% Mizar ND problem: t9_membered,membered,325,14 
fof(dh_c1_20__membered,definition,
    ( ( ( ~ v1_xboole_0(c1_20__membered)
        & v4_membered(c1_20__membered) )
     => ? [A] : 
          ( v1_int_1(A)
          & r2_hidden(A,c1_20__membered) ) )
   => ! [B] : 
        ( ( ~ v1_xboole_0(B)
          & v4_membered(B) )
       => ? [C] : 
            ( v1_int_1(C)
            & r2_hidden(C,B) ) ) ),
    introduced(definition,[new_symbol(c1_20__membered),file(membered,c1_20__membered)]),
    [interesting(0.8),axiom,file(membered,c1_20__membered)]).

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(cc4_membered,theorem,(
    ! [A] : 
      ( v2_membered(A)
     => v1_membered(A) ) ),
    file(membered,cc4_membered),
    [interesting(0.9),axiom,file(membered,cc4_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(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(cc3_membered,theorem,(
    ! [A] : 
      ( v3_membered(A)
     => v2_membered(A) ) ),
    file(membered,cc3_membered),
    [interesting(0.9),axiom,file(membered,cc3_membered)]).

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(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(cc1_rat_1,theorem,(
    ! [A] : 
      ( v1_rat_1(A)
     => ( v1_xcmplx_0(A)
        & v1_xreal_0(A) ) ) ),
    file(rat_1,cc1_rat_1),
    [interesting(0.9),axiom,file(rat_1,cc1_rat_1)]).

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

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(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_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(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(rc2_rat_1,theorem,(
    ? [A] : v1_rat_1(A) ),
    file(rat_1,rc2_rat_1),
    [interesting(0.9),axiom,file(rat_1,rc2_rat_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(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_c1_20__membered,assumption,
    ( ~ v1_xboole_0(c1_20__membered)
    & v4_membered(c1_20__membered) ),
    introduced(assumption,[file(membered,c1_20__membered)]),
    [interesting(0.8),axiom,file(membered,c1_20__membered)]).

fof(dh_c2_20__membered,definition,
    ( ? [A] : r2_hidden(A,c1_20__membered)
   => r2_hidden(c2_20__membered,c1_20__membered) ),
    introduced(definition,[new_symbol(c2_20__membered),file(membered,c2_20__membered)]),
    [interesting(0.8),axiom,file(membered,c2_20__membered)]).

fof(cc2_rat_1,theorem,(
    ! [A] : 
      ( v1_int_1(A)
     => ( v1_xcmplx_0(A)
        & v1_xreal_0(A)
        & v1_rat_1(A) ) ) ),
    file(rat_1,cc2_rat_1),
    [interesting(0.9),axiom,file(rat_1,cc2_rat_1)]).

fof(cc4_int_1,theorem,(
    ! [A] : 
      ( v1_int_1(A)
     => ( v1_xcmplx_0(A)
        & v1_xreal_0(A) ) ) ),
    file(int_1,cc4_int_1),
    [interesting(0.9),axiom,file(int_1,cc4_int_1)]).

fof(rc2_int_1,theorem,(
    ? [A] : v1_int_1(A) ),
    file(int_1,rc2_int_1),
    [interesting(0.9),axiom,file(int_1,rc2_int_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(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(d1_xboole_0,definition,(
    ! [A] : 
      ( A = k1_xboole_0
    <=> ! [B] : ~ r2_hidden(B,A) ) ),
    file(xboole_0,d1_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,d1_xboole_0)]).

fof(e1_20__membered,plain,(
    ? [A] : r2_hidden(A,c1_20__membered) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_20__membered])],[cc10_membered,cc11_membered,cc3_xreal_0,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,cc12_membered,cc1_rat_1,cc2_rat_1,cc2_xreal_0,cc3_membered,cc4_int_1,cc7_xreal_0,rc1_xreal_0,rc2_int_1,rc2_rat_1,existence_m1_subset_1,dt_m1_subset_1,cc13_membered,cc2_membered,rc1_xboole_0,rc2_xboole_0,t2_subset,t8_boole,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_c1_20__membered,fc1_xboole_0,t1_subset,t6_boole,t7_boole,d1_xboole_0]),
    [interesting(0.8),file(membered,e1_20__membered),[file(membered,e1_20__membered)]]).

fof(dt_c2_20__membered,plain,(
    $true ),
    inference(consider,[status(thm),assumptions([dt_c1_20__membered])],[dh_c2_20__membered,e1_20__membered]),
    [interesting(0.8),file(membered,c2_20__membered),[file(membered,c2_20__membered)]]).

fof(e2_20__membered,plain,(
    r2_hidden(c2_20__membered,c1_20__membered) ),
    inference(consider,[status(thm),assumptions([dt_c1_20__membered])],[dh_c2_20__membered,e1_20__membered]),
    [interesting(0.8),file(membered,e2_20__membered),[file(membered,e2_20__membered)]]).

fof(d4_membered,definition,(
    ! [A] : 
      ( v4_membered(A)
    <=> ! [B] : 
          ( r2_hidden(B,A)
         => v1_int_1(B) ) ) ),
    file(membered,d4_membered),
    [interesting(0.9),axiom,file(membered,d4_membered)]).

fof(e3_20__membered,plain,(
    v1_int_1(c2_20__membered) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_20__membered])],[cc10_membered,dt_k1_xboole_0,cc11_membered,cc3_xreal_0,cc4_membered,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,cc12_membered,cc13_membered,cc1_rat_1,cc2_xreal_0,cc3_membered,cc7_xreal_0,rc1_xboole_0,rc1_xreal_0,rc2_rat_1,rc2_xboole_0,t2_subset,t6_boole,antisymmetry_r2_hidden,dt_c1_20__membered,dt_c2_20__membered,cc2_membered,cc2_rat_1,cc4_int_1,rc2_int_1,t1_subset,t7_boole,e2_20__membered,d4_membered]),
    [interesting(0.8),file(membered,e3_20__membered),[file(membered,e3_20__membered)]]).

fof(e4_20__membered,plain,(
    ? [A] : 
      ( v1_int_1(A)
      & r2_hidden(A,c1_20__membered) ) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_20__membered])],[cc10_membered,cc11_membered,cc4_membered,dt_k1_xboole_0,cc12_membered,cc3_membered,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,cc13_membered,cc1_rat_1,cc2_membered,cc2_xreal_0,cc7_xreal_0,rc1_xboole_0,rc1_xreal_0,rc2_rat_1,rc2_xboole_0,t2_subset,t6_boole,antisymmetry_r2_hidden,dt_c1_20__membered,dt_c2_20__membered,cc2_rat_1,cc4_int_1,rc2_int_1,t1_subset,t7_boole,e3_20__membered,e2_20__membered]),
    [interesting(0.8),file(membered,e4_20__membered),[file(membered,e4_20__membered)]]).

fof(i2_20__membered,theorem,(
    $true ),
    introduced(tautology,[file(membered,i2_20__membered)]),
    [interesting(0.8),trivial,file(membered,i2_20__membered)]).

fof(i1_20__membered,plain,(
    ? [A] : 
      ( v1_int_1(A)
      & r2_hidden(A,c1_20__membered) ) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_20__membered])],[e4_20__membered,i2_20__membered]),
    [interesting(0.8),file(membered,i1_20__membered),[file(membered,i1_20__membered)]]).

fof(i1_20_tmp__membered,plain,
    ( ( ~ v1_xboole_0(c1_20__membered)
      & v4_membered(c1_20__membered) )
   => ? [A] : 
        ( v1_int_1(A)
        & r2_hidden(A,c1_20__membered) ) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_20__membered])],[dt_c1_20__membered,i1_20__membered]),
    [interesting(1),t9_membered]).

fof(t9_membered,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v4_membered(A) )
     => ? [B] : 
          ( v1_int_1(B)
          & r2_hidden(B,A) ) ) ),
    inference(let,[status(thm),assumptions([])],[i1_20_tmp__membered,dh_c1_20__membered]),
    [interesting(1),file(membered,t9_membered),[file(membered,t9_membered)]]).