(A)
            & v1_classes2(A) )
         => ! [B] : 
              ( ( ~ v1_xboole_0(B)
                & v1_classes2(B) )
             => ( ( r2_hidden(c1_53__classes2,A)
                  & r2_hidden(A,B) )
               => r2_hidden(c1_53__classes2,B) ) ) ) )
   => ! [C] : 
        ( ( ~ v1_xboole_0(C)
          & v1_classes2(C) )
       => ! [D] : 
            ( ( ~ v1_xboole_0(D)
              & v1_classes2(D) )
           => ! [E] : 
                ( ( ~ v1_xboole_0(E)
                  & v1_classes2(E) )
               => ( ( r2_hidden(C,D)
                    & r2_hidden(D,E) )
                 => r2_hidden(C,E) ) ) ) ) ),
    introduced(definition,[new_symbol(c1_53__classes2),file(classes2,c1_53__classes2)]),
    [interesting(0.8),axiom,file(classes2,c1_53__classes2)]).

fof(dh_c2_53__classes2,definition,
    ( ( ( ~ v1_xboole_0(c2_53__classes2)
        & v1_classes2(c2_53__classes2) )
     => ! [A] : 
          ( ( ~ v1_xboole_0(A)
            & v1_classes2(A) )
         => ( ( r2_hidden(c1_53__classes2,c2_53__classes2)
              & r2_hidden(c2_53__classes2,A) )
           => r2_hidden(c1_53__classes2,A) ) ) )
   => ! [B] : 
        ( ( ~ v1_xboole_0(B)
          & v1_classes2(B) )
       => ! [C] : 
            ( ( ~ v1_xboole_0(C)
              & v1_classes2(C) )
           => ( ( r2_hidden(c1_53__classes2,B)
                & r2_hidden(B,C) )
             => r2_hidden(c1_53__classes2,C) ) ) ) ),
    introduced(definition,[new_symbol(c2_53__classes2),file(classes2,c2_53__classes2)]),
    [interesting(0.8),axiom,file(classes2,c2_53__classes2)]).

fof(dh_c3_53__classes2,definition,
    ( ( ( ~ v1_xboole_0(c3_53__classes2)
        & v1_classes2(c3_53__classes2) )
     => ( ( r2_hidden(c1_53__classes2,c2_53__classes2)
          & r2_hidden(c2_53__classes2,c3_53__classes2) )
       => r2_hidden(c1_53__classes2,c3_53__classes2) ) )
   => ! [A] : 
        ( ( ~ v1_xboole_0(A)
          & v1_classes2(A) )
       => ( ( r2_hidden(c1_53__classes2,c2_53__classes2)
            & r2_hidden(c2_53__classes2,A) )
         => r2_hidden(c1_53__classes2,A) ) ) ),
    introduced(definition,[new_symbol(c3_53__classes2),file(classes2,c3_53__classes2)]),
    [interesting(0.8),axiom,file(classes2,c3_53__classes2)]).

fof(e1_53__classes2,assumption,
    ( r2_hidden(c1_53__classes2,c2_53__classes2)
    & r2_hidden(c2_53__classes2,c3_53__classes2) ),
    introduced(assumption,[file(classes2,e1_53__classes2)]),
    [interesting(0.8),axiom,file(classes2,e1_53__classes2)]).

fof(rc1_partfun1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v2_funct_1(A)
      & v1_xboole_0(A) ) ),
    file(partfun1,rc1_partfun1),
    [interesting(0.9),axiom,file(partfun1,rc1_partfun1)]).

fof(rc2_ordinal1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v2_funct_1(A)
      & v1_xboole_0(A)
      & v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A) ) ),
    file(ordinal1,rc2_ordinal1),
    [interesting(0.9),axiom,file(ordinal1,rc2_ordinal1)]).

fof(dt_k1_xboole_0,axiom,(
    $true ),
    file(xboole_0,k1_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,k1_xboole_0)]).

fof(cc1_classes2,theorem,(
    ! [A] : 
      ( v2_classes1(A)
     => v1_classes1(A) ) ),
    file(classes2,cc1_classes2),
    [interesting(0.9),axiom,file(classes2,cc1_classes2)]).

fof(cc3_classes2,theorem,(
    ! [A] : 
      ( ( v1_ordinal1(A)
        & v2_classes1(A) )
     => v1_classes2(A) ) ),
    file(classes2,cc3_classes2),
    [interesting(0.9),axiom,file(classes2,cc3_classes2)]).

fof(fc2_ordinal1,theorem,
    ( v1_relat_1(k1_xboole_0)
    & v3_relat_1(k1_xboole_0)
    & v1_funct_1(k1_xboole_0)
    & v2_funct_1(k1_xboole_0)
    & v1_xboole_0(k1_xboole_0)
    & v1_ordinal1(k1_xboole_0)
    & v2_ordinal1(k1_xboole_0)
    & v3_ordinal1(k1_xboole_0) ),
    file(ordinal1,fc2_ordinal1),
    [interesting(0.9),axiom,file(ordinal1,fc2_ordinal1)]).

fof(rc1_classes2,theorem,(
    ? [A] : 
      ( v1_ordinal1(A)
      & ~ v1_xboole_0(A)
      & v1_classes1(A)
      & v2_classes1(A)
      & v1_classes2(A) ) ),
    file(classes2,rc1_classes2),
    [interesting(0.9),axiom,file(classes2,rc1_classes2)]).

fof(existence_m1_subset_1,axiom,(
    ! [A] : 
    ? [B] : m1_subset_1(B,A) ),
    file(subset_1,m1_subset_1),
    [interesting(0.9),axiom,file(subset_1,m1_subset_1)]).

fof(dt_m1_subset_1,axiom,(
    $true ),
    file(subset_1,m1_subset_1),
    [interesting(0.9),axiom,file(subset_1,m1_subset_1)]).

fof(cc2_classes2,theorem,(
    ! [A] : 
      ( v1_classes2(A)
     => ( v1_ordinal1(A)
        & v1_classes1(A)
        & v2_classes1(A) ) ) ),
    file(classes2,cc2_classes2),
    [interesting(0.9),axiom,file(classes2,cc2_classes2)]).

fof(cc2_ordinal1,theorem,(
    ! [A] : 
      ( ( v1_ordinal1(A)
        & v2_ordinal1(A) )
     => v3_ordinal1(A) ) ),
    file(ordinal1,cc2_ordinal1),
    [interesting(0.9),axiom,file(ordinal1,cc2_ordinal1)]).

fof(cc3_ordinal1,theorem,(
    ! [A] : 
      ( v1_xboole_0(A)
     => ( v1_ordinal1(A)
        & v2_ordinal1(A)
        & v3_ordinal1(A) ) ) ),
    file(ordinal1,cc3_ordinal1),
    [interesting(0.9),axiom,file(ordinal1,cc3_ordinal1)]).

fof(fc2_classes2,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ( v1_ordinal1(k2_ordinal2(A))
        & v2_ordinal1(k2_ordinal2(A))
        & v3_ordinal1(k2_ordinal2(A)) ) ) ),
    file(classes2,fc2_classes2),
    [interesting(0.9),axiom,file(classes2,fc2_classes2)]).

fof(rc1_ordinal1,theorem,(
    ? [A] : 
      ( v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A) ) ),
    file(ordinal1,rc1_ordinal1),
    [interesting(0.9),axiom,file(ordinal1,rc1_ordinal1)]).

fof(rc3_ordinal1,theorem,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A) ) ),
    file(ordinal1,rc3_ordinal1),
    [interesting(0.9),axiom,file(ordinal1,rc3_ordinal1)]).

fof(t2_subset,theorem,(
    ! [A,B] : 
      ( m1_subset_1(A,B)
     => ( v1_xboole_0(B)
        | r2_hidden(A,B) ) ) ),
    file(subset,t2_subset),
    [interesting(0.9),axiom,file(subset,t2_subset)]).

fof(t6_boole,theorem,(
    ! [A] : 
      ( v1_xboole_0(A)
     => A = k1_xboole_0 ) ),
    file(boole,t6_boole),
    [interesting(0.9),axiom,file(boole,t6_boole)]).

fof(t8_boole,theorem,(
    ! [A,B] : 
      ~ ( v1_xboole_0(A)
        & A != B
        & v1_xboole_0(B) ) ),
    file(boole,t8_boole),
    [interesting(0.9),axiom,file(boole,t8_boole)]).

fof(antisymmetry_r2_hidden,theorem,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => ~ r2_hidden(B,A) ) ),
    file(hidden,r2_hidden),
    [interesting(0.9),axiom,file(hidden,r2_hidden)]).

fof(dt_k2_ordinal2,axiom,(
    $true ),
    file(ordinal2,k2_ordinal2),
    [interesting(0.9),axiom,file(ordinal2,k2_ordinal2)]).

fof(dt_k4_classes1,axiom,(
    $true ),
    file(classes1,k4_classes1),
    [interesting(0.9),axiom,file(classes1,k4_classes1)]).

fof(dt_c1_53__classes2,assumption,
    ( ~ v1_xboole_0(c1_53__classes2)
    & v1_classes2(c1_53__classes2) ),
    introduced(assumption,[file(classes2,c1_53__classes2)]),
    [interesting(0.8),axiom,file(classes2,c1_53__classes2)]).

fof(dt_c2_53__classes2,assumption,
    ( ~ v1_xboole_0(c2_53__classes2)
    & v1_classes2(c2_53__classes2) ),
    introduced(assumption,[file(classes2,c2_53__classes2)]),
    [interesting(0.8),axiom,file(classes2,c2_53__classes2)]).

fof(dt_c3_53__classes2,assumption,
    ( ~ v1_xboole_0(c3_53__classes2)
    & v1_classes2(c3_53__classes2) ),
    introduced(assumption,[file(classes2,c3_53__classes2)]),
    [interesting(0.8),axiom,file(classes2,c3_53__classes2)]).

fof(cc1_ordinal1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ( v1_ordinal1(A)
        & v2_ordinal1(A) ) ) ),
    file(ordinal1,cc1_ordinal1),
    [interesting(0.9),axiom,file(ordinal1,cc1_ordinal1)]).

fof(t1_subset,theorem,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => m1_subset_1(A,B) ) ),
    file(subset,t1_subset),
    [interesting(0.9),axiom,file(subset,t1_subset)]).

fof(t7_boole,theorem,(
    ! [A,B] : 
      ~ ( r2_hidden(A,B)
        & v1_xboole_0(B) ) ),
    file(boole,t7_boole),
    [interesting(0.9),axiom,file(boole,t7_boole)]).

fof(t54_classes2,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => A = k4_classes1(k2_ordinal2(A)) ) ),
    file(classes2,t54_classes2),
    [interesting(0.9),axiom,file(classes2,t54_classes2)]).

fof(e2_53__classes2,plain,
    ( c1_53__classes2 = k4_classes1(k2_ordinal2(c1_53__classes2))
    & c2_53__classes2 = k4_classes1(k2_ordinal2(c2_53__classes2))
    & c3_53__classes2 = k4_classes1(k2_ordinal2(c3_53__classes2)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_53__classes2,dt_c2_53__classes2,dt_c3_53__classes2])],[existence_m1_subset_1,dt_m1_subset_1,rc1_partfun1,rc2_ordinal1,t2_subset,antisymmetry_r2_hidden,dt_k1_xboole_0,cc1_classes2,cc1_ordinal1,cc2_ordinal1,cc3_classes2,fc2_ordinal1,rc1_classes2,rc1_ordinal1,rc3_ordinal1,t1_subset,dt_k2_ordinal2,dt_k4_classes1,dt_c1_53__classes2,dt_c2_53__classes2,dt_c3_53__classes2,cc2_classes2,cc3_ordinal1,fc2_classes2,t6_boole,t7_boole,t8_boole,t54_classes2]),
    [interesting(0.8),file(classes2,e2_53__classes2),[file(classes2,e2_53__classes2)]]).

fof(t42_classes1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( r2_hidden(A,B)
          <=> r2_hidden(k4_classes1(A),k4_classes1(B)) ) ) ) ),
    file(classes1,t42_classes1),
    [interesting(0.9),axiom,file(classes1,t42_classes1)]).

fof(e3_53__classes2,plain,
    ( r2_hidden(k2_ordinal2(c1_53__classes2),k2_ordinal2(c2_53__classes2))
    & r2_hidden(k2_ordinal2(c2_53__classes2),k2_ordinal2(c3_53__classes2)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_53__classes2,dt_c2_53__classes2,dt_c3_53__classes2,e1_53__classes2])],[rc1_partfun1,rc2_ordinal1,dt_k1_xboole_0,cc1_classes2,cc3_classes2,fc2_ordinal1,rc1_classes2,existence_m1_subset_1,dt_m1_subset_1,cc2_classes2,cc2_ordinal1,cc3_ordinal1,fc2_classes2,rc1_ordinal1,rc3_ordinal1,t2_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,dt_k2_ordinal2,dt_k4_classes1,dt_c1_53__classes2,dt_c2_53__classes2,dt_c3_53__classes2,cc1_ordinal1,t1_subset,t7_boole,e2_53__classes2,e1_53__classes2,t42_classes1]),
    [interesting(0.8),file(classes2,e3_53__classes2),[file(classes2,e3_53__classes2)]]).

fof(t19_ordinal1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ! [C] : 
              ( v1_ordinal1(C)
             => ( ( r2_hidden(C,A)
                  & r2_hidden(A,B) )
               => r2_hidden(C,B) ) ) ) ) ),
    file(ordinal1,t19_ordinal1),
    [interesting(0.9),axiom,file(ordinal1,t19_ordinal1)]).

fof(e4_53__classes2,plain,(
    r2_hidden(k2_ordinal2(c1_53__classes2),k2_ordinal2(c3_53__classes2)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_53__classes2,dt_c2_53__classes2,dt_c3_53__classes2,e1_53__classes2])],[rc1_partfun1,rc2_ordinal1,dt_k1_xboole_0,cc1_classes2,cc3_classes2,fc2_ordinal1,rc1_classes2,t8_boole,existence_m1_subset_1,dt_m1_subset_1,cc2_classes2,cc2_ordinal1,cc3_ordinal1,fc2_classes2,rc1_ordinal1,rc3_ordinal1,t2_subset,t6_boole,antisymmetry_r2_hidden,dt_k2_ordinal2,dt_c1_53__classes2,dt_c2_53__classes2,dt_c3_53__classes2,cc1_ordinal1,t1_subset,t7_boole,e3_53__classes2,t19_ordinal1]),
    [interesting(0.8),file(classes2,e4_53__classes2),[file(classes2,e4_53__classes2)]]).

fof(e5_53__classes2,plain,(
    r2_hidden(c1_53__classes2,c3_53__classes2) ),
    inference(mizar_by,[status(thm),assumptions([e1_53__classes2,dt_c1_53__classes2,dt_c2_53__classes2,dt_c3_53__classes2])],[rc1_partfun1,rc2_ordinal1,dt_k1_xboole_0,cc1_classes2,cc3_classes2,fc2_ordinal1,rc1_classes2,existence_m1_subset_1,dt_m1_subset_1,cc2_classes2,cc2_ordinal1,cc3_ordinal1,fc2_classes2,rc1_ordinal1,rc3_ordinal1,t2_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,dt_k2_ordinal2,dt_k4_classes1,dt_c1_53__classes2,dt_c2_53__classes2,dt_c3_53__classes2,cc1_ordinal1,t1_subset,t7_boole,e4_53__classes2,e2_53__classes2,t42_classes1]),
    [interesting(0.8),file(classes2,e5_53__classes2),[file(classes2,e5_53__classes2)]]).

fof(i5_53__classes2,theorem,(
    $true ),
    introduced(tautology,[file(classes2,i5_53__classes2)]),
    [interesting(0.8),trivial,file(classes2,i5_53__classes2)]).

fof(i4_53__classes2,plain,(
    r2_hidden(c1_53__classes2,c3_53__classes2) ),
    inference(conclusion,[status(thm),assumptions([e1_53__classes2,dt_c1_53__classes2,dt_c2_53__classes2,dt_c3_53__classes2])],[e5_53__classes2,i5_53__classes2]),
    [interesting(0.8),file(classes2,i4_53__classes2),[file(classes2,i4_53__classes2)]]).

fof(i3_53__classes2,plain,
    ( ( r2_hidden(c1_53__classes2,c2_53__classes2)
      & r2_hidden(c2_53__classes2,c3_53__classes2) )
   => r2_hidden(c1_53__classes2,c3_53__classes2) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_53__classes2,dt_c2_53__classes2,dt_c3_53__classes2]),discharge_asm(discharge,[e1_53__classes2])],[e1_53__classes2,i4_53__classes2]),
    [interesting(0.8),file(classes2,i3_53__classes2),[file(classes2,i3_53__classes2)]]).

fof(i3_53_tmp__classes2,plain,
    ( ( ~ v1_xboole_0(c3_53__classes2)
      & v1_classes2(c3_53__classes2) )
   => ( ( r2_hidden(c1_53__classes2,c2_53__classes2)
        & r2_hidden(c2_53__classes2,c3_53__classes2) )
     => r2_hidden(c1_53__classes2,c3_53__classes2) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_53__classes2,dt_c2_53__classes2]),discharge_asm(discharge,[dt_c3_53__classes2])],[dt_c3_53__classes2,i3_53__classes2]),
    [interesting(0.8),i2_53__classes2]).

fof(i2_53__classes2,plain,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ( ( r2_hidden(c1_53__classes2,c2_53__classes2)
          & r2_hidden(c2_53__classes2,A) )
       => r2_hidden(c1_53__classes2,A) ) ) ),
    inference(let,[status(thm),assumptions([dt_c1_53__classes2,dt_c2_53__classes2])],[i3_53_tmp__classes2,dh_c3_53__classes2]),
    [interesting(0.8),file(classes2,i2_53__classes2),[file(classes2,i2_53__classes2)]]).

fof(i2_53_tmp__classes2,plain,
    ( ( ~ v1_xboole_0(c2_53__classes2)
      & v1_classes2(c2_53__classes2) )
   => ! [A] : 
        ( ( ~ v1_xboole_0(A)
          & v1_classes2(A) )
       => ( ( r2_hidden(c1_53__classes2,c2_53__classes2)
            & r2_hidden(c2_53__classes2,A) )
         => r2_hidden(c1_53__classes2,A) ) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_53__classes2]),discharge_asm(discharge,[dt_c2_53__classes2])],[dt_c2_53__classes2,i2_53__classes2]),
    [interesting(0.8),i1_53__classes2]).

fof(i1_53__classes2,plain,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] : 
          ( ( ~ v1_xboole_0(B)
            & v1_classes2(B) )
         => ( ( r2_hidden(c1_53__classes2,A)
              & r2_hidden(A,B) )
           => r2_hidden(c1_53__classes2,B) ) ) ) ),
    inference(let,[status(thm),assumptions([dt_c1_53__classes2])],[i2_53_tmp__classes2,dh_c2_53__classes2]),
    [interesting(0.8),file(classes2,i1_53__classes2),[file(classes2,i1_53__classes2)]]).

fof(i1_53_tmp__classes2,plain,
    ( ( ~ v1_xboole_0(c1_53__classes2)
      & v1_classes2(c1_53__classes2) )
   => ! [A] : 
        ( ( ~ v1_xboole_0(A)
          & v1_classes2(A) )
       => ! [B] : 
            ( ( ~ v1_xboole_0(B)
              & v1_classes2(B) )
           => ( ( r2_hidden(c1_53__classes2,A)
                & r2_hidden(A,B) )
             => r2_hidden(c1_53__classes2,B) ) ) ) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_53__classes2])],[dt_c1_53__classes2,i1_53__classes2]),
    [interesting(1),t59_classes2]).

fof(t59_classes2,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B] : 
          ( ( ~ v1_xboole_0(B)
            & v1_classes2(B) )
         => ! [C] : 
              ( ( ~ v1_xboole_0(C)
                & v1_classes2(C) )
             => ( ( r2_hidden(A,B)
                  & r2_hidden(B,C) )
               => r2_hidden(A,C) ) ) ) ) ),
    inference(let,[status(thm),assumptions([])],[i1_53_tmp__classes2,dh_c1_53__classes2]),
    [interesting(1),file(classes2,t59_classes2),[file(classes2,t59_classes2)]]).