% Mizar ND problem: t4_orders_1,orders_1,189,15 
fof(dh_c1_5__orders_1,definition,
    ( ( ~ v1_xboole_0(c1_5__orders_1)
     => ~ r2_hidden(k1_xboole_0,k1_orders_1(c1_5__orders_1)) )
   => ! [A] : 
        ( ~ v1_xboole_0(A)
       => ~ r2_hidden(k1_xboole_0,k1_orders_1(A)) ) ),
    introduced(definition,[new_symbol(c1_5__orders_1),file(orders_1,c1_5__orders_1)]),
    [interesting(0.8),axiom,file(orders_1,c1_5__orders_1)]).

fof(e1_5__orders_1,assumption,(
    r2_hidden(k1_xboole_0,k1_orders_1(c1_5__orders_1)) ),
    introduced(assumption,[file(orders_1,e1_5__orders_1)]),
    [interesting(0.8),axiom,file(orders_1,e1_5__orders_1)]).

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(cc1_finset_1,theorem,(
    ! [A] : 
      ( v1_xboole_0(A)
     => v1_finset_1(A) ) ),
    file(finset_1,cc1_finset_1),
    [interesting(0.9),axiom,file(finset_1,cc1_finset_1)]).

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(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(rc1_finset_1,theorem,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_finset_1(A) ) ),
    file(finset_1,rc1_finset_1),
    [interesting(0.9),axiom,file(finset_1,rc1_finset_1)]).

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(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(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(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_k1_tarski,axiom,(
    $true ),
    file(tarski,k1_tarski),
    [interesting(0.9),axiom,file(tarski,k1_tarski)]).

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

fof(fc1_finset_1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(k1_tarski(A))
      & v1_finset_1(k1_tarski(A)) ) ),
    file(finset_1,fc1_finset_1),
    [interesting(0.9),axiom,file(finset_1,fc1_finset_1)]).

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(fc2_subset_1,theorem,(
    ! [A] : ~ v1_xboole_0(k1_tarski(A)) ),
    file(subset_1,fc2_subset_1),
    [interesting(0.9),axiom,file(subset_1,fc2_subset_1)]).

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(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(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(reflexivity_r1_tarski,theorem,(
    ! [A,B] : r1_tarski(A,A) ),
    file(tarski,r1_tarski),
    [interesting(0.9),axiom,file(tarski,r1_tarski)]).

fof(cc2_finset_1,theorem,(
    ! [A] : 
      ( v1_finset_1(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => v1_finset_1(B) ) ) ),
    file(finset_1,cc2_finset_1),
    [interesting(0.9),axiom,file(finset_1,cc2_finset_1)]).

fof(fc12_finset_1,theorem,(
    ! [A,B] : 
      ( v1_finset_1(A)
     => v1_finset_1(k4_xboole_0(A,B)) ) ),
    file(finset_1,fc12_finset_1),
    [interesting(0.9),axiom,file(finset_1,fc12_finset_1)]).

fof(rc1_subset_1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ? [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
          & ~ v1_xboole_0(B) ) ) ),
    file(subset_1,rc1_subset_1),
    [interesting(0.9),axiom,file(subset_1,rc1_subset_1)]).

fof(rc2_subset_1,theorem,(
    ! [A] : 
    ? [B] : 
      ( m1_subset_1(B,k1_zfmisc_1(A))
      & v1_xboole_0(B) ) ),
    file(subset_1,rc2_subset_1),
    [interesting(0.9),axiom,file(subset_1,rc2_subset_1)]).

fof(rc3_finset_1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ? [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
          & ~ v1_xboole_0(B)
          & v1_finset_1(B) ) ) ),
    file(finset_1,rc3_finset_1),
    [interesting(0.9),axiom,file(finset_1,rc3_finset_1)]).

fof(rc4_finset_1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ? [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
          & ~ v1_xboole_0(B)
          & v1_finset_1(B) ) ) ),
    file(finset_1,rc4_finset_1),
    [interesting(0.9),axiom,file(finset_1,rc4_finset_1)]).

fof(t3_subset,theorem,(
    ! [A,B] : 
      ( m1_subset_1(A,k1_zfmisc_1(B))
    <=> r1_tarski(A,B) ) ),
    file(subset,t3_subset),
    [interesting(0.9),axiom,file(subset,t3_subset)]).

fof(t4_subset,theorem,(
    ! [A,B,C] : 
      ( ( r2_hidden(A,B)
        & m1_subset_1(B,k1_zfmisc_1(C)) )
     => m1_subset_1(A,C) ) ),
    file(subset,t4_subset),
    [interesting(0.9),axiom,file(subset,t4_subset)]).

fof(t5_subset,theorem,(
    ! [A,B,C] : 
      ~ ( r2_hidden(A,B)
        & m1_subset_1(B,k1_zfmisc_1(C))
        & v1_xboole_0(C) ) ),
    file(subset,t5_subset),
    [interesting(0.9),axiom,file(subset,t5_subset)]).

fof(dt_k1_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k1_zfmisc_1),
    [interesting(0.9),axiom,file(zfmisc_1,k1_zfmisc_1)]).

fof(dt_k4_xboole_0,axiom,(
    $true ),
    file(xboole_0,k4_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,k4_xboole_0)]).

fof(dt_c1_5__orders_1,assumption,(
    ~ v1_xboole_0(c1_5__orders_1) ),
    introduced(assumption,[file(orders_1,c1_5__orders_1)]),
    [interesting(0.8),axiom,file(orders_1,c1_5__orders_1)]).

fof(fc1_subset_1,theorem,(
    ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ),
    file(subset_1,fc1_subset_1),
    [interesting(0.9),axiom,file(subset_1,fc1_subset_1)]).

fof(t3_boole,theorem,(
    ! [A] : k4_xboole_0(A,k1_xboole_0) = A ),
    file(boole,t3_boole),
    [interesting(0.9),axiom,file(boole,t3_boole)]).

fof(t4_boole,theorem,(
    ! [A] : k4_xboole_0(k1_xboole_0,A) = k1_xboole_0 ),
    file(boole,t4_boole),
    [interesting(0.9),axiom,file(boole,t4_boole)]).

fof(fc1_orders_1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ~ v1_xboole_0(k1_orders_1(A)) ) ),
    file(orders_1,fc1_orders_1),
    [interesting(0.9),axiom,file(orders_1,fc1_orders_1)]).

fof(dt_k1_orders_1,axiom,(
    $true ),
    file(orders_1,k1_orders_1),
    [interesting(0.9),axiom,file(orders_1,k1_orders_1)]).

fof(d2_orders_1,definition,(
    ! [A] : k1_orders_1(A) = k4_xboole_0(k1_zfmisc_1(A),k1_tarski(k1_xboole_0)) ),
    file(orders_1,d2_orders_1),
    [interesting(0.9),axiom,file(orders_1,d2_orders_1)]).

fof(e2_5__orders_1,plain,(
    r2_hidden(k1_xboole_0,k4_xboole_0(k1_zfmisc_1(c1_5__orders_1),k1_tarski(k1_xboole_0))) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__orders_1,e1_5__orders_1])],[reflexivity_r1_tarski,existence_m1_subset_1,dt_m1_subset_1,cc1_finset_1,cc1_ordinal1,cc2_finset_1,cc2_ordinal1,cc3_ordinal1,fc12_finset_1,fc1_orders_1,rc1_finset_1,rc1_ordinal1,rc1_partfun1,rc1_subset_1,rc2_ordinal1,rc2_subset_1,rc3_finset_1,rc3_ordinal1,rc4_finset_1,t2_subset,t3_boole,t3_subset,t4_boole,t4_subset,t5_subset,t8_boole,antisymmetry_r2_hidden,dt_k1_orders_1,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k4_xboole_0,dt_c1_5__orders_1,fc1_finset_1,fc1_subset_1,fc2_ordinal1,fc2_subset_1,t1_subset,t6_boole,t7_boole,d2_orders_1,e1_5__orders_1]),
    [interesting(0.8),file(orders_1,e2_5__orders_1),[file(orders_1,e2_5__orders_1)]]).

fof(d4_xboole_0,definition,(
    ! [A,B,C] : 
      ( C = k4_xboole_0(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ( r2_hidden(D,A)
            & ~ r2_hidden(D,B) ) ) ) ),
    file(xboole_0,d4_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,d4_xboole_0)]).

fof(e3_5__orders_1,plain,(
    ~ r2_hidden(k1_xboole_0,k1_tarski(k1_xboole_0)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__orders_1,e1_5__orders_1])],[reflexivity_r1_tarski,existence_m1_subset_1,dt_m1_subset_1,cc1_finset_1,cc1_ordinal1,cc2_finset_1,cc2_ordinal1,cc3_ordinal1,fc12_finset_1,rc1_finset_1,rc1_ordinal1,rc1_partfun1,rc1_subset_1,rc2_ordinal1,rc2_subset_1,rc3_finset_1,rc3_ordinal1,rc4_finset_1,t2_subset,t3_subset,t4_subset,t5_subset,t8_boole,antisymmetry_r2_hidden,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k4_xboole_0,dt_c1_5__orders_1,fc1_finset_1,fc1_subset_1,fc2_ordinal1,fc2_subset_1,t1_subset,t3_boole,t4_boole,t6_boole,t7_boole,e2_5__orders_1,d4_xboole_0]),
    [interesting(0.8),file(orders_1,e3_5__orders_1),[file(orders_1,e3_5__orders_1)]]).

fof(d1_tarski,definition,(
    ! [A,B] : 
      ( B = k1_tarski(A)
    <=> ! [C] : 
          ( r2_hidden(C,B)
        <=> C = A ) ) ),
    file(tarski,d1_tarski),
    [interesting(0.9),axiom,file(tarski,d1_tarski)]).

fof(e4_5__orders_1,plain,(
    ~ $true ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__orders_1,e1_5__orders_1])],[existence_m1_subset_1,dt_m1_subset_1,cc1_finset_1,cc1_ordinal1,cc2_ordinal1,cc3_ordinal1,rc1_finset_1,rc1_ordinal1,rc1_partfun1,rc2_ordinal1,rc3_ordinal1,t2_subset,t8_boole,antisymmetry_r2_hidden,dt_k1_tarski,dt_k1_xboole_0,fc1_finset_1,fc2_ordinal1,fc2_subset_1,t1_subset,t6_boole,t7_boole,e3_5__orders_1,d1_tarski]),
    [interesting(0.8),file(orders_1,e4_5__orders_1),[file(orders_1,e4_5__orders_1)]]).

fof(i3_5__orders_1,theorem,(
    $true ),
    introduced(tautology,[file(orders_1,i3_5__orders_1)]),
    [interesting(0.8),trivial,file(orders_1,i3_5__orders_1)]).

fof(i2_5__orders_1,plain,(
    ~ $true ),
    inference(conclusion,[status(thm),assumptions([dt_c1_5__orders_1,e1_5__orders_1])],[e4_5__orders_1,i3_5__orders_1]),
    [interesting(0.8),file(orders_1,i2_5__orders_1),[file(orders_1,i2_5__orders_1)]]).

fof(i1_5__orders_1,plain,(
    ~ r2_hidden(k1_xboole_0,k1_orders_1(c1_5__orders_1)) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_5__orders_1]),discharge_asm(discharge,[e1_5__orders_1])],[e1_5__orders_1,i2_5__orders_1]),
    [interesting(0.8),file(orders_1,i1_5__orders_1),[file(orders_1,i1_5__orders_1)]]).

fof(i1_5_tmp__orders_1,plain,
    ( ~ v1_xboole_0(c1_5__orders_1)
   => ~ r2_hidden(k1_xboole_0,k1_orders_1(c1_5__orders_1)) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_5__orders_1])],[dt_c1_5__orders_1,i1_5__orders_1]),
    [interesting(1),t4_orders_1]).

fof(t4_orders_1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ~ r2_hidden(k1_xboole_0,k1_orders_1(A)) ) ),
    inference(let,[status(thm),assumptions([])],[i1_5_tmp__orders_1,dh_c1_5__orders_1]),
    [interesting(1),file(orders_1,t4_orders_1),[file(orders_1,t4_orders_1)]]).