SET007 Axioms: SET007+664.ax
%------------------------------------------------------------------------------
% File : SET007+664 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Canonical Formulae
% Version : [Urb08] axioms.
% English :
% Refs : [Mat90] Matuszewski (1990), Formalized Mathematics
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : hilbert3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 84 ( 4 unt; 0 def)
% Number of atoms : 555 ( 69 equ)
% Maximal formula atoms : 22 ( 6 avg)
% Number of connectives : 513 ( 42 ~; 3 |; 261 &)
% ( 8 <=>; 199 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 8 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 31 ( 29 usr; 1 prp; 0-4 aty)
% Number of functors : 47 ( 47 usr; 9 con; 0-6 aty)
% Number of variables : 267 ( 256 !; 11 ?)
% SPC :
% Comments : The individual reference can be found in [Mat90] by looking for
% the name provided by [Urb08].
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : These set theory axioms are used in encodings of problems in
% various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(cc1_hilbert3,axiom,
! [A,B,C,D] :
( m1_relset_1(D,A,k1_funct_2(B,C))
=> ( ( v1_funct_1(D)
& v1_funct_2(D,A,k1_funct_2(B,C)) )
=> ( v1_funct_1(D)
& v1_funct_2(D,A,k1_funct_2(B,C))
& v1_funcop_1(D) ) ) ) ).
fof(fc1_hilbert3,axiom,
( 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_funcop_1(k1_xboole_0) ) ).
fof(cc2_hilbert3,axiom,
! [A,B] :
( v1_fraenkel(B)
=> ! [C] :
( m1_relset_1(C,A,B)
=> ( ( v1_funct_1(C)
& v1_funct_2(C,A,B) )
=> ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& v1_funcop_1(C) ) ) ) ) ).
fof(fc2_hilbert3,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v2_funct_1(A)
& v1_relat_1(B)
& v1_funct_1(B)
& v2_funct_1(B) )
=> ( v1_relat_1(k15_funct_3(A,B))
& v1_funct_1(k15_funct_3(A,B))
& v2_funct_1(k15_funct_3(A,B)) ) ) ).
fof(fc3_hilbert3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& v1_funct_1(C)
& v1_funct_2(C,A,A)
& v3_funct_2(C,A,A)
& m1_relset_1(C,A,A)
& v1_funct_1(D)
& v1_funct_2(D,B,B)
& v3_funct_2(D,B,B)
& m1_relset_1(D,B,B) )
=> ( ~ v1_xboole_0(k1_hilbert3(A,B,C,D))
& v1_relat_1(k1_hilbert3(A,B,C,D))
& v1_funct_1(k1_hilbert3(A,B,C,D))
& v2_funct_1(k1_hilbert3(A,B,C,D))
& v1_funct_2(k1_hilbert3(A,B,C,D),k1_fraenkel(A,B),k1_fraenkel(A,B))
& v2_funct_2(k1_hilbert3(A,B,C,D),k1_fraenkel(A,B),k1_fraenkel(A,B))
& v3_funct_2(k1_hilbert3(A,B,C,D),k1_fraenkel(A,B),k1_fraenkel(A,B))
& v1_funcop_1(k1_hilbert3(A,B,C,D))
& v1_partfun1(k1_hilbert3(A,B,C,D),k1_fraenkel(A,B),k1_fraenkel(A,B)) ) ) ).
fof(fc4_hilbert3,axiom,
! [A,B] :
( ( v2_relat_1(A)
& m1_pboole(A,k5_numbers)
& m1_subset_1(B,k1_hilbert1) )
=> ~ v1_xboole_0(k3_hilbert3(A,B)) ) ).
fof(fc5_hilbert3,axiom,
! [A,B,C] :
( ( v2_relat_1(A)
& m1_pboole(A,k5_numbers)
& m1_subset_1(B,k1_hilbert1)
& m1_subset_1(C,k1_hilbert1) )
=> ( ~ v1_xboole_0(k3_hilbert3(A,k3_hilbert1(B,C)))
& v1_fraenkel(k3_hilbert3(A,k3_hilbert1(B,C))) ) ) ).
fof(cc3_hilbert3,axiom,
! [A,B,C,D] :
( ( v2_relat_1(A)
& m1_pboole(A,k5_numbers)
& m1_subset_1(B,k1_hilbert1)
& m1_subset_1(C,k1_hilbert1)
& m1_subset_1(D,k1_hilbert1) )
=> ! [E] :
( m1_subset_1(E,k3_hilbert3(A,k3_hilbert1(B,k3_hilbert1(C,D))))
=> ( v1_relat_1(E)
& v1_funct_1(E)
& v1_funcop_1(E) ) ) ) ).
fof(rc1_hilbert3,axiom,
! [A,B,C,D] :
( ( v2_relat_1(A)
& m1_pboole(A,k5_numbers)
& m1_subset_1(B,k1_hilbert1)
& m1_subset_1(C,k1_hilbert1)
& m1_subset_1(D,k1_hilbert1) )
=> ? [E] :
( m1_relset_1(E,k3_hilbert3(A,k3_hilbert1(B,C)),k3_hilbert3(A,k3_hilbert1(B,D)))
& ~ v1_xboole_0(E)
& v1_relat_1(E)
& v1_funct_1(E)
& v1_funct_2(E,k3_hilbert3(A,k3_hilbert1(B,C)),k3_hilbert3(A,k3_hilbert1(B,D)))
& v1_funcop_1(E)
& v1_partfun1(E,k3_hilbert3(A,k3_hilbert1(B,C)),k3_hilbert3(A,k3_hilbert1(B,D))) ) ) ).
fof(rc2_hilbert3,axiom,
! [A,B,C,D] :
( ( v2_relat_1(A)
& m1_pboole(A,k5_numbers)
& m1_subset_1(B,k1_hilbert1)
& m1_subset_1(C,k1_hilbert1)
& m1_subset_1(D,k1_hilbert1) )
=> ? [E] :
( m1_subset_1(E,k3_hilbert3(A,k3_hilbert1(B,k3_hilbert1(C,D))))
& v1_relat_1(E)
& v1_funct_1(E)
& v1_funcop_1(E) ) ) ).
fof(fc6_hilbert3,axiom,
! [A,B,C] :
( ( v2_relat_1(A)
& m1_pboole(A,k5_numbers)
& m1_hilbert3(B,A)
& m1_subset_1(C,k1_hilbert1) )
=> ( ~ v1_xboole_0(k5_hilbert3(A,B,C))
& v1_relat_1(k5_hilbert3(A,B,C))
& v1_funct_1(k5_hilbert3(A,B,C))
& v2_funct_1(k5_hilbert3(A,B,C))
& v1_funct_2(k5_hilbert3(A,B,C),k3_hilbert3(A,C),k3_hilbert3(A,C))
& v2_funct_2(k5_hilbert3(A,B,C),k3_hilbert3(A,C),k3_hilbert3(A,C))
& v3_funct_2(k5_hilbert3(A,B,C),k3_hilbert3(A,C),k3_hilbert3(A,C))
& v1_partfun1(k5_hilbert3(A,B,C),k3_hilbert3(A,C),k3_hilbert3(A,C)) ) ) ).
fof(fc7_hilbert3,axiom,
( v1_relat_1(k2_hilbert1)
& v1_funct_1(k2_hilbert1)
& v1_finset_1(k2_hilbert1)
& v1_finseq_1(k2_hilbert1)
& v1_hilbert3(k2_hilbert1) ) ).
fof(rc3_hilbert3,axiom,
? [A] :
( m1_subset_1(A,k1_hilbert1)
& v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A)
& v1_finseq_1(A)
& v1_hilbert3(A) ) ).
fof(cc4_hilbert3,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ( v1_hilbert3(A)
=> ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A)
& v1_finseq_1(A)
& v2_hilbert3(A) ) ) ) ).
fof(t1_hilbert3,axiom,
! [A] :
( v1_int_1(A)
=> ( v1_abian(A)
<=> ~ v1_abian(k6_xcmplx_0(A,np__1)) ) ) ).
fof(t2_hilbert3,axiom,
! [A] :
( v1_int_1(A)
=> ( ~ v1_abian(A)
<=> v1_abian(k6_xcmplx_0(A,np__1)) ) ) ).
fof(t3_hilbert3,axiom,
! [A] :
( v1_realset1(A)
=> ! [B] :
( r2_hidden(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,A)
& m2_relset_1(C,A,A) )
=> r1_abian(B,C) ) ) ) ).
fof(t4_hilbert3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_funcop_1(A) )
=> k1_funct_6(k2_relat_1(A)) = k2_relat_1(A) ) ).
fof(t5_hilbert3,axiom,
! [A,B,C,D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( ( r2_hidden(C,A)
& r2_hidden(D,k1_funct_2(A,B)) )
=> r2_hidden(k1_funct_1(D,C),B) ) ) ).
fof(t6_hilbert3,axiom,
! [A,B,C] :
( ~ ( C = k1_xboole_0
& B != k1_xboole_0
& A != k1_xboole_0 )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,k1_funct_2(B,C))
& m2_relset_1(D,A,k1_funct_2(B,C)) )
=> k2_funct_6(D) = k2_pre_circ(A,B) ) ) ).
fof(t7_hilbert3,axiom,
! [A] : k1_funct_1(k1_xboole_0,A) = k1_xboole_0 ).
fof(t8_hilbert3,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> k7_funct_2(A,k2_tarski(np__0,np__1),k5_numbers,k5_funct_3(B,A),k5_funct_4(k5_numbers,np__0,np__1,np__1,np__0)) = k5_funct_3(k3_subset_1(A,B),A) ) ).
fof(t9_hilbert3,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> k7_funct_2(A,k2_tarski(np__0,np__1),k5_numbers,k5_funct_3(k3_subset_1(A,B),A),k5_funct_4(k5_numbers,np__0,np__1,np__1,np__0)) = k5_funct_3(B,A) ) ).
fof(t10_hilbert3,axiom,
! [A,B,C,D,E,F] :
( k4_funct_4(A,B,C,D) = k4_funct_4(A,B,E,F)
=> ( A = B
| ( C = E
& D = F ) ) ) ).
fof(t11_hilbert3,axiom,
! [A,B,C,D,E,F] :
( ( r2_hidden(C,E)
& r2_hidden(D,F) )
=> ( A = B
| r2_hidden(k4_funct_4(A,B,C,D),k4_card_3(k4_funct_4(A,B,E,F))) ) ) ).
fof(t12_hilbert3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,np__2,A)
& m2_relset_1(B,np__2,A) )
=> ? [C] :
( m1_subset_1(C,A)
& ? [D] :
( m1_subset_1(D,A)
& B = k5_funct_4(A,np__0,np__1,C,D) ) ) ) ) ).
fof(t13_hilbert3,axiom,
! [A,B,C,D] :
( A != B
=> k5_relat_1(k4_funct_4(A,B,B,A),k4_funct_4(A,B,C,D)) = k4_funct_4(A,B,D,C) ) ).
fof(t14_hilbert3,axiom,
! [A,B,C,D,E] :
( ( v1_relat_1(E)
& v1_funct_1(E) )
=> ( ( r2_hidden(C,k1_relat_1(E))
& r2_hidden(D,k1_relat_1(E)) )
=> ( A = B
| k5_relat_1(k4_funct_4(A,B,C,D),E) = k4_funct_4(A,B,k1_funct_1(E,C),k1_funct_1(E,D)) ) ) ) ).
fof(t15_hilbert3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C,D,E] :
( ( v1_funct_1(E)
& v1_funct_2(E,C,A)
& m2_relset_1(E,C,A) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,D,B)
& m2_relset_1(F,D,B) )
=> k7_funct_2(k2_zfmisc_1(C,D),k2_zfmisc_1(A,B),A,k16_funct_3(C,D,A,B,E,F),k9_funct_3(A,B)) = k1_partfun1(k2_zfmisc_1(C,D),C,C,A,k9_funct_3(C,D),E) ) ) ) ) ).
fof(t16_hilbert3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C,D,E] :
( ( v1_funct_1(E)
& v1_funct_2(E,C,A)
& m2_relset_1(E,C,A) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,D,B)
& m2_relset_1(F,D,B) )
=> k7_funct_2(k2_zfmisc_1(C,D),k2_zfmisc_1(A,B),B,k16_funct_3(C,D,A,B,E,F),k10_funct_3(A,B)) = k1_partfun1(k2_zfmisc_1(C,D),D,D,B,k10_funct_3(C,D),F) ) ) ) ) ).
fof(t17_hilbert3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> k15_pralg_1(k1_xboole_0,A) = k1_xboole_0 ) ).
fof(t18_hilbert3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_funcop_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> k5_relat_1(C,k15_pralg_1(A,B)) = k15_pralg_1(k5_relat_1(C,A),k5_relat_1(C,B)) ) ) ) ).
fof(t19_hilbert3,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,k1_funct_2(k1_xboole_0,A))
& m2_relset_1(C,B,k1_funct_2(k1_xboole_0,A)) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,B,k1_xboole_0)
& m2_relset_1(D,B,k1_xboole_0) )
=> k2_relat_1(k15_pralg_1(C,D)) = k1_tarski(k1_xboole_0) ) ) ) ).
fof(t20_hilbert3,axiom,
! [A,B,C] :
( ( B = k1_xboole_0
=> A = k1_xboole_0 )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,k1_funct_2(B,C))
& m2_relset_1(D,A,k1_funct_2(B,C)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,B)
& m2_relset_1(E,A,B) )
=> r1_tarski(k2_relat_1(k15_pralg_1(D,E)),C) ) ) ) ).
fof(t21_hilbert3,axiom,
! [A,B,C] :
( ~ ( C = k1_xboole_0
& B != k1_xboole_0
& A != k1_xboole_0 )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,k1_funct_2(B,C))
& m2_relset_1(D,A,k1_funct_2(B,C)) )
=> k1_relat_1(k3_pralg_2(D)) = k1_funct_2(A,B) ) ) ).
fof(t22_hilbert3,axiom,
$true ).
fof(t23_hilbert3,axiom,
! [A,B,C] :
( ~ ( C = k1_xboole_0
& B != k1_xboole_0
& A != k1_xboole_0 )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,k1_funct_2(B,C))
& m2_relset_1(D,A,k1_funct_2(B,C)) )
=> r1_tarski(k2_relat_1(k3_pralg_2(D)),k1_funct_2(A,C)) ) ) ).
fof(t24_hilbert3,axiom,
! [A,B,C] :
( ~ ( C = k1_xboole_0
& B != k1_xboole_0
& A != k1_xboole_0 )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,k1_funct_2(B,C))
& m2_relset_1(D,A,k1_funct_2(B,C)) )
=> ( v1_funct_1(k3_pralg_2(D))
& v1_funct_2(k3_pralg_2(D),k1_funct_2(A,B),k1_funct_2(A,C))
& m2_relset_1(k3_pralg_2(D),k1_funct_2(A,B),k1_funct_2(A,C)) ) ) ) ).
fof(t25_hilbert3,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,A)
& v3_funct_2(C,A,A)
& m2_relset_1(C,A,A) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,B,B)
& v3_funct_2(D,B,B)
& m2_relset_1(D,B,B) )
=> v3_funct_2(k16_funct_3(A,B,A,B,C,D),k2_zfmisc_1(A,B),k2_zfmisc_1(A,B)) ) ) ).
fof(d1_hilbert3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,A)
& v3_funct_2(C,A,A)
& m2_relset_1(C,A,A) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,B,B)
& m2_relset_1(D,B,B) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k1_fraenkel(A,B),k1_fraenkel(A,B))
& m2_relset_1(E,k1_fraenkel(A,B),k1_fraenkel(A,B)) )
=> ( E = k1_hilbert3(A,B,C,D)
<=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,A,B)
& m2_relset_1(F,A,B) )
=> k1_funct_1(E,F) = k7_funct_2(A,A,B,k6_funct_2(A,C),k7_funct_2(A,B,B,F,D)) ) ) ) ) ) ) ) ).
fof(t26_hilbert3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,A)
& v3_funct_2(C,A,A)
& m2_relset_1(C,A,A) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,B,B)
& v3_funct_2(D,B,B)
& m2_relset_1(D,B,B) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,B)
& m2_relset_1(E,A,B) )
=> k1_funct_1(k6_funct_2(k1_fraenkel(A,B),k1_hilbert3(A,B,C,D)),E) = k7_funct_2(A,A,B,C,k7_funct_2(A,B,B,E,k6_funct_2(B,D))) ) ) ) ) ) ).
fof(t27_hilbert3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,A)
& v3_funct_2(C,A,A)
& m2_relset_1(C,A,A) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,B,B)
& v3_funct_2(D,B,B)
& m2_relset_1(D,B,B) )
=> k6_funct_2(k1_fraenkel(A,B),k1_hilbert3(A,B,C,D)) = k1_hilbert3(A,B,k6_funct_2(A,C),k6_funct_2(B,D)) ) ) ) ) ).
fof(t28_hilbert3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,k1_fraenkel(B,C))
& m2_relset_1(D,A,k1_fraenkel(B,C)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,B)
& m2_relset_1(E,A,B) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,B,B)
& v3_funct_2(F,B,B)
& m2_relset_1(F,B,B) )
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,C,C)
& v3_funct_2(G,C,C)
& m2_relset_1(G,C,C) )
=> k15_pralg_1(k7_funct_2(A,k1_fraenkel(B,C),k1_fraenkel(B,C),D,k1_hilbert3(B,C,F,G)),k7_funct_2(A,B,B,E,F)) = k5_relat_1(k15_pralg_1(D,E),G) ) ) ) ) ) ) ) ).
fof(d2_hilbert3,axiom,
! [A] :
( ( v2_relat_1(A)
& m1_pboole(A,k5_numbers) )
=> ! [B] :
( m1_pboole(B,k1_hilbert1)
=> ( B = k2_hilbert3(A)
<=> ( k1_funct_1(B,k2_hilbert1) = np__1
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> k1_funct_1(B,k1_hilbert2(C)) = k1_funct_1(A,C) )
& ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> ! [D] :
( m1_subset_1(D,k1_hilbert1)
=> ( k1_funct_1(B,k4_hilbert1(C,D)) = k2_zfmisc_1(k1_funct_1(B,C),k1_funct_1(B,D))
& k1_funct_1(B,k3_hilbert1(C,D)) = k1_funct_2(k1_funct_1(B,C),k1_funct_1(B,D)) ) ) ) ) ) ) ) ).
fof(d3_hilbert3,axiom,
! [A] :
( ( v2_relat_1(A)
& m1_pboole(A,k5_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> k3_hilbert3(A,B) = k1_funct_1(k2_hilbert3(A),B) ) ) ).
fof(t29_hilbert3,axiom,
! [A] :
( ( v2_relat_1(A)
& m1_pboole(A,k5_numbers) )
=> k3_hilbert3(A,k2_hilbert1) = np__1 ) ).
fof(t30_hilbert3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v2_relat_1(B)
& m1_pboole(B,k5_numbers) )
=> k3_hilbert3(B,k1_hilbert2(A)) = k1_funct_1(B,A) ) ) ).
fof(t31_hilbert3,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,k5_numbers) )
=> k3_hilbert3(C,k4_hilbert1(A,B)) = k2_zfmisc_1(k3_hilbert3(C,A),k3_hilbert3(C,B)) ) ) ) ).
fof(t32_hilbert3,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,k5_numbers) )
=> k3_hilbert3(C,k3_hilbert1(A,B)) = k1_fraenkel(k3_hilbert3(C,A),k3_hilbert3(C,B)) ) ) ) ).
fof(d4_hilbert3,axiom,
! [A] :
( ( v2_relat_1(A)
& m1_pboole(A,k5_numbers) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( m1_hilbert3(B,A)
<=> ( k1_relat_1(B) = k5_numbers
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( v1_funct_1(k1_funct_1(B,C))
& v1_funct_2(k1_funct_1(B,C),k1_funct_1(A,C),k1_funct_1(A,C))
& v3_funct_2(k1_funct_1(B,C),k1_funct_1(A,C),k1_funct_1(A,C))
& m2_relset_1(k1_funct_1(B,C),k1_funct_1(A,C),k1_funct_1(A,C)) ) ) ) ) ) ) ).
fof(d5_hilbert3,axiom,
! [A] :
( ( v2_relat_1(A)
& m1_pboole(A,k5_numbers) )
=> ! [B] :
( m1_hilbert3(B,A)
=> ! [C] :
( m3_pboole(C,k1_hilbert1,k2_hilbert3(A),k2_hilbert3(A))
=> ( C = k4_hilbert3(A,B)
<=> ( k1_msualg_3(k1_hilbert1,k2_hilbert3(A),k2_hilbert3(A),C,k2_hilbert1) = k6_partfun1(np__1)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k1_msualg_3(k1_hilbert1,k2_hilbert3(A),k2_hilbert3(A),C,k1_hilbert2(D)) = k1_funct_1(B,D) )
& ! [D] :
( m1_subset_1(D,k1_hilbert1)
=> ! [E] :
( m1_subset_1(E,k1_hilbert1)
=> ? [F] :
( v1_funct_1(F)
& v1_funct_2(F,k3_hilbert3(A,D),k3_hilbert3(A,D))
& v3_funct_2(F,k3_hilbert3(A,D),k3_hilbert3(A,D))
& m2_relset_1(F,k3_hilbert3(A,D),k3_hilbert3(A,D))
& ? [G] :
( v1_funct_1(G)
& v1_funct_2(G,k3_hilbert3(A,E),k3_hilbert3(A,E))
& v3_funct_2(G,k3_hilbert3(A,E),k3_hilbert3(A,E))
& m2_relset_1(G,k3_hilbert3(A,E),k3_hilbert3(A,E))
& F = k1_msualg_3(k1_hilbert1,k2_hilbert3(A),k2_hilbert3(A),C,D)
& G = k1_msualg_3(k1_hilbert1,k2_hilbert3(A),k2_hilbert3(A),C,E)
& k1_msualg_3(k1_hilbert1,k2_hilbert3(A),k2_hilbert3(A),C,k4_hilbert1(D,E)) = k16_funct_3(k3_hilbert3(A,D),k3_hilbert3(A,E),k3_hilbert3(A,D),k3_hilbert3(A,E),F,G)
& k1_msualg_3(k1_hilbert1,k2_hilbert3(A),k2_hilbert3(A),C,k3_hilbert1(D,E)) = k1_hilbert3(k3_hilbert3(A,D),k3_hilbert3(A,E),F,G) ) ) ) ) ) ) ) ) ) ).
fof(d6_hilbert3,axiom,
! [A] :
( ( v2_relat_1(A)
& m1_pboole(A,k5_numbers) )
=> ! [B] :
( m1_hilbert3(B,A)
=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> k5_hilbert3(A,B,C) = k1_msualg_3(k1_hilbert1,k2_hilbert3(A),k2_hilbert3(A),k4_hilbert3(A,B),C) ) ) ) ).
fof(t33_hilbert3,axiom,
! [A] :
( ( v2_relat_1(A)
& m1_pboole(A,k5_numbers) )
=> ! [B] :
( m1_hilbert3(B,A)
=> k5_hilbert3(A,B,k2_hilbert1) = k6_partfun1(k3_hilbert3(A,k2_hilbert1)) ) ) ).
fof(t34_hilbert3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v2_relat_1(B)
& m1_pboole(B,k5_numbers) )
=> ! [C] :
( m1_hilbert3(C,B)
=> k5_hilbert3(B,C,k1_hilbert2(A)) = k1_funct_1(C,A) ) ) ) ).
fof(t35_hilbert3,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,k5_numbers) )
=> ! [D] :
( m1_hilbert3(D,C)
=> k5_hilbert3(C,D,k4_hilbert1(A,B)) = k16_funct_3(k3_hilbert3(C,A),k3_hilbert3(C,B),k3_hilbert3(C,A),k3_hilbert3(C,B),k5_hilbert3(C,D,A),k5_hilbert3(C,D,B)) ) ) ) ) ).
fof(t36_hilbert3,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,k5_numbers) )
=> ! [D] :
( m1_hilbert3(D,C)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k3_hilbert3(C,A),k3_hilbert3(C,A))
& v3_funct_2(E,k3_hilbert3(C,A),k3_hilbert3(C,A))
& m2_relset_1(E,k3_hilbert3(C,A),k3_hilbert3(C,A)) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k3_hilbert3(C,B),k3_hilbert3(C,B))
& v3_funct_2(F,k3_hilbert3(C,B),k3_hilbert3(C,B))
& m2_relset_1(F,k3_hilbert3(C,B),k3_hilbert3(C,B)) )
=> ( ( E = k5_hilbert3(C,D,A)
& F = k5_hilbert3(C,D,B) )
=> k5_hilbert3(C,D,k3_hilbert1(A,B)) = k1_hilbert3(k3_hilbert3(C,A),k3_hilbert3(C,B),E,F) ) ) ) ) ) ) ) ).
fof(t37_hilbert3,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,k5_numbers) )
=> ! [D] :
( m1_hilbert3(D,C)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k3_hilbert3(C,A),k3_hilbert3(C,B))
& m2_relset_1(E,k3_hilbert3(C,A),k3_hilbert3(C,B)) )
=> k1_funct_1(k5_hilbert3(C,D,k3_hilbert1(A,B)),E) = k7_funct_2(k3_hilbert3(C,A),k3_hilbert3(C,A),k3_hilbert3(C,B),k6_funct_2(k3_hilbert3(C,A),k5_hilbert3(C,D,A)),k7_funct_2(k3_hilbert3(C,A),k3_hilbert3(C,B),k3_hilbert3(C,B),E,k5_hilbert3(C,D,B))) ) ) ) ) ) ).
fof(t38_hilbert3,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,k5_numbers) )
=> ! [D] :
( m1_hilbert3(D,C)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k3_hilbert3(C,A),k3_hilbert3(C,B))
& m2_relset_1(E,k3_hilbert3(C,A),k3_hilbert3(C,B)) )
=> k1_funct_1(k6_funct_2(k3_hilbert3(C,k3_hilbert1(A,B)),k5_hilbert3(C,D,k3_hilbert1(A,B))),E) = k7_funct_2(k3_hilbert3(C,A),k3_hilbert3(C,A),k3_hilbert3(C,B),k5_hilbert3(C,D,A),k7_funct_2(k3_hilbert3(C,A),k3_hilbert3(C,B),k3_hilbert3(C,B),E,k6_funct_2(k3_hilbert3(C,B),k5_hilbert3(C,D,B)))) ) ) ) ) ) ).
fof(t39_hilbert3,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,k5_numbers) )
=> ! [D] :
( m1_hilbert3(D,C)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k3_hilbert3(C,A),k3_hilbert3(C,B))
& m2_relset_1(E,k3_hilbert3(C,A),k3_hilbert3(C,B)) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k3_hilbert3(C,A),k3_hilbert3(C,B))
& m2_relset_1(F,k3_hilbert3(C,A),k3_hilbert3(C,B)) )
=> ( E = k1_funct_1(k5_hilbert3(C,D,k3_hilbert1(A,B)),F)
=> k7_funct_2(k3_hilbert3(C,A),k3_hilbert3(C,B),k3_hilbert3(C,B),F,k5_hilbert3(C,D,B)) = k7_funct_2(k3_hilbert3(C,A),k3_hilbert3(C,A),k3_hilbert3(C,B),k5_hilbert3(C,D,A),E) ) ) ) ) ) ) ) ).
fof(t40_hilbert3,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,k5_numbers) )
=> ! [D] :
( m1_hilbert3(D,C)
=> ! [E] :
( r1_abian(E,k5_hilbert3(C,D,A))
=> ! [F] :
( ( v1_relat_1(F)
& v1_funct_1(F) )
=> ( r1_abian(F,k5_hilbert3(C,D,k3_hilbert1(A,B)))
=> r1_abian(k1_funct_1(F,E),k5_hilbert3(C,D,B)) ) ) ) ) ) ) ) ).
fof(d7_hilbert3,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ( v1_hilbert3(A)
<=> ! [B] :
( ( v2_relat_1(B)
& m1_pboole(B,k5_numbers) )
=> ? [C] :
! [D] :
( m1_hilbert3(D,B)
=> r1_abian(C,k5_hilbert3(B,D,A)) ) ) ) ) ).
fof(t41_hilbert3,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> v1_hilbert3(k3_hilbert1(A,k3_hilbert1(B,A))) ) ) ).
fof(t42_hilbert3,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> v1_hilbert3(k3_hilbert1(k3_hilbert1(A,k3_hilbert1(B,C)),k3_hilbert1(k3_hilbert1(A,B),k3_hilbert1(A,C)))) ) ) ) ).
fof(t43_hilbert3,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> v1_hilbert3(k3_hilbert1(k4_hilbert1(A,B),A)) ) ) ).
fof(t44_hilbert3,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> v1_hilbert3(k3_hilbert1(k4_hilbert1(A,B),B)) ) ) ).
fof(t45_hilbert3,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> v1_hilbert3(k3_hilbert1(A,k3_hilbert1(B,k4_hilbert1(A,B)))) ) ) ).
fof(t46_hilbert3,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ( ( v1_hilbert3(A)
& v1_hilbert3(k3_hilbert1(A,B)) )
=> v1_hilbert3(B) ) ) ) ).
fof(t47_hilbert3,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ( r2_hidden(A,k6_hilbert1)
=> v1_hilbert3(A) ) ) ).
fof(d8_hilbert3,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ( v2_hilbert3(A)
<=> ! [B] :
( ( v2_relat_1(B)
& m1_pboole(B,k5_numbers) )
=> ! [C] :
( m1_hilbert3(C,B)
=> ? [D] : r1_abian(D,k5_hilbert3(B,C,A)) ) ) ) ) ).
fof(t48_hilbert3,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> v2_hilbert3(k3_hilbert1(A,k3_hilbert1(B,A))) ) ) ).
fof(t49_hilbert3,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> v2_hilbert3(k3_hilbert1(k3_hilbert1(A,k3_hilbert1(B,C)),k3_hilbert1(k3_hilbert1(A,B),k3_hilbert1(A,C)))) ) ) ) ).
fof(t50_hilbert3,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> v2_hilbert3(k3_hilbert1(k4_hilbert1(A,B),A)) ) ) ).
fof(t51_hilbert3,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> v2_hilbert3(k3_hilbert1(k4_hilbert1(A,B),B)) ) ) ).
fof(t52_hilbert3,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> v2_hilbert3(k3_hilbert1(A,k3_hilbert1(B,k4_hilbert1(A,B)))) ) ) ).
fof(t53_hilbert3,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ( ( v2_hilbert3(A)
& v2_hilbert3(k3_hilbert1(A,B)) )
=> v2_hilbert3(B) ) ) ) ).
fof(t54_hilbert3,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,k5_numbers) )
=> ! [D] :
( m1_hilbert3(D,C)
=> ~ ( ? [E] : r1_abian(E,k5_hilbert3(C,D,A))
& ! [E] : ~ r1_abian(E,k5_hilbert3(C,D,B))
& v2_hilbert3(k3_hilbert1(A,B)) ) ) ) ) ) ).
fof(t55_hilbert3,axiom,
~ v2_hilbert3(k3_hilbert1(k3_hilbert1(k3_hilbert1(k1_hilbert2(np__0),k1_hilbert2(np__1)),k1_hilbert2(np__0)),k1_hilbert2(np__0))) ).
fof(dt_m1_hilbert3,axiom,
! [A] :
( ( v2_relat_1(A)
& m1_pboole(A,k5_numbers) )
=> ! [B] :
( m1_hilbert3(B,A)
=> ( v1_relat_1(B)
& v1_funct_1(B) ) ) ) ).
fof(existence_m1_hilbert3,axiom,
! [A] :
( ( v2_relat_1(A)
& m1_pboole(A,k5_numbers) )
=> ? [B] : m1_hilbert3(B,A) ) ).
fof(dt_k1_hilbert3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& v1_funct_1(C)
& v1_funct_2(C,A,A)
& v3_funct_2(C,A,A)
& m1_relset_1(C,A,A)
& v1_funct_1(D)
& v1_funct_2(D,B,B)
& m1_relset_1(D,B,B) )
=> ( v1_funct_1(k1_hilbert3(A,B,C,D))
& v1_funct_2(k1_hilbert3(A,B,C,D),k1_fraenkel(A,B),k1_fraenkel(A,B))
& m2_relset_1(k1_hilbert3(A,B,C,D),k1_fraenkel(A,B),k1_fraenkel(A,B)) ) ) ).
fof(dt_k2_hilbert3,axiom,
! [A] :
( ( v2_relat_1(A)
& m1_pboole(A,k5_numbers) )
=> m1_pboole(k2_hilbert3(A),k1_hilbert1) ) ).
fof(dt_k3_hilbert3,axiom,
$true ).
fof(dt_k4_hilbert3,axiom,
! [A,B] :
( ( v2_relat_1(A)
& m1_pboole(A,k5_numbers)
& m1_hilbert3(B,A) )
=> m3_pboole(k4_hilbert3(A,B),k1_hilbert1,k2_hilbert3(A),k2_hilbert3(A)) ) ).
fof(dt_k5_hilbert3,axiom,
! [A,B,C] :
( ( v2_relat_1(A)
& m1_pboole(A,k5_numbers)
& m1_hilbert3(B,A)
& m1_subset_1(C,k1_hilbert1) )
=> ( v1_funct_1(k5_hilbert3(A,B,C))
& v1_funct_2(k5_hilbert3(A,B,C),k3_hilbert3(A,C),k3_hilbert3(A,C))
& m2_relset_1(k5_hilbert3(A,B,C),k3_hilbert3(A,C),k3_hilbert3(A,C)) ) ) ).
%------------------------------------------------------------------------------