% Mizar problem: t23_cat_5,cat_5,1901,20 
fof(t23_cat_5,conjecture,(
    ! [A] : 
      ( ( v2_cat_1(A)
        & l1_cat_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_cat_1(A))
         => ! [C] : 
              ( m1_subset_1(C,u2_cat_1(A))
             => ( r2_hidden(C,k12_cat_5(A,B))
              <=> k3_cat_1(A,C) = B ) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,cc1_funct_1,cc1_realset1,cc1_relset_1,cc2_funct_1,d1_tarski,d3_cat_1,d9_cat_5,dt_k10_relat_1,dt_k12_cat_5,dt_k1_funct_1,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k3_cat_1,dt_k3_funct_2,dt_k6_domain_1,dt_k8_funct_2,dt_l1_cat_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_u1_cat_1,dt_u2_cat_1,dt_u4_cat_1,existence_l1_cat_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,fc1_cat_5,fc1_subset_1,fc1_xboole_0,fc2_subset_1,fc3_realset1,fc4_subset_1,rc1_funct_1,rc1_realset1,rc1_subset_1,rc1_xboole_0,rc2_cat_1,rc2_funct_1,rc2_realset1,rc2_subset_1,rc2_xboole_0,rc3_funct_1,redefinition_k3_funct_2,redefinition_k6_domain_1,redefinition_k8_funct_2,redefinition_m2_relset_1,reflexivity_r1_tarski,t1_subset,t2_subset,t3_subset,t46_funct_2,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole]),
    [file(cat_5,t23_cat_5)]).

fof(antisymmetry_r2_hidden,axiom,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => ~ r2_hidden(B,A) ) ),
    file(hidden,r2_hidden),
    []).

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

fof(cc1_realset1,axiom,(
    ! [A] : 
      ( ~ v1_realset1(A)
     => ~ v1_xboole_0(A) ) ),
    file(realset1,cc1_realset1),
    []).

fof(cc1_relset_1,axiom,(
    ! [A,B,C] : 
      ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
     => v1_relat_1(C) ) ),
    file(relset_1,cc1_relset_1),
    []).

fof(cc2_funct_1,axiom,(
    ! [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),
    []).

fof(d1_tarski,axiom,(
    ! [A,B] : 
      ( B = k1_tarski(A)
    <=> ! [C] : 
          ( r2_hidden(C,B)
        <=> C = A ) ) ),
    file(tarski,d1_tarski),
    []).

fof(d3_cat_1,axiom,(
    ! [A] : 
      ( l1_cat_1(A)
     => ! [B] : 
          ( m1_subset_1(B,u2_cat_1(A))
         => k3_cat_1(A,B) = k8_funct_2(u2_cat_1(A),u1_cat_1(A),u4_cat_1(A),B) ) ) ),
    file(cat_1,d3_cat_1),
    []).

fof(d9_cat_5,axiom,(
    ! [A] : 
      ( ( v2_cat_1(A)
        & l1_cat_1(A) )
     => ! [B] : 
          ( m1_subset_1(B,u1_cat_1(A))
         => k12_cat_5(A,B) = k3_funct_2(u2_cat_1(A),u1_cat_1(A),u4_cat_1(A),k6_domain_1(u1_cat_1(A),B)) ) ) ),
    file(cat_5,d9_cat_5),
    []).

fof(dt_k10_relat_1,axiom,(
    $true ),
    file(relat_1,k10_relat_1),
    []).

fof(dt_k12_cat_5,axiom,(
    ! [A,B] : 
      ( ( v2_cat_1(A)
        & l1_cat_1(A)
        & m1_subset_1(B,u1_cat_1(A)) )
     => m1_subset_1(k12_cat_5(A,B),k1_zfmisc_1(u2_cat_1(A))) ) ),
    file(cat_5,k12_cat_5),
    []).

fof(dt_k1_funct_1,axiom,(
    $true ),
    file(funct_1,k1_funct_1),
    []).

fof(dt_k1_tarski,axiom,(
    $true ),
    file(tarski,k1_tarski),
    []).

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

fof(dt_k1_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k1_zfmisc_1),
    []).

fof(dt_k2_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k2_zfmisc_1),
    []).

fof(dt_k3_cat_1,axiom,(
    ! [A,B] : 
      ( ( l1_cat_1(A)
        & m1_subset_1(B,u2_cat_1(A)) )
     => m1_subset_1(k3_cat_1(A,B),u1_cat_1(A)) ) ),
    file(cat_1,k3_cat_1),
    []).

fof(dt_k3_funct_2,axiom,(
    ! [A,B,C,D] : 
      ( ( v1_funct_1(C)
        & m1_relset_1(C,A,B) )
     => m1_subset_1(k3_funct_2(A,B,C,D),k1_zfmisc_1(A)) ) ),
    file(funct_2,k3_funct_2),
    []).

fof(dt_k6_domain_1,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A) )
     => m1_subset_1(k6_domain_1(A,B),k1_zfmisc_1(A)) ) ),
    file(domain_1,k6_domain_1),
    []).

fof(dt_k8_funct_2,axiom,(
    ! [A,B,C,D] : 
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(C)
        & v1_funct_2(C,A,B)
        & m1_relset_1(C,A,B)
        & m1_subset_1(D,A) )
     => m1_subset_1(k8_funct_2(A,B,C,D),B) ) ),
    file(funct_2,k8_funct_2),
    []).

fof(dt_l1_cat_1,axiom,(
    $true ),
    file(cat_1,l1_cat_1),
    []).

fof(dt_m1_relset_1,axiom,(
    $true ),
    file(relset_1,m1_relset_1),
    []).

fof(dt_m1_subset_1,axiom,(
    $true ),
    file(subset_1,m1_subset_1),
    []).

fof(dt_m2_relset_1,axiom,(
    ! [A,B,C] : 
      ( m2_relset_1(C,A,B)
     => m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ),
    file(relset_1,m2_relset_1),
    []).

fof(dt_u1_cat_1,axiom,(
    ! [A] : 
      ( l1_cat_1(A)
     => ~ v1_xboole_0(u1_cat_1(A)) ) ),
    file(cat_1,u1_cat_1),
    []).

fof(dt_u2_cat_1,axiom,(
    ! [A] : 
      ( l1_cat_1(A)
     => ~ v1_xboole_0(u2_cat_1(A)) ) ),
    file(cat_1,u2_cat_1),
    []).

fof(dt_u4_cat_1,axiom,(
    ! [A] : 
      ( l1_cat_1(A)
     => ( v1_funct_1(u4_cat_1(A))
        & v1_funct_2(u4_cat_1(A),u2_cat_1(A),u1_cat_1(A))
        & m2_relset_1(u4_cat_1(A),u2_cat_1(A),u1_cat_1(A)) ) ) ),
    file(cat_1,u4_cat_1),
    []).

fof(existence_l1_cat_1,axiom,(
    ? [A] : l1_cat_1(A) ),
    file(cat_1,l1_cat_1),
    []).

fof(existence_m1_relset_1,axiom,(
    ! [A,B] : 
    ? [C] : m1_relset_1(C,A,B) ),
    file(relset_1,m1_relset_1),
    []).

fof(existence_m1_subset_1,axiom,(
    ! [A] : 
    ? [B] : m1_subset_1(B,A) ),
    file(subset_1,m1_subset_1),
    []).

fof(existence_m2_relset_1,axiom,(
    ! [A,B] : 
    ? [C] : m2_relset_1(C,A,B) ),
    file(relset_1,m2_relset_1),
    []).

fof(fc1_cat_5,axiom,(
    ! [A,B] : 
      ( ( v2_cat_1(A)
        & l1_cat_1(A)
        & m1_subset_1(B,u1_cat_1(A)) )
     => ~ v1_xboole_0(k12_cat_5(A,B)) ) ),
    file(cat_5,fc1_cat_5),
    []).

fof(fc1_subset_1,axiom,(
    ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ),
    file(subset_1,fc1_subset_1),
    []).

fof(fc1_xboole_0,axiom,(
    v1_xboole_0(k1_xboole_0) ),
    file(xboole_0,fc1_xboole_0),
    []).

fof(fc2_subset_1,axiom,(
    ! [A] : ~ v1_xboole_0(k1_tarski(A)) ),
    file(subset_1,fc2_subset_1),
    []).

fof(fc3_realset1,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(k1_tarski(A))
      & v1_finset_1(k1_tarski(A))
      & v1_realset1(k1_tarski(A)) ) ),
    file(realset1,fc3_realset1),
    []).

fof(fc4_subset_1,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B) )
     => ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ),
    file(subset_1,fc4_subset_1),
    []).

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

fof(rc1_realset1,axiom,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_realset1(A) ) ),
    file(realset1,rc1_realset1),
    []).

fof(rc1_subset_1,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ? [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
          & ~ v1_xboole_0(B) ) ) ),
    file(subset_1,rc1_subset_1),
    []).

fof(rc1_xboole_0,axiom,(
    ? [A] : v1_xboole_0(A) ),
    file(xboole_0,rc1_xboole_0),
    []).

fof(rc2_cat_1,axiom,(
    ? [A] : 
      ( l1_cat_1(A)
      & v2_cat_1(A) ) ),
    file(cat_1,rc2_cat_1),
    []).

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

fof(rc2_realset1,axiom,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & ~ v1_realset1(A) ) ),
    file(realset1,rc2_realset1),
    []).

fof(rc2_subset_1,axiom,(
    ! [A] : 
    ? [B] : 
      ( m1_subset_1(B,k1_zfmisc_1(A))
      & v1_xboole_0(B) ) ),
    file(subset_1,rc2_subset_1),
    []).

fof(rc2_xboole_0,axiom,(
    ? [A] : ~ v1_xboole_0(A) ),
    file(xboole_0,rc2_xboole_0),
    []).

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

fof(redefinition_k3_funct_2,axiom,(
    ! [A,B,C,D] : 
      ( ( v1_funct_1(C)
        & m1_relset_1(C,A,B) )
     => k3_funct_2(A,B,C,D) = k10_relat_1(C,D) ) ),
    file(funct_2,k3_funct_2),
    []).

fof(redefinition_k6_domain_1,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A) )
     => k6_domain_1(A,B) = k1_tarski(B) ) ),
    file(domain_1,k6_domain_1),
    []).

fof(redefinition_k8_funct_2,axiom,(
    ! [A,B,C,D] : 
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(C)
        & v1_funct_2(C,A,B)
        & m1_relset_1(C,A,B)
        & m1_subset_1(D,A) )
     => k8_funct_2(A,B,C,D) = k1_funct_1(C,D) ) ),
    file(funct_2,k8_funct_2),
    []).

fof(redefinition_m2_relset_1,axiom,(
    ! [A,B,C] : 
      ( m2_relset_1(C,A,B)
    <=> m1_relset_1(C,A,B) ) ),
    file(relset_1,m2_relset_1),
    []).

fof(reflexivity_r1_tarski,axiom,(
    ! [A,B] : r1_tarski(A,A) ),
    file(tarski,r1_tarski),
    []).

fof(t1_subset,axiom,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => m1_subset_1(A,B) ) ),
    file(subset,t1_subset),
    []).

fof(t2_subset,axiom,(
    ! [A,B] : 
      ( m1_subset_1(A,B)
     => ( v1_xboole_0(B)
        | r2_hidden(A,B) ) ) ),
    file(subset,t2_subset),
    []).

fof(t3_subset,axiom,(
    ! [A,B] : 
      ( m1_subset_1(A,k1_zfmisc_1(B))
    <=> r1_tarski(A,B) ) ),
    file(subset,t3_subset),
    []).

fof(t46_funct_2,axiom,(
    ! [A,B,C,D] : 
      ( ( v1_funct_1(D)
        & v1_funct_2(D,A,B)
        & m2_relset_1(D,A,B) )
     => ( B != k1_xboole_0
       => ! [E] : 
            ( r2_hidden(E,k10_relat_1(D,C))
          <=> ( r2_hidden(E,A)
              & r2_hidden(k1_funct_1(D,E),C) ) ) ) ) ),
    file(funct_2,t46_funct_2),
    []).

fof(t4_subset,axiom,(
    ! [A,B,C] : 
      ( ( r2_hidden(A,B)
        & m1_subset_1(B,k1_zfmisc_1(C)) )
     => m1_subset_1(A,C) ) ),
    file(subset,t4_subset),
    []).

fof(t5_subset,axiom,(
    ! [A,B,C] : ~ ( r2_hidden(A,B)
      & m1_subset_1(B,k1_zfmisc_1(C))
      & v1_xboole_0(C) ) ),
    file(subset,t5_subset),
    []).

fof(t6_boole,axiom,(
    ! [A] : 
      ( v1_xboole_0(A)
     => A = k1_xboole_0 ) ),
    file(boole,t6_boole),
    []).

fof(t7_boole,axiom,(
    ! [A,B] : ~ ( r2_hidden(A,B)
      & v1_xboole_0(B) ) ),
    file(boole,t7_boole),
    []).

fof(t8_boole,axiom,(
    ! [A,B] : ~ ( v1_xboole_0(A)
      & A != B
      & v1_xboole_0(B) ) ),
    file(boole,t8_boole),
    []).