% Mizar ND problem: t5_partit1,partit1,88,34 
fof(dh_c1_5__partit1,definition,
    ( ( ~ v1_xboole_0(c1_5__partit1)
     => ! [A] : 
          ( m1_eqrel_1(A,c1_5__partit1)
         => ! [B] : 
              ( m1_eqrel_1(B,c1_5__partit1)
             => ( ( r1_setfam_1(B,A)
                  & r1_setfam_1(A,B) )
               => A = B ) ) ) )
   => ! [C] : 
        ( ~ v1_xboole_0(C)
       => ! [D] : 
            ( m1_eqrel_1(D,C)
           => ! [E] : 
                ( m1_eqrel_1(E,C)
               => ( ( r1_setfam_1(E,D)
                    & r1_setfam_1(D,E) )
                 => D = E ) ) ) ) ),
    introduced(definition,[new_symbol(c1_5__partit1),file(partit1,c1_5__partit1)]),
    [interesting(0.8),axiom,file(partit1,c1_5__partit1)]).

fof(dh_c2_5__partit1,definition,
    ( ( m1_eqrel_1(c2_5__partit1,c1_5__partit1)
     => ! [A] : 
          ( m1_eqrel_1(A,c1_5__partit1)
         => ( ( r1_setfam_1(A,c2_5__partit1)
              & r1_setfam_1(c2_5__partit1,A) )
           => c2_5__partit1 = A ) ) )
   => ! [B] : 
        ( m1_eqrel_1(B,c1_5__partit1)
       => ! [C] : 
            ( m1_eqrel_1(C,c1_5__partit1)
           => ( ( r1_setfam_1(C,B)
                & r1_setfam_1(B,C) )
             => B = C ) ) ) ),
    introduced(definition,[new_symbol(c2_5__partit1),file(partit1,c2_5__partit1)]),
    [interesting(0.8),axiom,file(partit1,c2_5__partit1)]).

fof(dh_c3_5__partit1,definition,
    ( ( m1_eqrel_1(c3_5__partit1,c1_5__partit1)
     => ( ( r1_setfam_1(c3_5__partit1,c2_5__partit1)
          & r1_setfam_1(c2_5__partit1,c3_5__partit1) )
       => c2_5__partit1 = c3_5__partit1 ) )
   => ! [A] : 
        ( m1_eqrel_1(A,c1_5__partit1)
       => ( ( r1_setfam_1(A,c2_5__partit1)
            & r1_setfam_1(c2_5__partit1,A) )
         => c2_5__partit1 = A ) ) ),
    introduced(definition,[new_symbol(c3_5__partit1),file(partit1,c3_5__partit1)]),
    [interesting(0.8),axiom,file(partit1,c3_5__partit1)]).

fof(e1_5__partit1,assumption,
    ( r1_setfam_1(c3_5__partit1,c2_5__partit1)
    & r1_setfam_1(c2_5__partit1,c3_5__partit1) ),
    introduced(assumption,[file(partit1,e1_5__partit1)]),
    [interesting(0.8),axiom,file(partit1,e1_5__partit1)]).

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_xboole_0,axiom,(
    $true ),
    file(xboole_0,k1_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,k1_xboole_0)]).

fof(cc10_membered,theorem,(
    ! [A] : 
      ( v1_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,A)
         => v1_xcmplx_0(B) ) ) ),
    file(membered,cc10_membered),
    [interesting(0.9),axiom,file(membered,cc10_membered)]).

fof(cc11_membered,theorem,(
    ! [A] : 
      ( v2_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,A)
         => ( v1_xcmplx_0(B)
            & v1_xreal_0(B) ) ) ) ),
    file(membered,cc11_membered),
    [interesting(0.9),axiom,file(membered,cc11_membered)]).

fof(cc12_membered,theorem,(
    ! [A] : 
      ( v3_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,A)
         => ( v1_xcmplx_0(B)
            & v1_xreal_0(B)
            & v1_rat_1(B) ) ) ) ),
    file(membered,cc12_membered),
    [interesting(0.9),axiom,file(membered,cc12_membered)]).

fof(cc13_membered,theorem,(
    ! [A] : 
      ( v4_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,A)
         => ( v1_xcmplx_0(B)
            & v1_xreal_0(B)
            & v1_int_1(B)
            & v1_rat_1(B) ) ) ) ),
    file(membered,cc13_membered),
    [interesting(0.9),axiom,file(membered,cc13_membered)]).

fof(cc14_membered,theorem,(
    ! [A] : 
      ( v5_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,A)
         => ( v1_xcmplx_0(B)
            & v4_ordinal2(B)
            & v1_xreal_0(B)
            & v1_int_1(B)
            & v1_rat_1(B) ) ) ) ),
    file(membered,cc14_membered),
    [interesting(0.9),axiom,file(membered,cc14_membered)]).

fof(cc16_membered,theorem,(
    ! [A] : 
      ( v1_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => v1_membered(B) ) ) ),
    file(membered,cc16_membered),
    [interesting(0.9),axiom,file(membered,cc16_membered)]).

fof(cc17_membered,theorem,(
    ! [A] : 
      ( v2_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( v1_membered(B)
            & v2_membered(B) ) ) ) ),
    file(membered,cc17_membered),
    [interesting(0.9),axiom,file(membered,cc17_membered)]).

fof(cc18_membered,theorem,(
    ! [A] : 
      ( v3_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( v1_membered(B)
            & v2_membered(B)
            & v3_membered(B) ) ) ) ),
    file(membered,cc18_membered),
    [interesting(0.9),axiom,file(membered,cc18_membered)]).

fof(cc19_membered,theorem,(
    ! [A] : 
      ( v4_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( v1_membered(B)
            & v2_membered(B)
            & v3_membered(B)
            & v4_membered(B) ) ) ) ),
    file(membered,cc19_membered),
    [interesting(0.9),axiom,file(membered,cc19_membered)]).

fof(cc1_membered,theorem,(
    ! [A] : 
      ( v5_membered(A)
     => v4_membered(A) ) ),
    file(membered,cc1_membered),
    [interesting(0.9),axiom,file(membered,cc1_membered)]).

fof(cc20_membered,theorem,(
    ! [A] : 
      ( v5_membered(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( v1_membered(B)
            & v2_membered(B)
            & v3_membered(B)
            & v4_membered(B)
            & v5_membered(B) ) ) ) ),
    file(membered,cc20_membered),
    [interesting(0.9),axiom,file(membered,cc20_membered)]).

fof(cc2_membered,theorem,(
    ! [A] : 
      ( v4_membered(A)
     => v3_membered(A) ) ),
    file(membered,cc2_membered),
    [interesting(0.9),axiom,file(membered,cc2_membered)]).

fof(cc3_membered,theorem,(
    ! [A] : 
      ( v3_membered(A)
     => v2_membered(A) ) ),
    file(membered,cc3_membered),
    [interesting(0.9),axiom,file(membered,cc3_membered)]).

fof(cc4_membered,theorem,(
    ! [A] : 
      ( v2_membered(A)
     => v1_membered(A) ) ),
    file(membered,cc4_membered),
    [interesting(0.9),axiom,file(membered,cc4_membered)]).

fof(fc6_membered,theorem,
    ( v1_xboole_0(k1_xboole_0)
    & v1_membered(k1_xboole_0)
    & v2_membered(k1_xboole_0)
    & v3_membered(k1_xboole_0)
    & v4_membered(k1_xboole_0)
    & v5_membered(k1_xboole_0) ),
    file(membered,fc6_membered),
    [interesting(0.9),axiom,file(membered,fc6_membered)]).

fof(rc1_membered,theorem,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_membered(A)
      & v2_membered(A)
      & v3_membered(A)
      & v4_membered(A)
      & v5_membered(A) ) ),
    file(membered,rc1_membered),
    [interesting(0.9),axiom,file(membered,rc1_membered)]).

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(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(cc15_membered,theorem,(
    ! [A] : 
      ( v1_xboole_0(A)
     => ( v1_membered(A)
        & v2_membered(A)
        & v3_membered(A)
        & v4_membered(A)
        & v5_membered(A) ) ) ),
    file(membered,cc15_membered),
    [interesting(0.9),axiom,file(membered,cc15_membered)]).

fof(cc1_eqrel_1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ~ v1_xboole_0(B) ) ) ),
    file(eqrel_1,cc1_eqrel_1),
    [interesting(0.9),axiom,file(eqrel_1,cc1_eqrel_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(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(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(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_eqrel_1,axiom,(
    ! [A] : 
    ? [B] : m1_eqrel_1(B,A) ),
    file(eqrel_1,m1_eqrel_1),
    [interesting(0.9),axiom,file(eqrel_1,m1_eqrel_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_k1_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k1_zfmisc_1),
    [interesting(0.9),axiom,file(zfmisc_1,k1_zfmisc_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),
    [interesting(0.9),axiom,file(eqrel_1,m1_eqrel_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(dt_c1_5__partit1,assumption,(
    ~ v1_xboole_0(c1_5__partit1) ),
    introduced(assumption,[file(partit1,c1_5__partit1)]),
    [interesting(0.8),axiom,file(partit1,c1_5__partit1)]).

fof(cc2_eqrel_1,theorem,(
    ! [A,B] : 
      ( m1_eqrel_1(B,A)
     => v1_setfam_1(B) ) ),
    file(eqrel_1,cc2_eqrel_1),
    [interesting(0.9),axiom,file(eqrel_1,cc2_eqrel_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(reflexivity_r1_tarski,theorem,(
    ! [A,B] : r1_tarski(A,A) ),
    file(tarski,r1_tarski),
    [interesting(0.9),axiom,file(tarski,r1_tarski)]).

fof(dt_c2_5__partit1,assumption,(
    m1_eqrel_1(c2_5__partit1,c1_5__partit1) ),
    introduced(assumption,[file(partit1,c2_5__partit1)]),
    [interesting(0.8),axiom,file(partit1,c2_5__partit1)]).

fof(dt_c3_5__partit1,assumption,(
    m1_eqrel_1(c3_5__partit1,c1_5__partit1) ),
    introduced(assumption,[file(partit1,c3_5__partit1)]),
    [interesting(0.8),axiom,file(partit1,c3_5__partit1)]).

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(reflexivity_r1_setfam_1,theorem,(
    ! [A,B] : r1_setfam_1(A,A) ),
    file(setfam_1,r1_setfam_1),
    [interesting(0.9),axiom,file(setfam_1,r1_setfam_1)]).

fof(t4_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ! [C] : 
              ( m1_eqrel_1(C,A)
             => ( ( r1_setfam_1(C,B)
                  & r1_setfam_1(B,C) )
               => r1_tarski(C,B) ) ) ) ) ),
    file(partit1,t4_partit1),
    [interesting(0.9),axiom,file(partit1,t4_partit1)]).

fof(e3_5__partit1,plain,(
    r1_tarski(c2_5__partit1,c3_5__partit1) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__partit1,dt_c2_5__partit1,dt_c3_5__partit1,e1_5__partit1])],[antisymmetry_r2_hidden,existence_m1_subset_1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_m1_subset_1,dt_c1_5__partit1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_membered,cc4_membered,fc1_subset_1,fc6_membered,rc1_membered,rc1_subset_1,rc2_subset_1,t1_subset,t2_subset,t4_subset,t5_subset,t8_boole,reflexivity_r1_setfam_1,reflexivity_r1_tarski,existence_m1_eqrel_1,dt_m1_eqrel_1,dt_c2_5__partit1,dt_c3_5__partit1,cc15_membered,cc1_eqrel_1,cc2_eqrel_1,t3_subset,t6_boole,t7_boole,e1_5__partit1,t4_partit1]),
    [interesting(0.8),file(partit1,e3_5__partit1),[file(partit1,e3_5__partit1)]]).

fof(e2_5__partit1,plain,(
    r1_tarski(c3_5__partit1,c2_5__partit1) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__partit1,dt_c2_5__partit1,dt_c3_5__partit1,e1_5__partit1])],[antisymmetry_r2_hidden,existence_m1_subset_1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_m1_subset_1,dt_c1_5__partit1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_membered,cc4_membered,fc1_subset_1,fc6_membered,rc1_membered,rc1_subset_1,rc2_subset_1,t1_subset,t2_subset,t4_subset,t5_subset,t8_boole,reflexivity_r1_setfam_1,reflexivity_r1_tarski,existence_m1_eqrel_1,dt_m1_eqrel_1,dt_c2_5__partit1,dt_c3_5__partit1,cc15_membered,cc1_eqrel_1,cc2_eqrel_1,t3_subset,t6_boole,t7_boole,e1_5__partit1,t4_partit1]),
    [interesting(0.8),file(partit1,e2_5__partit1),[file(partit1,e2_5__partit1)]]).

fof(d10_xboole_0,definition,(
    ! [A,B] : 
      ( A = B
    <=> ( r1_tarski(A,B)
        & r1_tarski(B,A) ) ) ),
    file(xboole_0,d10_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,d10_xboole_0)]).

fof(e4_5__partit1,plain,(
    c2_5__partit1 = c3_5__partit1 ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__partit1,dt_c2_5__partit1,dt_c3_5__partit1,e1_5__partit1])],[antisymmetry_r2_hidden,dt_k1_xboole_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_membered,cc4_membered,fc6_membered,rc1_membered,t1_subset,t4_subset,t5_subset,cc15_membered,cc1_eqrel_1,rc1_subset_1,rc2_subset_1,t2_subset,t6_boole,t7_boole,t8_boole,existence_m1_eqrel_1,existence_m1_subset_1,dt_k1_zfmisc_1,dt_m1_eqrel_1,dt_m1_subset_1,dt_c1_5__partit1,cc2_eqrel_1,fc1_subset_1,reflexivity_r1_tarski,dt_c2_5__partit1,dt_c3_5__partit1,t3_subset,e3_5__partit1,e2_5__partit1,d10_xboole_0]),
    [interesting(0.8),file(partit1,e4_5__partit1),[file(partit1,e4_5__partit1)]]).

fof(i4_5__partit1,theorem,(
    $true ),
    introduced(tautology,[file(partit1,i4_5__partit1)]),
    [interesting(0.8),trivial,file(partit1,i4_5__partit1)]).

fof(i3_5__partit1,plain,(
    c2_5__partit1 = c3_5__partit1 ),
    inference(conclusion,[status(thm),assumptions([dt_c1_5__partit1,dt_c2_5__partit1,dt_c3_5__partit1,e1_5__partit1])],[e4_5__partit1,i4_5__partit1]),
    [interesting(0.8),file(partit1,i3_5__partit1),[file(partit1,i3_5__partit1)]]).

fof(i2_5__partit1,plain,
    ( ( r1_setfam_1(c3_5__partit1,c2_5__partit1)
      & r1_setfam_1(c2_5__partit1,c3_5__partit1) )
   => c2_5__partit1 = c3_5__partit1 ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_5__partit1,dt_c2_5__partit1,dt_c3_5__partit1]),discharge_asm(discharge,[e1_5__partit1])],[e1_5__partit1,i3_5__partit1]),
    [interesting(0.8),file(partit1,i2_5__partit1),[file(partit1,i2_5__partit1)]]).

fof(i2_5_tmp__partit1,plain,
    ( ( m1_eqrel_1(c2_5__partit1,c1_5__partit1)
      & m1_eqrel_1(c3_5__partit1,c1_5__partit1) )
   => ( ( r1_setfam_1(c3_5__partit1,c2_5__partit1)
        & r1_setfam_1(c2_5__partit1,c3_5__partit1) )
     => c2_5__partit1 = c3_5__partit1 ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_5__partit1]),discharge_asm(discharge,[dt_c2_5__partit1,dt_c3_5__partit1])],[dt_c2_5__partit1,dt_c3_5__partit1,i2_5__partit1]),
    [interesting(0.8),i1_5__partit1]).

fof(i1_5__partit1,plain,(
    ! [A] : 
      ( m1_eqrel_1(A,c1_5__partit1)
     => ! [B] : 
          ( m1_eqrel_1(B,c1_5__partit1)
         => ( ( r1_setfam_1(B,A)
              & r1_setfam_1(A,B) )
           => A = B ) ) ) ),
    inference(let,[status(thm),assumptions([dt_c1_5__partit1])],[i2_5_tmp__partit1,dh_c2_5__partit1,dh_c3_5__partit1]),
    [interesting(0.8),file(partit1,i1_5__partit1),[file(partit1,i1_5__partit1)]]).

fof(i1_5_tmp__partit1,plain,
    ( ~ v1_xboole_0(c1_5__partit1)
   => ! [A] : 
        ( m1_eqrel_1(A,c1_5__partit1)
       => ! [B] : 
            ( m1_eqrel_1(B,c1_5__partit1)
           => ( ( r1_setfam_1(B,A)
                & r1_setfam_1(A,B) )
             => A = B ) ) ) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_5__partit1])],[dt_c1_5__partit1,i1_5__partit1]),
    [interesting(1),t5_partit1]).

fof(t5_partit1,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( m1_eqrel_1(B,A)
         => ! [C] : 
              ( m1_eqrel_1(C,A)
             => ( ( r1_setfam_1(C,B)
                  & r1_setfam_1(B,C) )
               => B = C ) ) ) ) ),
    inference(let,[status(thm),assumptions([])],[i1_5_tmp__partit1,dh_c1_5__partit1]),
    [interesting(1),file(partit1,t5_partit1),[file(partit1,t5_partit1)]]).