SET007 Axioms: SET007+196.ax
%------------------------------------------------------------------------------
% File : SET007+196 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Abian's Fixed Point Theorem
% 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 : abian [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 43 ( 0 unt; 0 def)
% Number of atoms : 277 ( 14 equ)
% Maximal formula atoms : 14 ( 6 avg)
% Number of connectives : 281 ( 47 ~; 0 |; 169 &)
% ( 10 <=>; 55 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 8 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 31 ( 30 usr; 0 prp; 1-3 aty)
% Number of functors : 23 ( 23 usr; 5 con; 0-4 aty)
% Number of variables : 98 ( 86 !; 12 ?)
% 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_abian,axiom,
? [A] :
( m1_subset_1(A,k5_numbers)
& v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A)
& v4_ordinal2(A)
& v1_xcmplx_0(A)
& v1_xreal_0(A)
& ~ v3_xreal_0(A)
& v1_int_1(A)
& v1_abian(A) ) ).
fof(rc2_abian,axiom,
? [A] :
( m1_subset_1(A,k5_numbers)
& v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A)
& v4_ordinal2(A)
& v1_xcmplx_0(A)
& v1_xreal_0(A)
& ~ v3_xreal_0(A)
& v1_int_1(A)
& ~ v1_abian(A) ) ).
fof(rc3_abian,axiom,
? [A] :
( v1_xcmplx_0(A)
& v1_xreal_0(A)
& v1_int_1(A)
& v1_abian(A) ) ).
fof(rc4_abian,axiom,
? [A] :
( v1_xcmplx_0(A)
& v1_xreal_0(A)
& v1_int_1(A)
& ~ v1_abian(A) ) ).
fof(fc1_abian,axiom,
! [A] :
( v1_int_1(A)
=> ( v1_xcmplx_0(k3_xcmplx_0(np__2,A))
& v1_xreal_0(k3_xcmplx_0(np__2,A))
& v1_int_1(k3_xcmplx_0(np__2,A))
& v1_abian(k3_xcmplx_0(np__2,A)) ) ) ).
fof(fc2_abian,axiom,
! [A] :
( ( v1_int_1(A)
& v1_abian(A) )
=> ( v1_xcmplx_0(k2_xcmplx_0(A,np__1))
& v1_xreal_0(k2_xcmplx_0(A,np__1))
& v1_int_1(k2_xcmplx_0(A,np__1))
& ~ v1_abian(k2_xcmplx_0(A,np__1)) ) ) ).
fof(fc3_abian,axiom,
! [A] :
( ( v1_int_1(A)
& ~ v1_abian(A) )
=> ( v1_xcmplx_0(k2_xcmplx_0(A,np__1))
& v1_xreal_0(k2_xcmplx_0(A,np__1))
& v1_int_1(k2_xcmplx_0(A,np__1))
& v1_abian(k2_xcmplx_0(A,np__1)) ) ) ).
fof(fc4_abian,axiom,
! [A] :
( ( v1_int_1(A)
& v1_abian(A) )
=> ( v1_xcmplx_0(k6_xcmplx_0(A,np__1))
& v1_xreal_0(k6_xcmplx_0(A,np__1))
& v1_int_1(k6_xcmplx_0(A,np__1))
& ~ v1_abian(k6_xcmplx_0(A,np__1)) ) ) ).
fof(fc5_abian,axiom,
! [A] :
( ( v1_int_1(A)
& ~ v1_abian(A) )
=> ( v1_xcmplx_0(k6_xcmplx_0(A,np__1))
& v1_xreal_0(k6_xcmplx_0(A,np__1))
& v1_int_1(k6_xcmplx_0(A,np__1))
& v1_abian(k6_xcmplx_0(A,np__1)) ) ) ).
fof(fc6_abian,axiom,
! [A,B] :
( ( v1_int_1(A)
& v1_abian(A)
& v1_int_1(B) )
=> ( v1_xcmplx_0(k3_xcmplx_0(A,B))
& v1_xreal_0(k3_xcmplx_0(A,B))
& v1_int_1(k3_xcmplx_0(A,B))
& v1_abian(k3_xcmplx_0(A,B)) ) ) ).
fof(fc7_abian,axiom,
! [A,B] :
( ( v1_int_1(A)
& v1_abian(A)
& v1_int_1(B) )
=> ( v1_xcmplx_0(k3_xcmplx_0(B,A))
& v1_xreal_0(k3_xcmplx_0(B,A))
& v1_int_1(k3_xcmplx_0(B,A))
& v1_abian(k3_xcmplx_0(B,A)) ) ) ).
fof(fc8_abian,axiom,
! [A,B] :
( ( v1_int_1(A)
& ~ v1_abian(A)
& v1_int_1(B)
& ~ v1_abian(B) )
=> ( v1_xcmplx_0(k3_xcmplx_0(A,B))
& v1_xreal_0(k3_xcmplx_0(A,B))
& v1_int_1(k3_xcmplx_0(A,B))
& ~ v1_abian(k3_xcmplx_0(A,B)) ) ) ).
fof(fc9_abian,axiom,
! [A,B] :
( ( v1_int_1(A)
& v1_abian(A)
& v1_int_1(B)
& v1_abian(B) )
=> ( v1_xcmplx_0(k2_xcmplx_0(A,B))
& v1_xreal_0(k2_xcmplx_0(A,B))
& v1_int_1(k2_xcmplx_0(A,B))
& v1_abian(k2_xcmplx_0(A,B)) ) ) ).
fof(fc10_abian,axiom,
! [A,B] :
( ( v1_int_1(A)
& v1_abian(A)
& v1_int_1(B)
& ~ v1_abian(B) )
=> ( v1_xcmplx_0(k2_xcmplx_0(A,B))
& v1_xreal_0(k2_xcmplx_0(A,B))
& v1_int_1(k2_xcmplx_0(A,B))
& ~ v1_abian(k2_xcmplx_0(A,B)) ) ) ).
fof(fc11_abian,axiom,
! [A,B] :
( ( v1_int_1(A)
& v1_abian(A)
& v1_int_1(B)
& ~ v1_abian(B) )
=> ( v1_xcmplx_0(k2_xcmplx_0(B,A))
& v1_xreal_0(k2_xcmplx_0(B,A))
& v1_int_1(k2_xcmplx_0(B,A))
& ~ v1_abian(k2_xcmplx_0(B,A)) ) ) ).
fof(fc12_abian,axiom,
! [A,B] :
( ( v1_int_1(A)
& ~ v1_abian(A)
& v1_int_1(B)
& ~ v1_abian(B) )
=> ( v1_xcmplx_0(k2_xcmplx_0(A,B))
& v1_xreal_0(k2_xcmplx_0(A,B))
& v1_int_1(k2_xcmplx_0(A,B))
& v1_abian(k2_xcmplx_0(A,B)) ) ) ).
fof(fc13_abian,axiom,
! [A,B] :
( ( v1_int_1(A)
& v1_abian(A)
& v1_int_1(B)
& ~ v1_abian(B) )
=> ( v1_xcmplx_0(k6_xcmplx_0(A,B))
& v1_xreal_0(k6_xcmplx_0(A,B))
& v1_int_1(k6_xcmplx_0(A,B))
& ~ v1_abian(k6_xcmplx_0(A,B)) ) ) ).
fof(fc14_abian,axiom,
! [A,B] :
( ( v1_int_1(A)
& v1_abian(A)
& v1_int_1(B)
& ~ v1_abian(B) )
=> ( v1_xcmplx_0(k6_xcmplx_0(B,A))
& v1_xreal_0(k6_xcmplx_0(B,A))
& v1_int_1(k6_xcmplx_0(B,A))
& ~ v1_abian(k6_xcmplx_0(B,A)) ) ) ).
fof(fc15_abian,axiom,
! [A,B] :
( ( v1_int_1(A)
& ~ v1_abian(A)
& v1_int_1(B)
& ~ v1_abian(B) )
=> ( v1_xcmplx_0(k6_xcmplx_0(A,B))
& v1_xreal_0(k6_xcmplx_0(A,B))
& v1_int_1(k6_xcmplx_0(A,B))
& v1_abian(k6_xcmplx_0(A,B)) ) ) ).
fof(rc5_abian,axiom,
! [A] :
? [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
& ~ v1_xboole_0(B)
& v1_finset_1(B)
& v2_abian(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) ).
fof(cc1_abian,axiom,
! [A] :
( v2_setfam_1(A)
=> v1_realset1(A) ) ).
fof(rc6_abian,axiom,
! [A,B] :
( ~ v2_setfam_1(B)
=> ? [C] :
( m1_relset_1(C,A,B)
& v1_relat_1(C)
& v2_relat_1(C)
& v1_funct_1(C)
& v1_funct_2(C,A,B) ) ) ).
fof(fc16_abian,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v2_setfam_1(B)
& v2_relat_1(C)
& v1_funct_1(C)
& v1_funct_2(C,A,B)
& m1_relset_1(C,A,B)
& m1_subset_1(D,A) )
=> ~ v1_xboole_0(k1_funct_1(C,D)) ) ).
fof(fc17_abian,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ~ v1_xboole_0(k1_zfmisc_1(A))
& ~ v2_setfam_1(k1_zfmisc_1(A)) ) ) ).
fof(d1_abian,axiom,
! [A] :
( v1_abian(A)
<=> ? [B] :
( v1_int_1(B)
& A = k3_xcmplx_0(np__2,B) ) ) ).
fof(d2_abian,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( v1_abian(A)
<=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& A = k2_nat_1(np__2,B) ) ) ) ).
fof(t1_abian,axiom,
! [A] :
( v1_int_1(A)
=> ( ~ ( ~ v1_abian(A)
& ! [B] :
( v1_int_1(B)
=> A != k2_xcmplx_0(k3_xcmplx_0(np__2,B),np__1) ) )
& ~ ( ? [B] :
( v1_int_1(B)
& A = k2_xcmplx_0(k3_xcmplx_0(np__2,B),np__1) )
& v1_abian(A) ) ) ) ).
fof(t2_abian,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k5_numbers)) )
=> ( r2_hidden(np__0,A)
=> k10_cqc_sim1(A) = np__0 ) ) ).
fof(t3_abian,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( m1_subset_1(C,A)
=> k8_funct_2(A,A,k1_abian(A,B,np__0),C) = C ) ) ) ).
fof(d3_abian,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_abian(A,B)
<=> ( r2_hidden(A,k1_relat_1(B))
& A = k1_funct_1(B,A) ) ) ) ).
fof(d4_abian,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,A)
& m2_relset_1(C,A,A) )
=> ( r2_abian(A,B,C)
<=> B = k8_funct_2(A,A,C,B) ) ) ) ) ).
fof(d5_abian,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( r3_abian(A)
<=> ? [B] : r1_abian(B,A) ) ) ).
fof(d6_abian,axiom,
! [A,B] :
( m1_subset_1(B,A)
=> ( v2_abian(B,A)
<=> k3_tarski(B) = k3_tarski(k3_tarski(A)) ) ) ).
fof(t4_abian,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( v2_abian(B,k1_zfmisc_1(k1_zfmisc_1(A)))
<=> k5_setfam_1(A,B) = A ) ) ).
fof(t5_abian,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v2_abian(C,k1_zfmisc_1(k1_zfmisc_1(A)))
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ~ ( ! [D] :
( m2_subset_1(D,k1_zfmisc_1(A),C)
=> r1_xboole_0(D,k2_funct_2(A,A,B,D)) )
& r3_abian(B) ) ) ) ).
fof(d7_abian,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( ( v3_relat_2(C)
& v8_relat_2(C)
& v1_partfun1(C,A,A)
& m2_relset_1(C,A,A) )
=> ( C = k2_abian(A,B)
<=> ! [D,E] :
( ( r2_hidden(D,A)
& r2_hidden(E,A) )
=> ( r2_hidden(k4_tarski(D,E),C)
<=> ? [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
& ? [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
& k1_funct_1(k1_abian(A,B,F),D) = k1_funct_1(k1_abian(A,B,G),E) ) ) ) ) ) ) ) ).
fof(t6_abian,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(A),k8_eqrel_1(A,k2_abian(A,B)))
=> ! [D] :
( m2_subset_1(D,A,C)
=> r2_hidden(k8_funct_2(A,A,B,D),C) ) ) ) ) ).
fof(t7_abian,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(A),k8_eqrel_1(A,k2_abian(A,B)))
=> ! [D] :
( m2_subset_1(D,A,C)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> r2_hidden(k8_funct_2(A,A,k1_abian(A,B,E),D),C) ) ) ) ) ) ).
fof(t8_abian,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,A)
& m2_relset_1(B,A,A) )
=> ~ ( ~ r3_abian(B)
& ! [C,D,E] :
~ ( k2_xboole_0(k2_xboole_0(C,D),E) = A
& r1_xboole_0(k2_funct_2(A,A,B,C),C)
& r1_xboole_0(k2_funct_2(A,A,B,D),D)
& r1_xboole_0(k2_funct_2(A,A,B,E),E) ) ) ) ) ).
fof(redefinition_r2_abian,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A)
& v1_funct_1(C)
& v1_funct_2(C,A,A)
& m1_relset_1(C,A,A) )
=> ( r2_abian(A,B,C)
<=> r1_abian(B,C) ) ) ).
fof(dt_k1_abian,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,A)
& m1_relset_1(B,A,A)
& m1_subset_1(C,k5_numbers) )
=> ( v1_funct_1(k1_abian(A,B,C))
& v1_funct_2(k1_abian(A,B,C),A,A)
& m2_relset_1(k1_abian(A,B,C),A,A) ) ) ).
fof(redefinition_k1_abian,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,A)
& m1_relset_1(B,A,A)
& m1_subset_1(C,k5_numbers) )
=> k1_abian(A,B,C) = k9_funct_7(B,C) ) ).
fof(dt_k2_abian,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,A)
& m1_relset_1(B,A,A) )
=> ( v3_relat_2(k2_abian(A,B))
& v8_relat_2(k2_abian(A,B))
& v1_partfun1(k2_abian(A,B),A,A)
& m2_relset_1(k2_abian(A,B),A,A) ) ) ).
%------------------------------------------------------------------------------