% Mizar ND problem: t2_cantor_1,cantor_1,95,56 
fof(dh_c1_5__cantor_1,definition,
    ( ( l1_pre_topc(c1_5__cantor_1)
     => m1_cantor_1(u1_pre_topc(c1_5__cantor_1),c1_5__cantor_1) )
   => ! [A] : 
        ( l1_pre_topc(A)
       => m1_cantor_1(u1_pre_topc(A),A) ) ),
    introduced(definition,[new_symbol(c1_5__cantor_1),file(cantor_1,c1_5__cantor_1)]),
    [interesting(0.8),axiom,file(cantor_1,c1_5__cantor_1)]).

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(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(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(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(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(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(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_l1_struct_0,axiom,(
    ? [A] : l1_struct_0(A) ),
    file(struct_0,l1_struct_0),
    [interesting(0.9),axiom,file(struct_0,l1_struct_0)]).

fof(dt_l1_struct_0,axiom,(
    $true ),
    file(struct_0,l1_struct_0),
    [interesting(0.9),axiom,file(struct_0,l1_struct_0)]).

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(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(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(existence_l1_pre_topc,axiom,(
    ? [A] : l1_pre_topc(A) ),
    file(pre_topc,l1_pre_topc),
    [interesting(0.9),axiom,file(pre_topc,l1_pre_topc)]).

fof(existence_m1_cantor_1,axiom,(
    ! [A] : 
      ( l1_pre_topc(A)
     => ? [B] : m1_cantor_1(B,A) ) ),
    file(cantor_1,m1_cantor_1),
    [interesting(0.9),axiom,file(cantor_1,m1_cantor_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_cantor_1,axiom,(
    ! [A,B] : 
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => m1_subset_1(k1_cantor_1(A,B),k1_zfmisc_1(k1_zfmisc_1(A))) ) ),
    file(cantor_1,k1_cantor_1),
    [interesting(0.9),axiom,file(cantor_1,k1_cantor_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_l1_pre_topc,axiom,(
    ! [A] : 
      ( l1_pre_topc(A)
     => l1_struct_0(A) ) ),
    file(pre_topc,l1_pre_topc),
    [interesting(0.9),axiom,file(pre_topc,l1_pre_topc)]).

fof(dt_m1_cantor_1,axiom,(
    ! [A] : 
      ( l1_pre_topc(A)
     => ! [B] : 
          ( m1_cantor_1(B,A)
         => m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ) ),
    file(cantor_1,m1_cantor_1),
    [interesting(0.9),axiom,file(cantor_1,m1_cantor_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_u1_pre_topc,axiom,(
    ! [A] : 
      ( l1_pre_topc(A)
     => m1_subset_1(u1_pre_topc(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ),
    file(pre_topc,u1_pre_topc),
    [interesting(0.9),axiom,file(pre_topc,u1_pre_topc)]).

fof(dt_u1_struct_0,axiom,(
    $true ),
    file(struct_0,u1_struct_0),
    [interesting(0.9),axiom,file(struct_0,u1_struct_0)]).

fof(dt_c1_5__cantor_1,assumption,(
    l1_pre_topc(c1_5__cantor_1) ),
    introduced(assumption,[file(cantor_1,c1_5__cantor_1)]),
    [interesting(0.8),axiom,file(cantor_1,c1_5__cantor_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_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(t1_cantor_1,theorem,(
    ! [A,B] : 
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => r1_tarski(B,k1_cantor_1(A,B)) ) ),
    file(cantor_1,t1_cantor_1),
    [interesting(0.9),axiom,file(cantor_1,t1_cantor_1)]).

fof(e1_5__cantor_1,plain,(
    r1_tarski(u1_pre_topc(c1_5__cantor_1),k1_cantor_1(u1_struct_0(c1_5__cantor_1),u1_pre_topc(c1_5__cantor_1))) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__cantor_1])],[antisymmetry_r2_hidden,dt_k1_xboole_0,cc2_finset_1,fc1_xboole_0,rc1_finset_1,rc3_finset_1,rc4_finset_1,t1_subset,t4_subset,t5_subset,t8_boole,existence_l1_pre_topc,existence_l1_struct_0,dt_l1_pre_topc,dt_l1_struct_0,cc1_finset_1,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,t2_subset,t6_boole,t7_boole,reflexivity_r1_tarski,existence_m1_subset_1,dt_k1_cantor_1,dt_k1_zfmisc_1,dt_m1_subset_1,dt_u1_pre_topc,dt_u1_struct_0,dt_c1_5__cantor_1,fc1_subset_1,t3_subset,t1_cantor_1]),
    [interesting(0.8),file(cantor_1,e1_5__cantor_1),[file(cantor_1,e1_5__cantor_1)]]).

fof(d2_cantor_1,definition,(
    ! [A] : 
      ( l1_pre_topc(A)
     => ! [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( m1_cantor_1(B,A)
          <=> ( r1_tarski(B,u1_pre_topc(A))
              & r1_tarski(u1_pre_topc(A),k1_cantor_1(u1_struct_0(A),B)) ) ) ) ) ),
    file(cantor_1,d2_cantor_1),
    [interesting(0.9),axiom,file(cantor_1,d2_cantor_1)]).

fof(e2_5__cantor_1,plain,(
    m1_cantor_1(u1_pre_topc(c1_5__cantor_1),c1_5__cantor_1) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__cantor_1])],[antisymmetry_r2_hidden,dt_k1_xboole_0,cc2_finset_1,fc1_xboole_0,rc1_finset_1,rc3_finset_1,rc4_finset_1,t1_subset,t4_subset,t5_subset,t8_boole,existence_l1_struct_0,dt_l1_struct_0,cc1_finset_1,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,t2_subset,t6_boole,t7_boole,reflexivity_r1_tarski,existence_l1_pre_topc,existence_m1_cantor_1,existence_m1_subset_1,dt_k1_cantor_1,dt_k1_zfmisc_1,dt_l1_pre_topc,dt_m1_cantor_1,dt_m1_subset_1,dt_u1_pre_topc,dt_u1_struct_0,dt_c1_5__cantor_1,fc1_subset_1,t3_subset,e1_5__cantor_1,d2_cantor_1]),
    [interesting(0.8),file(cantor_1,e2_5__cantor_1),[file(cantor_1,e2_5__cantor_1)]]).

fof(i2_5__cantor_1,theorem,(
    $true ),
    introduced(tautology,[file(cantor_1,i2_5__cantor_1)]),
    [interesting(0.8),trivial,file(cantor_1,i2_5__cantor_1)]).

fof(i1_5__cantor_1,plain,(
    m1_cantor_1(u1_pre_topc(c1_5__cantor_1),c1_5__cantor_1) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_5__cantor_1])],[e2_5__cantor_1,i2_5__cantor_1]),
    [interesting(0.8),file(cantor_1,i1_5__cantor_1),[file(cantor_1,i1_5__cantor_1)]]).

fof(i1_5_tmp__cantor_1,plain,
    ( l1_pre_topc(c1_5__cantor_1)
   => m1_cantor_1(u1_pre_topc(c1_5__cantor_1),c1_5__cantor_1) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_5__cantor_1])],[dt_c1_5__cantor_1,i1_5__cantor_1]),
    [interesting(1),t2_cantor_1]).

fof(t2_cantor_1,theorem,(
    ! [A] : 
      ( l1_pre_topc(A)
     => m1_cantor_1(u1_pre_topc(A),A) ) ),
    inference(let,[status(thm),assumptions([])],[i1_5_tmp__cantor_1,dh_c1_5__cantor_1]),
    [interesting(1),file(cantor_1,t2_cantor_1),[file(cantor_1,t2_cantor_1)]]).