SET007 Axioms: SET007+864.ax
%------------------------------------------------------------------------------
% File : SET007+864 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Equivalences of Inconsistency and Henkin Models
% 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 : henmodel [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 32 ( 6 unt; 0 def)
% Number of atoms : 174 ( 12 equ)
% Maximal formula atoms : 18 ( 5 avg)
% Number of connectives : 169 ( 27 ~; 4 |; 50 &)
% ( 12 <=>; 76 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 7 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 25 ( 24 usr; 0 prp; 1-4 aty)
% Number of functors : 33 ( 33 usr; 10 con; 0-4 aty)
% Number of variables : 74 ( 70 !; 4 ?)
% 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(rc1_henmodel,axiom,
? [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
& v1_henmodel(A) ) ).
fof(t1_henmodel,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_finset_1(B)
& m1_subset_1(B,k1_zfmisc_1(k5_numbers)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m2_relset_1(C,A,B) )
=> ( ( v2_funct_1(C)
& k5_relset_1(A,B,C) = B
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( r2_hidden(E,k4_relset_1(A,B,C))
& r2_hidden(D,k4_relset_1(A,B,C)) )
=> ( r1_xreal_0(E,D)
| r2_hidden(k1_funct_1(C,D),k1_funct_1(C,E)) ) ) ) ) )
=> ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> k1_ordinal1(D) != A )
| k1_funct_1(C,k3_tarski(A)) = k3_tarski(k5_relset_1(A,B,C)) ) ) ) ) ) ).
fof(t2_henmodel,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& m1_subset_1(A,k1_zfmisc_1(k5_numbers)) )
=> ( r2_hidden(k3_tarski(A),A)
& ! [B] :
~ ( r2_hidden(B,A)
& ~ r2_hidden(B,k3_tarski(A))
& B != k3_tarski(A) ) ) ) ).
fof(d1_henmodel,axiom,
! [A,B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k5_numbers)) )
=> ( B = k1_henmodel(A)
<=> ( r2_hidden(B,A)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,A)
=> r1_xreal_0(B,C) ) ) ) ) )
& ( ~ ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k5_numbers)) )
=> ( B = k1_henmodel(A)
<=> B = np__0 ) ) ) ) ).
fof(t3_henmodel,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,A)
& m2_relset_1(B,k5_numbers,A) )
=> ! [C] :
( v1_finset_1(C)
=> ~ ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( r2_hidden(E,k4_relset_1(k5_numbers,A,B))
& r2_hidden(D,k4_relset_1(k5_numbers,A,B)) )
=> ( r1_xreal_0(E,D)
| r1_tarski(k8_funct_2(k5_numbers,A,B,D),k8_funct_2(k5_numbers,A,B,E)) ) ) ) )
& r1_tarski(C,k3_tarski(k5_relset_1(k5_numbers,A,B)))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ r1_tarski(C,k8_funct_2(k5_numbers,A,B,D)) ) ) ) ) ) ).
fof(d2_henmodel,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ( r1_henmodel(A,B)
<=> ? [C] :
( m2_finseq_1(C,k7_cqc_lang)
& r1_tarski(k5_relset_1(k5_numbers,k7_cqc_lang,C),A)
& r4_calcul_1(k8_finseq_1(k7_cqc_lang,C,k12_finseq_1(k7_cqc_lang,B))) ) ) ) ) ).
fof(d3_henmodel,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ( v1_henmodel(A)
<=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ~ ( r1_henmodel(A,B)
& r1_henmodel(A,k10_cqc_lang(B)) ) ) ) ) ).
fof(d4_henmodel,axiom,
! [A] :
( m2_finseq_1(A,k7_cqc_lang)
=> ( v2_henmodel(A)
<=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ~ ( r4_calcul_1(k8_finseq_1(k7_cqc_lang,A,k12_finseq_1(k7_cqc_lang,B)))
& r4_calcul_1(k8_finseq_1(k7_cqc_lang,A,k12_finseq_1(k7_cqc_lang,k10_cqc_lang(B)))) ) ) ) ) ).
fof(t4_henmodel,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m2_finseq_1(B,k7_cqc_lang)
=> ( ( v1_henmodel(A)
& r1_tarski(k5_relset_1(k5_numbers,k7_cqc_lang,B),A) )
=> v2_henmodel(B) ) ) ) ).
fof(t5_henmodel,axiom,
! [A] :
( m2_subset_1(A,k8_qc_lang1,k7_cqc_lang)
=> ! [B] :
( m2_finseq_1(B,k7_cqc_lang)
=> ! [C] :
( m2_finseq_1(C,k7_cqc_lang)
=> ( r4_calcul_1(k8_finseq_1(k7_cqc_lang,B,k12_finseq_1(k7_cqc_lang,A)))
=> r4_calcul_1(k8_finseq_1(k7_cqc_lang,k8_finseq_1(k7_cqc_lang,B,C),k12_finseq_1(k7_cqc_lang,A))) ) ) ) ) ).
fof(t6_henmodel,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ( ~ v1_henmodel(A)
<=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> r1_henmodel(A,B) ) ) ) ).
fof(t7_henmodel,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ~ ( ~ v1_henmodel(A)
& ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k7_cqc_lang))
=> ~ ( r1_tarski(B,A)
& v1_finset_1(B)
& ~ v1_henmodel(B) ) ) ) ) ).
fof(t8_henmodel,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ! [C] :
( m2_subset_1(C,k8_qc_lang1,k7_cqc_lang)
=> ~ ( r1_henmodel(k4_subset_1(k7_cqc_lang,A,k6_domain_1(k7_cqc_lang,B)),C)
& ! [D] :
( m2_finseq_1(D,k7_cqc_lang)
=> ~ ( r1_tarski(k5_relset_1(k5_numbers,k7_cqc_lang,D),A)
& r4_calcul_1(k8_finseq_1(k7_cqc_lang,k8_finseq_1(k7_cqc_lang,D,k12_finseq_1(k7_cqc_lang,B)),k12_finseq_1(k7_cqc_lang,C))) ) ) ) ) ) ) ).
fof(t9_henmodel,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ( r1_henmodel(A,B)
<=> ~ v1_henmodel(k4_subset_1(k7_cqc_lang,A,k6_domain_1(k7_cqc_lang,k10_cqc_lang(B)))) ) ) ) ).
fof(t10_henmodel,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( m2_subset_1(B,k8_qc_lang1,k7_cqc_lang)
=> ( r1_henmodel(A,k10_cqc_lang(B))
<=> ~ v1_henmodel(k4_subset_1(k7_cqc_lang,A,k6_domain_1(k7_cqc_lang,B))) ) ) ) ).
fof(t11_henmodel,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_zfmisc_1(k7_cqc_lang))
& m2_relset_1(A,k5_numbers,k1_zfmisc_1(k7_cqc_lang)) )
=> ( ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r2_hidden(C,k4_relset_1(k5_numbers,k1_zfmisc_1(k7_cqc_lang),A))
& r2_hidden(B,k4_relset_1(k5_numbers,k1_zfmisc_1(k7_cqc_lang),A)) )
=> ( r1_xreal_0(C,B)
| ( v1_henmodel(k8_funct_2(k5_numbers,k1_zfmisc_1(k7_cqc_lang),A,B))
& r1_tarski(k8_funct_2(k5_numbers,k1_zfmisc_1(k7_cqc_lang),A,B),k8_funct_2(k5_numbers,k1_zfmisc_1(k7_cqc_lang),A,C)) ) ) ) ) )
=> v1_henmodel(k5_setfam_1(k7_cqc_lang,k5_relset_1(k5_numbers,k1_zfmisc_1(k7_cqc_lang),A))) ) ) ).
fof(t12_henmodel,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang))
=> ! [B] :
( ~ v1_xboole_0(B)
=> ( ~ v1_henmodel(A)
=> ! [C] :
( m1_valuat_1(C,B)
=> ! [D] :
( m2_fraenkel(D,k2_qc_lang1,B,k2_valuat_1(B))
=> ~ r6_calcul_1(A,B,C,D) ) ) ) ) ) ).
fof(t13_henmodel,axiom,
v1_henmodel(k6_domain_1(k7_cqc_lang,k9_cqc_lang)) ).
fof(d5_henmodel,axiom,
k2_henmodel = k2_qc_lang1 ).
fof(d6_henmodel,axiom,
! [A] :
( ( v1_henmodel(A)
& m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang)) )
=> ! [B] :
( m1_valuat_1(B,k2_henmodel)
=> ( m1_henmodel(B,A)
<=> ! [C] :
( m1_subset_1(C,k5_qc_lang1)
=> ! [D] :
( m1_subset_1(D,k3_margrel1(k2_henmodel))
=> ( r1_margrel1(k2_henmodel,k8_funct_2(k5_qc_lang1,k3_margrel1(k2_henmodel),B,C),D)
=> ! [E] :
( r2_hidden(E,D)
<=> ? [F] :
( v1_cqc_lang(F,k6_qc_lang1(C))
& m1_qc_lang1(F,k6_qc_lang1(C))
& E = F
& r1_henmodel(A,k3_henmodel(C,F)) ) ) ) ) ) ) ) ) ).
fof(d7_henmodel,axiom,
k4_henmodel = k13_cqc_sim1(k2_qc_lang1) ).
fof(t14_henmodel,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_cqc_lang(B,A)
& m1_qc_lang1(B,A) )
=> k8_valuat_1(k2_henmodel,A,B,k4_henmodel) = B ) ) ).
fof(t15_henmodel,axiom,
! [A] :
( m2_finseq_1(A,k7_cqc_lang)
=> r4_calcul_1(k8_finseq_1(k7_cqc_lang,A,k12_finseq_1(k7_cqc_lang,k9_cqc_lang))) ) ).
fof(t16_henmodel,axiom,
! [A] :
( ( v1_henmodel(A)
& m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang)) )
=> ! [B] :
( m1_henmodel(B,A)
=> ( r1_valuat_1(k2_henmodel,k9_cqc_lang,B,k4_henmodel)
<=> r1_henmodel(A,k9_cqc_lang) ) ) ) ).
fof(t17_henmodel,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k5_qc_lang1,k7_qc_lang1(A))
=> ! [C] :
( ( v1_cqc_lang(C,A)
& m1_qc_lang1(C,A) )
=> ! [D] :
( ( v1_henmodel(D)
& m1_subset_1(D,k1_zfmisc_1(k7_cqc_lang)) )
=> ! [E] :
( m1_henmodel(E,D)
=> ( r1_valuat_1(k2_henmodel,k8_cqc_lang(A,B,C),E,k4_henmodel)
<=> r1_henmodel(D,k8_cqc_lang(A,B,C)) ) ) ) ) ) ) ).
fof(dt_m1_henmodel,axiom,
! [A] :
( ( v1_henmodel(A)
& m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang)) )
=> ! [B] :
( m1_henmodel(B,A)
=> m1_valuat_1(B,k2_henmodel) ) ) ).
fof(existence_m1_henmodel,axiom,
! [A] :
( ( v1_henmodel(A)
& m1_subset_1(A,k1_zfmisc_1(k7_cqc_lang)) )
=> ? [B] : m1_henmodel(B,A) ) ).
fof(dt_k1_henmodel,axiom,
! [A] : m2_subset_1(k1_henmodel(A),k1_numbers,k5_numbers) ).
fof(dt_k2_henmodel,axiom,
~ v1_xboole_0(k2_henmodel) ).
fof(dt_k3_henmodel,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_qc_lang1)
& v1_cqc_lang(B,k6_qc_lang1(A))
& m1_qc_lang1(B,k6_qc_lang1(A)) )
=> m2_subset_1(k3_henmodel(A,B),k8_qc_lang1,k7_cqc_lang) ) ).
fof(redefinition_k3_henmodel,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_qc_lang1)
& v1_cqc_lang(B,k6_qc_lang1(A))
& m1_qc_lang1(B,k6_qc_lang1(A)) )
=> k3_henmodel(A,B) = k9_qc_lang1(A,B) ) ).
fof(dt_k4_henmodel,axiom,
m2_fraenkel(k4_henmodel,k2_qc_lang1,k2_henmodel,k2_valuat_1(k2_henmodel)) ).
%------------------------------------------------------------------------------