% Mizar ND problem: t5_ordinal2,ordinal2,95,11 
fof(dh_c1_6__ordinal2,definition,
    ( ( v3_ordinal1(c1_6__ordinal2)
     => r1_ordinal1(k3_tarski(c1_6__ordinal2),c1_6__ordinal2) )
   => ! [A] : 
        ( v3_ordinal1(A)
       => r1_ordinal1(k3_tarski(A),A) ) ),
    introduced(definition,[new_symbol(c1_6__ordinal2),file(ordinal2,c1_6__ordinal2)]),
    [interesting(0.8),axiom,file(ordinal2,c1_6__ordinal2)]).

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(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(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(fc4_ordinal1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ( v1_ordinal1(k3_tarski(A))
        & v2_ordinal1(k3_tarski(A))
        & v3_ordinal1(k3_tarski(A)) ) ) ),
    file(ordinal1,fc4_ordinal1),
    [interesting(0.9),axiom,file(ordinal1,fc4_ordinal1)]).

fof(reflexivity_r1_ordinal1,theorem,(
    ! [A,B] : 
      ( ( v3_ordinal1(A)
        & v3_ordinal1(B) )
     => r1_ordinal1(A,A) ) ),
    file(ordinal1,r1_ordinal1),
    [interesting(0.9),axiom,file(ordinal1,r1_ordinal1)]).

fof(connectedness_r1_ordinal1,theorem,(
    ! [A,B] : 
      ( ( v3_ordinal1(A)
        & v3_ordinal1(B) )
     => ( r1_ordinal1(A,B)
        | r1_ordinal1(B,A) ) ) ),
    file(ordinal1,r1_ordinal1),
    [interesting(0.9),axiom,file(ordinal1,r1_ordinal1)]).

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

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(redefinition_r1_ordinal1,definition,(
    ! [A,B] : 
      ( ( v3_ordinal1(A)
        & v3_ordinal1(B) )
     => ( r1_ordinal1(A,B)
      <=> r1_tarski(A,B) ) ) ),
    file(ordinal1,r1_ordinal1),
    [interesting(0.9),axiom,file(ordinal1,r1_ordinal1)]).

fof(dt_k3_tarski,axiom,(
    $true ),
    file(tarski,k3_tarski),
    [interesting(0.9),axiom,file(tarski,k3_tarski)]).

fof(dt_c1_6__ordinal2,assumption,(
    v3_ordinal1(c1_6__ordinal2) ),
    introduced(assumption,[file(ordinal2,c1_6__ordinal2)]),
    [interesting(0.8),axiom,file(ordinal2,c1_6__ordinal2)]).

fof(dt_c2_6__ordinal2,assumption,(
    $true ),
    introduced(assumption,[file(ordinal2,c2_6__ordinal2)]),
    [interesting(0.8),axiom,file(ordinal2,c2_6__ordinal2)]).

fof(d3_tarski,definition,(
    ! [A,B] : 
      ( r1_tarski(A,B)
    <=> ! [C] : 
          ( r2_hidden(C,A)
         => r2_hidden(C,B) ) ) ),
    file(tarski,d3_tarski),
    [interesting(0.9),axiom,file(tarski,d3_tarski)]).

fof(dh_c2_6__ordinal2,definition,
    ( ~ ( r2_hidden(c2_6__ordinal2,k3_tarski(c1_6__ordinal2))
        & ~ r2_hidden(c2_6__ordinal2,c1_6__ordinal2) )
   => ! [A] : 
        ~ ( r2_hidden(A,k3_tarski(c1_6__ordinal2))
          & ~ r2_hidden(A,c1_6__ordinal2) ) ),
    introduced(definition,[new_symbol(c2_6__ordinal2),file(ordinal2,c2_6__ordinal2)]),
    [interesting(0.8),axiom,file(ordinal2,c2_6__ordinal2)]).

fof(e1_6__ordinal2,assumption,(
    r2_hidden(c2_6__ordinal2,k3_tarski(c1_6__ordinal2)) ),
    introduced(assumption,[file(ordinal2,e1_6__ordinal2)]),
    [interesting(0.8),axiom,file(ordinal2,e1_6__ordinal2)]).

fof(cc2_funct_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_xboole_0(A)
        & v1_funct_1(A) )
     => ( v1_relat_1(A)
        & v1_funct_1(A)
        & v2_funct_1(A) ) ) ),
    file(funct_1,cc2_funct_1),
    [interesting(0.9),axiom,file(funct_1,cc2_funct_1)]).

fof(rc1_funct_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A) ) ),
    file(funct_1,rc1_funct_1),
    [interesting(0.9),axiom,file(funct_1,rc1_funct_1)]).

fof(rc2_funct_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_xboole_0(A)
      & v1_funct_1(A) ) ),
    file(funct_1,rc2_funct_1),
    [interesting(0.9),axiom,file(funct_1,rc2_funct_1)]).

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_funct_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v2_funct_1(A) ) ),
    file(funct_1,rc3_funct_1),
    [interesting(0.9),axiom,file(funct_1,rc3_funct_1)]).

fof(rc4_funct_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v3_relat_1(A)
      & v1_funct_1(A) ) ),
    file(funct_1,rc4_funct_1),
    [interesting(0.9),axiom,file(funct_1,rc4_funct_1)]).

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_xboole_0,theorem,(
    v1_xboole_0(k1_xboole_0) ),
    file(xboole_0,fc1_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,fc1_xboole_0)]).

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(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(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(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_k1_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k1_zfmisc_1),
    [interesting(0.9),axiom,file(zfmisc_1,k1_zfmisc_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_funct_1,theorem,(
    ! [A] : 
      ( v1_xboole_0(A)
     => v1_funct_1(A) ) ),
    file(funct_1,cc1_funct_1),
    [interesting(0.9),axiom,file(funct_1,cc1_funct_1)]).

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(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(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(rc1_xboole_0,theorem,(
    ? [A] : v1_xboole_0(A) ),
    file(xboole_0,rc1_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,rc1_xboole_0)]).

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(rc2_xboole_0,theorem,(
    ? [A] : ~ v1_xboole_0(A) ),
    file(xboole_0,rc2_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,rc2_xboole_0)]).

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(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(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(dh_c3_6__ordinal2,definition,
    ( ? [A] : 
        ( r2_hidden(c2_6__ordinal2,A)
        & r2_hidden(A,c1_6__ordinal2) )
   => ( r2_hidden(c2_6__ordinal2,c3_6__ordinal2)
      & r2_hidden(c3_6__ordinal2,c1_6__ordinal2) ) ),
    introduced(definition,[new_symbol(c3_6__ordinal2),file(ordinal2,c3_6__ordinal2)]),
    [interesting(0.8),axiom,file(ordinal2,c3_6__ordinal2)]).

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(d4_tarski,definition,(
    ! [A,B] : 
      ( B = k3_tarski(A)
    <=> ! [C] : 
          ( r2_hidden(C,B)
        <=> ? [D] : 
              ( r2_hidden(C,D)
              & r2_hidden(D,A) ) ) ) ),
    file(tarski,d4_tarski),
    [interesting(0.9),axiom,file(tarski,d4_tarski)]).

fof(e2_6__ordinal2,plain,(
    ? [A] : 
      ( r2_hidden(c2_6__ordinal2,A)
      & r2_hidden(A,c1_6__ordinal2) ) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_6__ordinal2,dt_c2_6__ordinal2,e1_6__ordinal2])],[cc2_funct_1,rc1_funct_1,rc2_funct_1,rc2_ordinal1,rc3_funct_1,rc4_funct_1,dt_k1_xboole_0,cc2_ordinal1,fc1_xboole_0,fc2_ordinal1,rc1_ordinal1,rc3_ordinal1,existence_m1_subset_1,dt_m1_subset_1,cc1_funct_1,cc1_ordinal1,cc3_ordinal1,fc4_ordinal1,rc1_xboole_0,rc2_xboole_0,t2_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,dt_k3_tarski,dt_c1_6__ordinal2,dt_c2_6__ordinal2,t1_subset,t7_boole,e1_6__ordinal2,d4_tarski]),
    [interesting(0.8),file(ordinal2,e2_6__ordinal2),[file(ordinal2,e2_6__ordinal2)]]).

fof(dt_c3_6__ordinal2,plain,(
    $true ),
    inference(consider,[status(thm),assumptions([dt_c1_6__ordinal2,dt_c2_6__ordinal2,e1_6__ordinal2])],[dh_c3_6__ordinal2,e2_6__ordinal2]),
    [interesting(0.8),file(ordinal2,c3_6__ordinal2),[file(ordinal2,c3_6__ordinal2)]]).

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(e3_6__ordinal2,plain,
    ( r2_hidden(c2_6__ordinal2,c3_6__ordinal2)
    & r2_hidden(c3_6__ordinal2,c1_6__ordinal2) ),
    inference(consider,[status(thm),assumptions([dt_c1_6__ordinal2,dt_c2_6__ordinal2,e1_6__ordinal2])],[dh_c3_6__ordinal2,e2_6__ordinal2]),
    [interesting(0.8),file(ordinal2,e3_6__ordinal2),[file(ordinal2,e3_6__ordinal2)]]).

fof(d2_ordinal1,definition,(
    ! [A] : 
      ( v1_ordinal1(A)
    <=> ! [B] : 
          ( r2_hidden(B,A)
         => r1_tarski(B,A) ) ) ),
    file(ordinal1,d2_ordinal1),
    [interesting(0.9),axiom,file(ordinal1,d2_ordinal1)]).

fof(e4_6__ordinal2,plain,(
    r1_tarski(c3_6__ordinal2,c1_6__ordinal2) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_6__ordinal2,dt_c2_6__ordinal2,e1_6__ordinal2])],[cc2_funct_1,rc1_funct_1,rc2_funct_1,rc2_ordinal1,rc3_funct_1,rc4_funct_1,dt_k1_xboole_0,cc2_ordinal1,fc1_xboole_0,fc2_ordinal1,rc1_ordinal1,rc3_ordinal1,t8_boole,existence_m1_subset_1,dt_k1_zfmisc_1,dt_m1_subset_1,cc1_funct_1,cc1_ordinal1,cc3_ordinal1,fc1_subset_1,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,t2_subset,t4_subset,t5_subset,t6_boole,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_c1_6__ordinal2,dt_c2_6__ordinal2,dt_c3_6__ordinal2,t1_subset,t3_subset,t7_boole,e3_6__ordinal2,d2_ordinal1]),
    [interesting(0.8),file(ordinal2,e4_6__ordinal2),[file(ordinal2,e4_6__ordinal2)]]).

fof(e5_6__ordinal2,plain,(
    r2_hidden(c2_6__ordinal2,c1_6__ordinal2) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_6__ordinal2,dt_c2_6__ordinal2,e1_6__ordinal2])],[cc2_funct_1,rc1_funct_1,rc2_funct_1,rc2_ordinal1,rc3_funct_1,rc4_funct_1,dt_k1_xboole_0,cc2_ordinal1,fc1_xboole_0,fc2_ordinal1,rc1_ordinal1,rc3_ordinal1,t8_boole,existence_m1_subset_1,dt_k1_zfmisc_1,dt_m1_subset_1,cc1_funct_1,cc1_ordinal1,cc3_ordinal1,fc1_subset_1,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,t2_subset,t4_subset,t5_subset,t6_boole,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_c1_6__ordinal2,dt_c2_6__ordinal2,dt_c3_6__ordinal2,t1_subset,t3_subset,t7_boole,e4_6__ordinal2,e3_6__ordinal2]),
    [interesting(0.8),file(ordinal2,e5_6__ordinal2),[file(ordinal2,e5_6__ordinal2)]]).

fof(i4_6__ordinal2,theorem,(
    $true ),
    introduced(tautology,[file(ordinal2,i4_6__ordinal2)]),
    [interesting(0.8),trivial,file(ordinal2,i4_6__ordinal2)]).

fof(i3_6__ordinal2,plain,(
    r2_hidden(c2_6__ordinal2,c1_6__ordinal2) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_6__ordinal2,dt_c2_6__ordinal2,e1_6__ordinal2])],[e5_6__ordinal2,i4_6__ordinal2]),
    [interesting(0.8),file(ordinal2,i3_6__ordinal2),[file(ordinal2,i3_6__ordinal2)]]).

fof(i2_6__ordinal2,plain,(
    ~ ( r2_hidden(c2_6__ordinal2,k3_tarski(c1_6__ordinal2))
      & ~ r2_hidden(c2_6__ordinal2,c1_6__ordinal2) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_6__ordinal2,dt_c2_6__ordinal2]),discharge_asm(discharge,[e1_6__ordinal2])],[e1_6__ordinal2,i3_6__ordinal2]),
    [interesting(0.8),file(ordinal2,i2_6__ordinal2),[file(ordinal2,i2_6__ordinal2)]]).

fof(i2_6_tmp__ordinal2,plain,(
    ~ ( r2_hidden(c2_6__ordinal2,k3_tarski(c1_6__ordinal2))
      & ~ r2_hidden(c2_6__ordinal2,c1_6__ordinal2) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_6__ordinal2]),discharge_asm(discharge,[dt_c2_6__ordinal2])],[dt_c2_6__ordinal2,i2_6__ordinal2]),
    [interesting(0.8),i1_6__ordinal2]).

fof(i1_6__ordinal2,plain,(
    r1_ordinal1(k3_tarski(c1_6__ordinal2),c1_6__ordinal2) ),
    inference(let,[status(thm),assumptions([dt_c1_6__ordinal2])],[i2_6_tmp__ordinal2,cc2_ordinal1,rc1_ordinal1,cc1_ordinal1,fc4_ordinal1,reflexivity_r1_ordinal1,connectedness_r1_ordinal1,reflexivity_r1_tarski,antisymmetry_r2_hidden,redefinition_r1_ordinal1,dt_k3_tarski,dt_c1_6__ordinal2,d3_tarski,dh_c2_6__ordinal2]),
    [interesting(0.8),file(ordinal2,i1_6__ordinal2),[file(ordinal2,i1_6__ordinal2)]]).

fof(i1_6_tmp__ordinal2,plain,
    ( v3_ordinal1(c1_6__ordinal2)
   => r1_ordinal1(k3_tarski(c1_6__ordinal2),c1_6__ordinal2) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_6__ordinal2])],[dt_c1_6__ordinal2,i1_6__ordinal2]),
    [interesting(1),t5_ordinal2]).

fof(t5_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => r1_ordinal1(k3_tarski(A),A) ) ),
    inference(let,[status(thm),assumptions([])],[i1_6_tmp__ordinal2,dh_c1_6__ordinal2]),
    [interesting(1),file(ordinal2,t5_ordinal2),[file(ordinal2,t5_ordinal2)]]).