% Mizar ND problem: t2_membered,membered,172,37 
fof(dh_c1_8__membered,definition,
    ( ( v2_membered(c1_8__membered)
     => r1_tarski(c1_8__membered,k1_numbers) )
   => ! [A] : 
        ( v2_membered(A)
       => r1_tarski(A,k1_numbers) ) ),
    introduced(definition,[new_symbol(c1_8__membered),file(membered,c1_8__membered)]),
    [interesting(0.8),axiom,file(membered,c1_8__membered)]).

fof(e1_8__membered,assumption,(
    v2_membered(c1_8__membered) ),
    introduced(assumption,[file(membered,e1_8__membered)]),
    [interesting(0.8),axiom,file(membered,e1_8__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(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(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(dt_k1_numbers,axiom,(
    $true ),
    file(numbers,k1_numbers),
    [interesting(0.9),axiom,file(numbers,k1_numbers)]).

fof(dt_c1_8__membered,assumption,(
    $true ),
    introduced(assumption,[file(membered,c1_8__membered)]),
    [interesting(0.8),axiom,file(membered,c1_8__membered)]).

fof(dt_c2_8__membered,assumption,(
    $true ),
    introduced(assumption,[file(membered,c2_8__membered)]),
    [interesting(0.8),axiom,file(membered,c2_8__membered)]).

fof(fc1_numbers,theorem,(
    ~ v1_xboole_0(k1_numbers) ),
    file(numbers,fc1_numbers),
    [interesting(0.9),axiom,file(numbers,fc1_numbers)]).

fof(fc2_membered,theorem,
    ( ~ v1_xboole_0(k1_numbers)
    & v1_membered(k1_numbers)
    & v2_membered(k1_numbers) ),
    file(membered,fc2_membered),
    [interesting(0.9),axiom,file(membered,fc2_membered)]).

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_8__membered,definition,
    ( ~ ( r2_hidden(c2_8__membered,c1_8__membered)
        & ~ r2_hidden(c2_8__membered,k1_numbers) )
   => ! [A] : 
        ~ ( r2_hidden(A,c1_8__membered)
          & ~ r2_hidden(A,k1_numbers) ) ),
    introduced(definition,[new_symbol(c2_8__membered),file(membered,c2_8__membered)]),
    [interesting(0.8),axiom,file(membered,c2_8__membered)]).

fof(e2_8__membered,assumption,(
    r2_hidden(c2_8__membered,c1_8__membered) ),
    introduced(assumption,[file(membered,e2_8__membered)]),
    [interesting(0.8),axiom,file(membered,e2_8__membered)]).

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

fof(cc3_xreal_0,theorem,(
    ! [A] : 
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A) )
     => ( ~ v1_xboole_0(A)
        & v1_xcmplx_0(A)
        & v1_xreal_0(A)
        & ~ v3_xreal_0(A) ) ) ),
    file(xreal_0,cc3_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,cc3_xreal_0)]).

fof(cc4_xreal_0,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_xreal_0(A)
        & ~ v3_xreal_0(A) )
     => ( v1_xcmplx_0(A)
        & v1_xreal_0(A)
        & v2_xreal_0(A) ) ) ),
    file(xreal_0,cc4_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,cc4_xreal_0)]).

fof(cc5_xreal_0,theorem,(
    ! [A] : 
      ( ( v1_xreal_0(A)
        & v3_xreal_0(A) )
     => ( ~ v1_xboole_0(A)
        & v1_xcmplx_0(A)
        & v1_xreal_0(A)
        & ~ v2_xreal_0(A) ) ) ),
    file(xreal_0,cc5_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,cc5_xreal_0)]).

fof(cc6_xreal_0,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_xreal_0(A)
        & ~ v2_xreal_0(A) )
     => ( v1_xcmplx_0(A)
        & v1_xreal_0(A)
        & v3_xreal_0(A) ) ) ),
    file(xreal_0,cc6_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,cc6_xreal_0)]).

fof(cc8_xreal_0,theorem,(
    ! [A] : 
      ( ( v1_xreal_0(A)
        & ~ v2_xreal_0(A)
        & ~ v3_xreal_0(A) )
     => ( v1_xboole_0(A)
        & v1_xcmplx_0(A)
        & v1_xreal_0(A) ) ) ),
    file(xreal_0,cc8_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,cc8_xreal_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(rc2_xreal_0,theorem,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_xcmplx_0(A)
      & v1_xreal_0(A)
      & v2_xreal_0(A)
      & ~ v3_xreal_0(A) ) ),
    file(xreal_0,rc2_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,rc2_xreal_0)]).

fof(rc3_xreal_0,theorem,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_xcmplx_0(A)
      & v1_xreal_0(A)
      & ~ v2_xreal_0(A)
      & v3_xreal_0(A) ) ),
    file(xreal_0,rc3_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,rc3_xreal_0)]).

fof(rc4_xreal_0,theorem,(
    ? [A] : 
      ( v1_xboole_0(A)
      & v1_xcmplx_0(A)
      & v1_xreal_0(A)
      & ~ v2_xreal_0(A)
      & ~ v3_xreal_0(A) ) ),
    file(xreal_0,rc4_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,rc4_xreal_0)]).

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_m1_subset_1,axiom,(
    $true ),
    file(subset_1,m1_subset_1),
    [interesting(0.9),axiom,file(subset_1,m1_subset_1)]).

fof(cc7_xreal_0,theorem,(
    ! [A] : 
      ( ( v1_xboole_0(A)
        & v1_xreal_0(A) )
     => ( v1_xcmplx_0(A)
        & v1_xreal_0(A)
        & ~ v2_xreal_0(A)
        & ~ v3_xreal_0(A) ) ) ),
    file(xreal_0,cc7_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,cc7_xreal_0)]).

fof(rc1_xreal_0,theorem,(
    ? [A] : 
      ( v1_xcmplx_0(A)
      & v1_xreal_0(A) ) ),
    file(xreal_0,rc1_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,rc1_xreal_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(cc2_xreal_0,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => v1_xcmplx_0(A) ) ),
    file(xreal_0,cc2_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,cc2_xreal_0)]).

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(d2_membered,definition,(
    ! [A] : 
      ( v2_membered(A)
    <=> ! [B] : 
          ( r2_hidden(B,A)
         => v1_xreal_0(B) ) ) ),
    file(membered,d2_membered),
    [interesting(0.9),axiom,file(membered,d2_membered)]).

fof(e3_8__membered,plain,(
    v1_xreal_0(c2_8__membered) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_8__membered,dt_c2_8__membered,e2_8__membered,e1_8__membered])],[dt_k1_xboole_0,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc1_xboole_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t8_boole,existence_m1_subset_1,dt_m1_subset_1,cc7_xreal_0,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,t2_subset,t6_boole,antisymmetry_r2_hidden,dt_c1_8__membered,dt_c2_8__membered,cc2_xreal_0,cc4_membered,t1_subset,t7_boole,e2_8__membered,e1_8__membered,d2_membered]),
    [interesting(0.8),file(membered,e3_8__membered),[file(membered,e3_8__membered)]]).

fof(d1_xreal_0,definition,(
    ! [A] : 
      ( v1_xreal_0(A)
    <=> r2_hidden(A,k1_numbers) ) ),
    file(xreal_0,d1_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,d1_xreal_0)]).

fof(e4_8__membered,plain,(
    r2_hidden(c2_8__membered,k1_numbers) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_8__membered,dt_c2_8__membered,e2_8__membered,e1_8__membered])],[dt_k1_xboole_0,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc1_xboole_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t8_boole,existence_m1_subset_1,dt_m1_subset_1,cc4_membered,cc7_xreal_0,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,t2_subset,t6_boole,antisymmetry_r2_hidden,dt_k1_numbers,dt_c2_8__membered,cc2_xreal_0,fc1_numbers,fc2_membered,t1_subset,t7_boole,e3_8__membered,d1_xreal_0]),
    [interesting(0.8),file(membered,e4_8__membered),[file(membered,e4_8__membered)]]).

fof(i5_8__membered,theorem,(
    $true ),
    introduced(tautology,[file(membered,i5_8__membered)]),
    [interesting(0.8),trivial,file(membered,i5_8__membered)]).

fof(i4_8__membered,plain,(
    r2_hidden(c2_8__membered,k1_numbers) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_8__membered,dt_c2_8__membered,e2_8__membered,e1_8__membered])],[e4_8__membered,i5_8__membered]),
    [interesting(0.8),file(membered,i4_8__membered),[file(membered,i4_8__membered)]]).

fof(i3_8__membered,plain,(
    ~ ( r2_hidden(c2_8__membered,c1_8__membered)
      & ~ r2_hidden(c2_8__membered,k1_numbers) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_8__membered,dt_c2_8__membered,e1_8__membered]),discharge_asm(discharge,[e2_8__membered])],[e2_8__membered,i4_8__membered]),
    [interesting(0.8),file(membered,i3_8__membered),[file(membered,i3_8__membered)]]).

fof(i3_8_tmp__membered,plain,(
    ~ ( r2_hidden(c2_8__membered,c1_8__membered)
      & ~ r2_hidden(c2_8__membered,k1_numbers) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_8__membered,e1_8__membered]),discharge_asm(discharge,[dt_c2_8__membered])],[dt_c2_8__membered,i3_8__membered]),
    [interesting(0.8),i2_8__membered]).

fof(i2_8__membered,plain,(
    r1_tarski(c1_8__membered,k1_numbers) ),
    inference(let,[status(thm),assumptions([dt_c1_8__membered,e1_8__membered])],[i3_8_tmp__membered,cc4_membered,rc1_xboole_0,rc2_xboole_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_numbers,dt_c1_8__membered,fc1_numbers,fc2_membered,d3_tarski,dh_c2_8__membered]),
    [interesting(0.8),file(membered,i2_8__membered),[file(membered,i2_8__membered)]]).

fof(i1_8__membered,plain,
    ( v2_membered(c1_8__membered)
   => r1_tarski(c1_8__membered,k1_numbers) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_8__membered]),discharge_asm(discharge,[e1_8__membered])],[e1_8__membered,i2_8__membered]),
    [interesting(0.8),file(membered,i1_8__membered),[file(membered,i1_8__membered)]]).

fof(i1_8_tmp__membered,plain,
    ( v2_membered(c1_8__membered)
   => r1_tarski(c1_8__membered,k1_numbers) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_8__membered])],[dt_c1_8__membered,i1_8__membered]),
    [interesting(1),t2_membered]).

fof(t2_membered,theorem,(
    ! [A] : 
      ( v2_membered(A)
     => r1_tarski(A,k1_numbers) ) ),
    inference(let,[status(thm),assumptions([])],[i1_8_tmp__membered,dh_c1_8__membered]),
    [interesting(1),file(membered,t2_membered),[file(membered,t2_membered)]]).