SET007 Axioms: SET007+147.ax
%------------------------------------------------------------------------------
% File : SET007+147 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Hilbert Positive Propositional Calculus
% 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 : hilbert1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 76 ( 6 unt; 0 def)
% Number of atoms : 308 ( 8 equ)
% Maximal formula atoms : 14 ( 4 avg)
% Number of connectives : 236 ( 4 ~; 0 |; 56 &)
% ( 12 <=>; 164 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 18 ( 16 usr; 1 prp; 0-3 aty)
% Number of functors : 18 ( 18 usr; 9 con; 0-2 aty)
% Number of variables : 159 ( 158 !; 1 ?)
% 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_hilbert1,axiom,
! [A] :
( v5_hilbert1(A)
=> ( ~ v1_xboole_0(A)
& v1_hilbert1(A)
& v2_hilbert1(A)
& v3_hilbert1(A)
& v4_hilbert1(A) ) ) ).
fof(cc2_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k13_finseq_1(k5_numbers)))
=> ( ( v1_hilbert1(A)
& v2_hilbert1(A)
& v3_hilbert1(A)
& v4_hilbert1(A) )
=> v5_hilbert1(A) ) ) ).
fof(fc1_hilbert1,axiom,
( ~ v1_xboole_0(k1_hilbert1)
& v1_hilbert1(k1_hilbert1)
& v2_hilbert1(k1_hilbert1)
& v3_hilbert1(k1_hilbert1)
& v4_hilbert1(k1_hilbert1)
& v5_hilbert1(k1_hilbert1) ) ).
fof(rc1_hilbert1,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_hilbert1(A)
& v2_hilbert1(A)
& v3_hilbert1(A)
& v4_hilbert1(A)
& v5_hilbert1(A) ) ).
fof(fc2_hilbert1,axiom,
( ~ v1_xboole_0(k1_hilbert1)
& v1_fraenkel(k1_hilbert1)
& v1_hilbert1(k1_hilbert1)
& v2_hilbert1(k1_hilbert1)
& v3_hilbert1(k1_hilbert1)
& v4_hilbert1(k1_hilbert1)
& v5_hilbert1(k1_hilbert1) ) ).
fof(cc3_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A)
& v1_finseq_1(A) ) ) ).
fof(fc3_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_hilbert1))
=> v6_hilbert1(k5_hilbert1(A)) ) ).
fof(fc4_hilbert1,axiom,
v6_hilbert1(k6_hilbert1) ).
fof(d1_hilbert1,axiom,
! [A] :
( v1_hilbert1(A)
<=> r2_hidden(k12_finseq_1(k5_numbers,np__0),A) ) ).
fof(d2_hilbert1,axiom,
! [A] :
( v2_hilbert1(A)
<=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( ( r2_hidden(B,A)
& r2_hidden(C,A) )
=> r2_hidden(k7_finseq_1(k7_finseq_1(k12_finseq_1(k5_numbers,np__1),B),C),A) ) ) ) ) ).
fof(d3_hilbert1,axiom,
! [A] :
( v3_hilbert1(A)
<=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( ( r2_hidden(B,A)
& r2_hidden(C,A) )
=> r2_hidden(k7_finseq_1(k7_finseq_1(k12_finseq_1(k5_numbers,np__2),B),C),A) ) ) ) ) ).
fof(d4_hilbert1,axiom,
! [A] :
( v4_hilbert1(A)
<=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r2_hidden(k12_finseq_1(k5_numbers,k1_nat_1(np__3,B)),A) ) ) ).
fof(d5_hilbert1,axiom,
! [A] :
( v5_hilbert1(A)
<=> ( r1_tarski(A,k13_finseq_1(k5_numbers))
& v1_hilbert1(A)
& v2_hilbert1(A)
& v3_hilbert1(A)
& v4_hilbert1(A) ) ) ).
fof(d6_hilbert1,axiom,
! [A] :
( A = k1_hilbert1
<=> ( v5_hilbert1(A)
& ! [B] :
( v5_hilbert1(B)
=> r1_tarski(A,B) ) ) ) ).
fof(d7_hilbert1,axiom,
k2_hilbert1 = k12_finseq_1(k5_numbers,np__0) ).
fof(d8_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> k3_hilbert1(A,B) = k7_finseq_1(k7_finseq_1(k12_finseq_1(k5_numbers,np__1),A),B) ) ) ).
fof(d9_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> k4_hilbert1(A,B) = k7_finseq_1(k7_finseq_1(k12_finseq_1(k5_numbers,np__2),A),B) ) ) ).
fof(d10_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_hilbert1))
=> ( v6_hilbert1(A)
<=> ( r2_hidden(k2_hilbert1,A)
& ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> ! [D] :
( m1_subset_1(D,k1_hilbert1)
=> ( r2_hidden(k3_hilbert1(B,k3_hilbert1(C,B)),A)
& r2_hidden(k3_hilbert1(k3_hilbert1(B,k3_hilbert1(C,D)),k3_hilbert1(k3_hilbert1(B,C),k3_hilbert1(B,D))),A)
& r2_hidden(k3_hilbert1(k4_hilbert1(B,C),B),A)
& r2_hidden(k3_hilbert1(k4_hilbert1(B,C),C),A)
& r2_hidden(k3_hilbert1(B,k3_hilbert1(C,k4_hilbert1(B,C))),A)
& ( ( r2_hidden(B,A)
& r2_hidden(k3_hilbert1(B,C),A) )
=> r2_hidden(C,A) ) ) ) ) ) ) ) ) ).
fof(d11_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_hilbert1))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_hilbert1))
=> ( B = k5_hilbert1(A)
<=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> ( r2_hidden(C,B)
<=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_hilbert1))
=> ( ( v6_hilbert1(D)
& r1_tarski(A,D) )
=> r2_hidden(C,D) ) ) ) ) ) ) ) ).
fof(d12_hilbert1,axiom,
k6_hilbert1 = k5_hilbert1(k1_subset_1(k1_hilbert1)) ).
fof(t1_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_hilbert1))
=> r2_hidden(k2_hilbert1,k5_hilbert1(A)) ) ).
fof(t2_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_hilbert1))
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> r2_hidden(k3_hilbert1(B,k3_hilbert1(C,k4_hilbert1(B,C))),k5_hilbert1(A)) ) ) ) ).
fof(t3_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_hilbert1))
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> ! [D] :
( m1_subset_1(D,k1_hilbert1)
=> r2_hidden(k3_hilbert1(k3_hilbert1(B,k3_hilbert1(C,D)),k3_hilbert1(k3_hilbert1(B,C),k3_hilbert1(B,D))),k5_hilbert1(A)) ) ) ) ) ).
fof(t4_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_hilbert1))
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> r2_hidden(k3_hilbert1(B,k3_hilbert1(C,B)),k5_hilbert1(A)) ) ) ) ).
fof(t5_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_hilbert1))
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> r2_hidden(k3_hilbert1(k4_hilbert1(B,C),B),k5_hilbert1(A)) ) ) ) ).
fof(t6_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_hilbert1))
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> r2_hidden(k3_hilbert1(k4_hilbert1(B,C),C),k5_hilbert1(A)) ) ) ) ).
fof(t7_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_hilbert1))
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> ( ( r2_hidden(B,k5_hilbert1(A))
& r2_hidden(k3_hilbert1(B,C),k5_hilbert1(A)) )
=> r2_hidden(C,k5_hilbert1(A)) ) ) ) ) ).
fof(t8_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_hilbert1))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_hilbert1))
=> ( ( v6_hilbert1(A)
& r1_tarski(B,A) )
=> r1_tarski(k5_hilbert1(B),A) ) ) ) ).
fof(t9_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_hilbert1))
=> r1_tarski(A,k5_hilbert1(A)) ) ).
fof(t10_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_hilbert1))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_hilbert1))
=> ( r1_tarski(A,B)
=> r1_tarski(k5_hilbert1(A),k5_hilbert1(B)) ) ) ) ).
fof(t11_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_hilbert1))
=> k5_hilbert1(k5_hilbert1(A)) = k5_hilbert1(A) ) ).
fof(t12_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_hilbert1))
=> ( v6_hilbert1(A)
<=> k5_hilbert1(A) = A ) ) ).
fof(t13_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_hilbert1))
=> ( v6_hilbert1(A)
=> r1_tarski(k6_hilbert1,A) ) ) ).
fof(t14_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> r2_hidden(k3_hilbert1(A,A),k6_hilbert1) ) ).
fof(t15_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ( r2_hidden(A,k6_hilbert1)
=> r2_hidden(k3_hilbert1(B,A),k6_hilbert1) ) ) ) ).
fof(t16_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> r2_hidden(k3_hilbert1(A,k2_hilbert1),k6_hilbert1) ) ).
fof(t17_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> r2_hidden(k3_hilbert1(k3_hilbert1(A,B),k3_hilbert1(A,A)),k6_hilbert1) ) ) ).
fof(t18_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> r2_hidden(k3_hilbert1(k3_hilbert1(A,B),k3_hilbert1(B,B)),k6_hilbert1) ) ) ).
fof(t19_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> r2_hidden(k3_hilbert1(k3_hilbert1(A,B),k3_hilbert1(k3_hilbert1(C,A),k3_hilbert1(C,B))),k6_hilbert1) ) ) ) ).
fof(t20_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> ( r2_hidden(k3_hilbert1(A,k3_hilbert1(B,C)),k6_hilbert1)
=> r2_hidden(k3_hilbert1(B,k3_hilbert1(A,C)),k6_hilbert1) ) ) ) ) ).
fof(t21_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> r2_hidden(k3_hilbert1(k3_hilbert1(A,B),k3_hilbert1(k3_hilbert1(B,C),k3_hilbert1(A,C))),k6_hilbert1) ) ) ) ).
fof(t22_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> ( r2_hidden(k3_hilbert1(A,B),k6_hilbert1)
=> r2_hidden(k3_hilbert1(k3_hilbert1(B,C),k3_hilbert1(A,C)),k6_hilbert1) ) ) ) ) ).
fof(t23_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> ( ( r2_hidden(k3_hilbert1(A,B),k6_hilbert1)
& r2_hidden(k3_hilbert1(B,C),k6_hilbert1) )
=> r2_hidden(k3_hilbert1(A,C),k6_hilbert1) ) ) ) ) ).
fof(t24_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> ! [D] :
( m1_subset_1(D,k1_hilbert1)
=> r2_hidden(k3_hilbert1(k3_hilbert1(A,k3_hilbert1(B,C)),k3_hilbert1(k3_hilbert1(D,B),k3_hilbert1(A,k3_hilbert1(D,C)))),k6_hilbert1) ) ) ) ) ).
fof(t25_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> r2_hidden(k3_hilbert1(k3_hilbert1(k3_hilbert1(A,B),C),k3_hilbert1(B,C)),k6_hilbert1) ) ) ) ).
fof(t26_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> r2_hidden(k3_hilbert1(k3_hilbert1(A,k3_hilbert1(B,C)),k3_hilbert1(B,k3_hilbert1(A,C))),k6_hilbert1) ) ) ) ).
fof(t27_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> r2_hidden(k3_hilbert1(k3_hilbert1(A,k3_hilbert1(A,B)),k3_hilbert1(A,B)),k6_hilbert1) ) ) ).
fof(t28_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> r2_hidden(k3_hilbert1(A,k3_hilbert1(k3_hilbert1(A,B),B)),k6_hilbert1) ) ) ).
fof(t29_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> ( ( r2_hidden(k3_hilbert1(A,k3_hilbert1(B,C)),k6_hilbert1)
& r2_hidden(B,k6_hilbert1) )
=> r2_hidden(k3_hilbert1(A,C),k6_hilbert1) ) ) ) ) ).
fof(t30_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> r2_hidden(k3_hilbert1(A,k4_hilbert1(A,A)),k6_hilbert1) ) ).
fof(t31_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ( r2_hidden(k4_hilbert1(A,B),k6_hilbert1)
<=> ( r2_hidden(A,k6_hilbert1)
& r2_hidden(B,k6_hilbert1) ) ) ) ) ).
fof(t32_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ( r2_hidden(k4_hilbert1(A,B),k6_hilbert1)
<=> r2_hidden(k4_hilbert1(B,A),k6_hilbert1) ) ) ) ).
fof(t33_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> r2_hidden(k3_hilbert1(k3_hilbert1(k4_hilbert1(A,B),C),k3_hilbert1(A,k3_hilbert1(B,C))),k6_hilbert1) ) ) ) ).
fof(t34_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> r2_hidden(k3_hilbert1(k3_hilbert1(A,k3_hilbert1(B,C)),k3_hilbert1(k4_hilbert1(A,B),C)),k6_hilbert1) ) ) ) ).
fof(t35_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> r2_hidden(k3_hilbert1(k3_hilbert1(A,B),k3_hilbert1(k3_hilbert1(A,C),k3_hilbert1(A,k4_hilbert1(B,C)))),k6_hilbert1) ) ) ) ).
fof(t36_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> r2_hidden(k3_hilbert1(k4_hilbert1(k3_hilbert1(A,B),A),B),k6_hilbert1) ) ) ).
fof(t37_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> r2_hidden(k3_hilbert1(k4_hilbert1(k4_hilbert1(k3_hilbert1(A,B),A),C),B),k6_hilbert1) ) ) ) ).
fof(t38_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> r2_hidden(k3_hilbert1(k3_hilbert1(A,B),k3_hilbert1(k4_hilbert1(C,A),B)),k6_hilbert1) ) ) ) ).
fof(t39_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> r2_hidden(k3_hilbert1(k3_hilbert1(A,B),k3_hilbert1(k4_hilbert1(A,C),B)),k6_hilbert1) ) ) ) ).
fof(t40_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> r2_hidden(k3_hilbert1(k3_hilbert1(k4_hilbert1(A,B),C),k3_hilbert1(k4_hilbert1(A,B),k4_hilbert1(C,B))),k6_hilbert1) ) ) ) ).
fof(t41_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> r2_hidden(k3_hilbert1(k3_hilbert1(A,B),k3_hilbert1(k4_hilbert1(A,C),k4_hilbert1(B,C))),k6_hilbert1) ) ) ) ).
fof(t42_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> r2_hidden(k3_hilbert1(k4_hilbert1(k3_hilbert1(A,B),k4_hilbert1(A,C)),k4_hilbert1(B,C)),k6_hilbert1) ) ) ) ).
fof(t43_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> r2_hidden(k3_hilbert1(k4_hilbert1(A,B),k4_hilbert1(B,A)),k6_hilbert1) ) ) ).
fof(t44_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> r2_hidden(k3_hilbert1(k4_hilbert1(k3_hilbert1(A,B),k4_hilbert1(A,C)),k4_hilbert1(C,B)),k6_hilbert1) ) ) ) ).
fof(t45_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> r2_hidden(k3_hilbert1(k3_hilbert1(A,B),k3_hilbert1(k4_hilbert1(A,C),k4_hilbert1(C,B))),k6_hilbert1) ) ) ) ).
fof(t46_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> r2_hidden(k3_hilbert1(k3_hilbert1(A,B),k3_hilbert1(k4_hilbert1(C,A),k4_hilbert1(C,B))),k6_hilbert1) ) ) ) ).
fof(t47_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> r2_hidden(k3_hilbert1(k4_hilbert1(A,k4_hilbert1(B,C)),k4_hilbert1(A,k4_hilbert1(C,B))),k6_hilbert1) ) ) ) ).
fof(t48_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> r2_hidden(k3_hilbert1(k4_hilbert1(k3_hilbert1(A,B),k3_hilbert1(A,C)),k3_hilbert1(A,k4_hilbert1(B,C))),k6_hilbert1) ) ) ) ).
fof(t49_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> r2_hidden(k3_hilbert1(k4_hilbert1(k4_hilbert1(A,B),C),k4_hilbert1(A,k4_hilbert1(B,C))),k6_hilbert1) ) ) ) ).
fof(t50_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_hilbert1)
=> ! [B] :
( m1_subset_1(B,k1_hilbert1)
=> ! [C] :
( m1_subset_1(C,k1_hilbert1)
=> r2_hidden(k3_hilbert1(k4_hilbert1(A,k4_hilbert1(B,C)),k4_hilbert1(k4_hilbert1(A,B),C)),k6_hilbert1) ) ) ) ).
fof(dt_k1_hilbert1,axiom,
$true ).
fof(dt_k2_hilbert1,axiom,
m1_subset_1(k2_hilbert1,k1_hilbert1) ).
fof(dt_k3_hilbert1,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_hilbert1)
& m1_subset_1(B,k1_hilbert1) )
=> m1_subset_1(k3_hilbert1(A,B),k1_hilbert1) ) ).
fof(dt_k4_hilbert1,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_hilbert1)
& m1_subset_1(B,k1_hilbert1) )
=> m1_subset_1(k4_hilbert1(A,B),k1_hilbert1) ) ).
fof(dt_k5_hilbert1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_hilbert1))
=> m1_subset_1(k5_hilbert1(A),k1_zfmisc_1(k1_hilbert1)) ) ).
fof(dt_k6_hilbert1,axiom,
m1_subset_1(k6_hilbert1,k1_zfmisc_1(k1_hilbert1)) ).
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