% Mizar ND problem: t1_pboole,pboole,34,40 
fof(dh_c1_2__pboole,definition,
    ( ( ( v1_relat_1(c1_2__pboole)
        & v1_funct_1(c1_2__pboole) )
     => ( v2_relat_1(c1_2__pboole)
       => v1_setfam_1(k2_relat_1(c1_2__pboole)) ) )
   => ! [A] : 
        ( ( v1_relat_1(A)
          & v1_funct_1(A) )
       => ( v2_relat_1(A)
         => v1_setfam_1(k2_relat_1(A)) ) ) ),
    introduced(definition,[new_symbol(c1_2__pboole),file(pboole,c1_2__pboole)]),
    [interesting(0.8),axiom,file(pboole,c1_2__pboole)]).

fof(e1_2__pboole,assumption,(
    v2_relat_1(c1_2__pboole) ),
    introduced(assumption,[file(pboole,e1_2__pboole)]),
    [interesting(0.8),axiom,file(pboole,e1_2__pboole)]).

fof(cc1_finseq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finset_1(A) ) ) ),
    file(finseq_1,cc1_finseq_1),
    [interesting(0.9),axiom,file(finseq_1,cc1_finseq_1)]).

fof(cc1_funct_1,theorem,(
    ! [A] : 
      ( v1_xboole_0(A)
     => v1_funct_1(A) ) ),
    file(funct_1,cc1_funct_1),
    [interesting(0.9),axiom,file(funct_1,cc1_funct_1)]).

fof(cc2_funct_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_xboole_0(A)
        & v1_funct_1(A) )
     => ( v1_relat_1(A)
        & v1_funct_1(A)
        & v2_funct_1(A) ) ) ),
    file(funct_1,cc2_funct_1),
    [interesting(0.9),axiom,file(funct_1,cc2_funct_1)]).

fof(fc11_finseq_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_finset_1(A) )
     => v1_finset_1(k2_relat_1(A)) ) ),
    file(finseq_1,fc11_finseq_1),
    [interesting(0.9),axiom,file(finseq_1,fc11_finseq_1)]).

fof(rc1_finseq_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v1_finseq_1(A) ) ),
    file(finseq_1,rc1_finseq_1),
    [interesting(0.9),axiom,file(finseq_1,rc1_finseq_1)]).

fof(rc1_funct_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A) ) ),
    file(funct_1,rc1_funct_1),
    [interesting(0.9),axiom,file(funct_1,rc1_funct_1)]).

fof(rc1_pboole,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v3_relat_1(A)
      & v1_funct_1(A) ) ),
    file(pboole,rc1_pboole),
    [interesting(0.9),axiom,file(pboole,rc1_pboole)]).

fof(rc2_funct_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_xboole_0(A)
      & v1_funct_1(A) ) ),
    file(funct_1,rc2_funct_1),
    [interesting(0.9),axiom,file(funct_1,rc2_funct_1)]).

fof(rc3_finseq_1,theorem,(
    ? [A] : 
      ( v1_xboole_0(A)
      & v1_relat_1(A)
      & v1_funct_1(A)
      & v2_funct_1(A)
      & v1_finset_1(A)
      & v1_finseq_1(A) ) ),
    file(finseq_1,rc3_finseq_1),
    [interesting(0.9),axiom,file(finseq_1,rc3_finseq_1)]).

fof(rc3_funct_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v2_funct_1(A) ) ),
    file(funct_1,rc3_funct_1),
    [interesting(0.9),axiom,file(funct_1,rc3_funct_1)]).

fof(rc4_funct_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v3_relat_1(A)
      & v1_funct_1(A) ) ),
    file(funct_1,rc4_funct_1),
    [interesting(0.9),axiom,file(funct_1,rc4_funct_1)]).

fof(rc6_finseq_1,theorem,(
    ? [A] : 
      ( v1_xboole_0(A)
      & v1_relat_1(A)
      & v1_funct_1(A)
      & v2_funct_1(A)
      & v1_finset_1(A) ) ),
    file(finseq_1,rc6_finseq_1),
    [interesting(0.9),axiom,file(finseq_1,rc6_finseq_1)]).

fof(rc7_finseq_1,theorem,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_relat_1(A)
      & v1_funct_1(A)
      & v1_finset_1(A) ) ),
    file(finseq_1,rc7_finseq_1),
    [interesting(0.9),axiom,file(finseq_1,rc7_finseq_1)]).

fof(rc8_finseq_1,theorem,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_relat_1(A)
      & v1_funct_1(A)
      & v1_finset_1(A)
      & v1_finseq_1(A) ) ),
    file(finseq_1,rc8_finseq_1),
    [interesting(0.9),axiom,file(finseq_1,rc8_finseq_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(dt_k2_relat_1,axiom,(
    $true ),
    file(relat_1,k2_relat_1),
    [interesting(0.9),axiom,file(relat_1,k2_relat_1)]).

fof(dt_c1_2__pboole,assumption,
    ( v1_relat_1(c1_2__pboole)
    & v1_funct_1(c1_2__pboole) ),
    introduced(assumption,[file(pboole,c1_2__pboole)]),
    [interesting(0.8),axiom,file(pboole,c1_2__pboole)]).

fof(fc2_finseq_1,theorem,
    ( v1_xboole_0(k1_xboole_0)
    & 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_finset_1(k1_xboole_0)
    & v1_finseq_1(k1_xboole_0) ),
    file(finseq_1,fc2_finseq_1),
    [interesting(0.9),axiom,file(finseq_1,fc2_finseq_1)]).

fof(d9_setfam_1,definition,(
    ! [A] : 
      ( v1_setfam_1(A)
    <=> ~ r2_hidden(k1_xboole_0,A) ) ),
    file(setfam_1,d9_setfam_1),
    [interesting(0.9),axiom,file(setfam_1,d9_setfam_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_m1_subset_1,axiom,(
    $true ),
    file(subset_1,m1_subset_1),
    [interesting(0.9),axiom,file(subset_1,m1_subset_1)]).

fof(fc6_funct_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v2_relat_1(A)
        & v1_funct_1(A) )
     => v1_setfam_1(k2_relat_1(A)) ) ),
    file(funct_1,fc6_funct_1),
    [interesting(0.9),axiom,file(funct_1,fc6_funct_1)]).

fof(rc5_funct_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v2_relat_1(A)
      & v1_funct_1(A) ) ),
    file(funct_1,rc5_funct_1),
    [interesting(0.9),axiom,file(funct_1,rc5_funct_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(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(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(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(d9_relat_1,definition,(
    ! [A] : 
      ( v1_relat_1(A)
     => ( v2_relat_1(A)
      <=> ~ r2_hidden(k1_xboole_0,k2_relat_1(A)) ) ) ),
    file(relat_1,d9_relat_1),
    [interesting(0.9),axiom,file(relat_1,d9_relat_1)]).

fof(e2_2__pboole,plain,(
    ~ r2_hidden(k1_xboole_0,k2_relat_1(c1_2__pboole)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_2__pboole,e1_2__pboole])],[existence_m1_subset_1,dt_m1_subset_1,cc1_finseq_1,cc1_funct_1,cc2_funct_1,fc11_finseq_1,fc6_funct_1,rc1_finseq_1,rc1_funct_1,rc1_pboole,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc5_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,t2_subset,t8_boole,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k2_relat_1,dt_c1_2__pboole,fc2_finseq_1,t1_subset,t6_boole,t7_boole,e1_2__pboole,d9_relat_1]),
    [interesting(0.8),file(pboole,e2_2__pboole),[file(pboole,e2_2__pboole)]]).

fof(i3_2__pboole,theorem,(
    $true ),
    introduced(tautology,[file(pboole,i3_2__pboole)]),
    [interesting(0.8),trivial,file(pboole,i3_2__pboole)]).

fof(i2_2__pboole,plain,(
    v1_setfam_1(k2_relat_1(c1_2__pboole)) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_2__pboole,e1_2__pboole])],[cc1_finseq_1,cc1_funct_1,cc2_funct_1,fc11_finseq_1,rc1_finseq_1,rc1_funct_1,rc1_pboole,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k2_relat_1,dt_c1_2__pboole,fc2_finseq_1,d9_setfam_1,e2_2__pboole,i3_2__pboole]),
    [interesting(0.8),file(pboole,i2_2__pboole),[file(pboole,i2_2__pboole)]]).

fof(i1_2__pboole,plain,
    ( v2_relat_1(c1_2__pboole)
   => v1_setfam_1(k2_relat_1(c1_2__pboole)) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_2__pboole]),discharge_asm(discharge,[e1_2__pboole])],[e1_2__pboole,i2_2__pboole]),
    [interesting(0.8),file(pboole,i1_2__pboole),[file(pboole,i1_2__pboole)]]).

fof(i1_2_tmp__pboole,plain,
    ( ( v1_relat_1(c1_2__pboole)
      & v1_funct_1(c1_2__pboole) )
   => ( v2_relat_1(c1_2__pboole)
     => v1_setfam_1(k2_relat_1(c1_2__pboole)) ) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_2__pboole])],[dt_c1_2__pboole,i1_2__pboole]),
    [interesting(1),t1_pboole]).

fof(t1_pboole,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v2_relat_1(A)
       => v1_setfam_1(k2_relat_1(A)) ) ) ),
    inference(let,[status(thm),assumptions([])],[i1_2_tmp__pboole,dh_c1_2__pboole]),
    [interesting(1),file(pboole,t1_pboole),[file(pboole,t1_pboole)]]).