% Mizar ND problem: t5_ordinal3,ordinal3,42,29 
fof(dh_c1_3__ordinal3,definition,
    ( ( v3_ordinal1(c1_3__ordinal3)
     => ! [A] : 
          ( v3_ordinal1(A)
         => ( r2_hidden(c1_3__ordinal3,A)
          <=> r2_hidden(k1_ordinal1(c1_3__ordinal3),k1_ordinal1(A)) ) ) )
   => ! [B] : 
        ( v3_ordinal1(B)
       => ! [C] : 
            ( v3_ordinal1(C)
           => ( r2_hidden(B,C)
            <=> r2_hidden(k1_ordinal1(B),k1_ordinal1(C)) ) ) ) ),
    introduced(definition,[new_symbol(c1_3__ordinal3),file(ordinal3,c1_3__ordinal3)]),
    [interesting(0.8),axiom,file(ordinal3,c1_3__ordinal3)]).

fof(dh_c2_3__ordinal3,definition,
    ( ( v3_ordinal1(c2_3__ordinal3)
     => ( r2_hidden(c1_3__ordinal3,c2_3__ordinal3)
      <=> r2_hidden(k1_ordinal1(c1_3__ordinal3),k1_ordinal1(c2_3__ordinal3)) ) )
   => ! [A] : 
        ( v3_ordinal1(A)
       => ( r2_hidden(c1_3__ordinal3,A)
        <=> r2_hidden(k1_ordinal1(c1_3__ordinal3),k1_ordinal1(A)) ) ) ),
    introduced(definition,[new_symbol(c2_3__ordinal3),file(ordinal3,c2_3__ordinal3)]),
    [interesting(0.8),axiom,file(ordinal3,c2_3__ordinal3)]).

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

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

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_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(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(fc3_ordinal1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ( ~ v1_xboole_0(k1_ordinal1(A))
        & v1_ordinal1(k1_ordinal1(A))
        & v2_ordinal1(k1_ordinal1(A))
        & v3_ordinal1(k1_ordinal1(A)) ) ) ),
    file(ordinal1,fc3_ordinal1),
    [interesting(0.9),axiom,file(ordinal1,fc3_ordinal1)]).

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_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(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(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(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(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_k1_ordinal1,axiom,(
    $true ),
    file(ordinal1,k1_ordinal1),
    [interesting(0.9),axiom,file(ordinal1,k1_ordinal1)]).

fof(dt_c1_3__ordinal3,assumption,(
    v3_ordinal1(c1_3__ordinal3) ),
    introduced(assumption,[file(ordinal3,c1_3__ordinal3)]),
    [interesting(0.8),axiom,file(ordinal3,c1_3__ordinal3)]).

fof(dt_c2_3__ordinal3,assumption,(
    v3_ordinal1(c2_3__ordinal3) ),
    introduced(assumption,[file(ordinal3,c2_3__ordinal3)]),
    [interesting(0.8),axiom,file(ordinal3,c2_3__ordinal3)]).

fof(fc1_ordinal1,theorem,(
    ! [A] : ~ v1_xboole_0(k1_ordinal1(A)) ),
    file(ordinal1,fc1_ordinal1),
    [interesting(0.9),axiom,file(ordinal1,fc1_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(t33_ordinal1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( r2_hidden(A,B)
          <=> r1_ordinal1(k1_ordinal1(A),B) ) ) ) ),
    file(ordinal1,t33_ordinal1),
    [interesting(0.9),axiom,file(ordinal1,t33_ordinal1)]).

fof(t34_ordinal1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( r2_hidden(A,k1_ordinal1(B))
          <=> r1_ordinal1(A,B) ) ) ) ),
    file(ordinal1,t34_ordinal1),
    [interesting(0.9),axiom,file(ordinal1,t34_ordinal1)]).

fof(e1_3__ordinal3,plain,
    ( ( r2_hidden(c1_3__ordinal3,c2_3__ordinal3)
     => r1_ordinal1(k1_ordinal1(c1_3__ordinal3),c2_3__ordinal3) )
    & ( r1_ordinal1(k1_ordinal1(c1_3__ordinal3),c2_3__ordinal3)
     => r2_hidden(c1_3__ordinal3,c2_3__ordinal3) )
    & ( r1_ordinal1(k1_ordinal1(c1_3__ordinal3),c2_3__ordinal3)
     => r2_hidden(k1_ordinal1(c1_3__ordinal3),k1_ordinal1(c2_3__ordinal3)) )
    & ( r2_hidden(k1_ordinal1(c1_3__ordinal3),k1_ordinal1(c2_3__ordinal3))
     => r1_ordinal1(k1_ordinal1(c1_3__ordinal3),c2_3__ordinal3) ) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_3__ordinal3,dt_c2_3__ordinal3])],[rc2_ordinal1,dt_k1_xboole_0,dt_k1_zfmisc_1,fc1_xboole_0,fc2_ordinal1,t4_subset,t5_subset,t8_boole,reflexivity_r1_tarski,existence_m1_subset_1,dt_m1_subset_1,cc2_ordinal1,cc3_ordinal1,rc1_ordinal1,rc1_xboole_0,rc2_xboole_0,rc3_ordinal1,t2_subset,t3_subset,t6_boole,reflexivity_r1_ordinal1,connectedness_r1_ordinal1,antisymmetry_r2_hidden,redefinition_r1_ordinal1,dt_k1_ordinal1,dt_c1_3__ordinal3,dt_c2_3__ordinal3,cc1_ordinal1,fc1_ordinal1,fc3_ordinal1,t1_subset,t7_boole,t33_ordinal1,t34_ordinal1]),
    [interesting(0.8),file(ordinal3,e1_3__ordinal3),[file(ordinal3,e1_3__ordinal3)]]).

fof(e2_3__ordinal3,plain,
    ( r2_hidden(c1_3__ordinal3,c2_3__ordinal3)
  <=> r2_hidden(k1_ordinal1(c1_3__ordinal3),k1_ordinal1(c2_3__ordinal3)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_3__ordinal3,dt_c2_3__ordinal3])],[rc2_ordinal1,dt_k1_xboole_0,dt_k1_zfmisc_1,cc2_ordinal1,fc1_xboole_0,fc2_ordinal1,rc1_ordinal1,rc3_ordinal1,t4_subset,t5_subset,t8_boole,reflexivity_r1_tarski,existence_m1_subset_1,dt_m1_subset_1,cc1_ordinal1,cc3_ordinal1,fc3_ordinal1,rc1_xboole_0,rc2_xboole_0,t2_subset,t3_subset,t6_boole,reflexivity_r1_ordinal1,connectedness_r1_ordinal1,antisymmetry_r2_hidden,redefinition_r1_ordinal1,dt_k1_ordinal1,dt_c1_3__ordinal3,dt_c2_3__ordinal3,fc1_ordinal1,t1_subset,t7_boole,e1_3__ordinal3]),
    [interesting(0.8),file(ordinal3,e2_3__ordinal3),[file(ordinal3,e2_3__ordinal3)]]).

fof(i3_3__ordinal3,theorem,(
    $true ),
    introduced(tautology,[file(ordinal3,i3_3__ordinal3)]),
    [interesting(0.8),trivial,file(ordinal3,i3_3__ordinal3)]).

fof(i2_3__ordinal3,plain,
    ( r2_hidden(c1_3__ordinal3,c2_3__ordinal3)
  <=> r2_hidden(k1_ordinal1(c1_3__ordinal3),k1_ordinal1(c2_3__ordinal3)) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_3__ordinal3,dt_c2_3__ordinal3])],[e2_3__ordinal3,i3_3__ordinal3]),
    [interesting(0.8),file(ordinal3,i2_3__ordinal3),[file(ordinal3,i2_3__ordinal3)]]).

fof(i2_3_tmp__ordinal3,plain,
    ( v3_ordinal1(c2_3__ordinal3)
   => ( r2_hidden(c1_3__ordinal3,c2_3__ordinal3)
    <=> r2_hidden(k1_ordinal1(c1_3__ordinal3),k1_ordinal1(c2_3__ordinal3)) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_3__ordinal3]),discharge_asm(discharge,[dt_c2_3__ordinal3])],[dt_c2_3__ordinal3,i2_3__ordinal3]),
    [interesting(0.8),i1_3__ordinal3]).

fof(i1_3__ordinal3,plain,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ( r2_hidden(c1_3__ordinal3,A)
      <=> r2_hidden(k1_ordinal1(c1_3__ordinal3),k1_ordinal1(A)) ) ) ),
    inference(let,[status(thm),assumptions([dt_c1_3__ordinal3])],[i2_3_tmp__ordinal3,dh_c2_3__ordinal3]),
    [interesting(0.8),file(ordinal3,i1_3__ordinal3),[file(ordinal3,i1_3__ordinal3)]]).

fof(i1_3_tmp__ordinal3,plain,
    ( v3_ordinal1(c1_3__ordinal3)
   => ! [A] : 
        ( v3_ordinal1(A)
       => ( r2_hidden(c1_3__ordinal3,A)
        <=> r2_hidden(k1_ordinal1(c1_3__ordinal3),k1_ordinal1(A)) ) ) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_3__ordinal3])],[dt_c1_3__ordinal3,i1_3__ordinal3]),
    [interesting(1),t5_ordinal3]).

fof(t5_ordinal3,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( r2_hidden(A,B)
          <=> r2_hidden(k1_ordinal1(A),k1_ordinal1(B)) ) ) ) ),
    inference(let,[status(thm),assumptions([])],[i1_3_tmp__ordinal3,dh_c1_3__ordinal3]),
    [interesting(1),file(ordinal3,t5_ordinal3),[file(ordinal3,t5_ordinal3)]]).