SET007 Axioms: SET007+117.ax
%------------------------------------------------------------------------------
% File : SET007+117 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Group and Field Definitions
% 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 : realset1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 48 ( 2 unt; 0 def)
% Number of atoms : 213 ( 11 equ)
% Maximal formula atoms : 13 ( 4 avg)
% Number of connectives : 212 ( 47 ~; 0 |; 85 &)
% ( 9 <=>; 71 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 7 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 19 ( 17 usr; 1 prp; 0-3 aty)
% Number of functors : 13 ( 13 usr; 1 con; 0-3 aty)
% Number of variables : 126 ( 110 !; 16 ?)
% 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(fc1_realset1,axiom,
! [A,B] :
( v1_relat_1(A)
=> v1_relat_1(k1_realset1(A,B)) ) ).
fof(fc2_realset1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v1_relat_1(k1_realset1(A,B))
& v1_funct_1(k1_realset1(A,B)) ) ) ).
fof(rc1_realset1,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_realset1(A) ) ).
fof(rc2_realset1,axiom,
? [A] :
( ~ v1_xboole_0(A)
& ~ v1_realset1(A) ) ).
fof(cc1_realset1,axiom,
! [A] :
( ~ v1_realset1(A)
=> ~ v1_xboole_0(A) ) ).
fof(fc3_realset1,axiom,
! [A] :
( ~ v1_xboole_0(k1_tarski(A))
& v1_finset_1(k1_tarski(A))
& v1_realset1(k1_tarski(A)) ) ).
fof(fc4_realset1,axiom,
! [A,B] :
( ( ~ v1_realset1(A)
& m4_realset1(B,A) )
=> ~ v1_xboole_0(k4_xboole_0(A,B)) ) ).
fof(rc3_realset1,axiom,
! [A] :
( ~ v1_realset1(A)
=> ? [B] :
( m4_realset1(B,A)
& ~ v1_xboole_0(B) ) ) ).
fof(t1_realset1,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ( r2_hidden(B,k2_zfmisc_1(A,A))
=> r2_hidden(k1_funct_1(C,B),A) ) ) ).
fof(t2_realset1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
& ! [D] :
( r2_hidden(D,k2_zfmisc_1(C,C))
=> r2_hidden(k1_funct_1(B,D),C) ) ) ) ).
fof(d1_realset1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r1_realset1(A,B,C)
<=> ! [D] :
( r2_hidden(D,k2_zfmisc_1(C,C))
=> r2_hidden(k1_funct_1(B,D),C) ) ) ) ) ).
fof(d2_realset1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( m1_realset1(C,A,B)
<=> ! [D] :
( r2_hidden(D,k2_zfmisc_1(C,C))
=> r2_hidden(k1_funct_1(B,D),C) ) ) ) ) ).
fof(d3_realset1,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] : k1_realset1(A,B) = k7_relat_1(A,k2_zfmisc_1(B,B)) ) ).
fof(t3_realset1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ! [C] :
( m1_realset1(C,A,B)
=> ( v1_funct_1(k1_realset1(B,C))
& v1_funct_2(k1_realset1(B,C),k2_zfmisc_1(C,C),C)
& m2_relset_1(k1_realset1(B,C),k2_zfmisc_1(C,C),C) ) ) ) ).
fof(d4_realset1,axiom,
! [A] :
( v1_realset1(A)
<=> ~ ( ~ v1_xboole_0(A)
& ! [B] : A != k1_tarski(B) ) ) ).
fof(t4_realset1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ~ v1_realset1(A)
<=> ! [B] : ~ v1_xboole_0(k4_xboole_0(A,k1_tarski(B))) ) ) ).
fof(t5_realset1,axiom,
? [A] :
( ~ v1_xboole_0(A)
& ! [B] :
( m1_subset_1(B,A)
=> ~ v1_xboole_0(k4_xboole_0(A,k1_tarski(B))) ) ) ).
fof(t6_realset1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ~ ( ! [B] :
( m1_subset_1(B,A)
=> ~ v1_xboole_0(k4_xboole_0(A,k1_tarski(B))) )
& v1_realset1(A) ) ) ).
fof(t7_realset1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ! [B] :
( m1_subset_1(B,A)
=> ~ v1_xboole_0(k4_xboole_0(A,k1_tarski(B))) )
=> ~ v1_realset1(A) ) ) ).
fof(d5_realset1,axiom,
! [A] :
( ~ v1_realset1(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ! [C] :
( m1_subset_1(C,A)
=> ( r2_realset1(A,B,C)
<=> ( m1_realset1(k4_xboole_0(A,k1_tarski(C)),A,B)
& v1_funct_1(k4_xboole_0(k1_realset1(B,A),k1_tarski(C)))
& v1_funct_2(k4_xboole_0(k1_realset1(B,A),k1_tarski(C)),k2_zfmisc_1(k4_xboole_0(A,k1_tarski(C)),k4_xboole_0(A,k1_tarski(C))),k4_xboole_0(A,k1_tarski(C)))
& m2_relset_1(k4_xboole_0(k1_realset1(B,A),k1_tarski(C)),k2_zfmisc_1(k4_xboole_0(A,k1_tarski(C)),k4_xboole_0(A,k1_tarski(C))),k4_xboole_0(A,k1_tarski(C))) ) ) ) ) ) ).
fof(t8_realset1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A)
& ! [D] :
( r2_hidden(D,k2_zfmisc_1(B,B))
=> r2_hidden(k1_funct_1(C,D),B) ) ) ) ).
fof(d6_realset1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ( m2_realset1(C,A,B)
<=> ! [D] :
( r2_hidden(D,k2_zfmisc_1(B,B))
=> r2_hidden(k1_funct_1(C,D),B) ) ) ) ) ).
fof(t9_realset1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m2_realset1(C,A,B)
=> ( v1_funct_1(k1_realset1(C,B))
& v1_funct_2(k1_realset1(C,B),k2_zfmisc_1(B,B),B)
& m2_relset_1(k1_realset1(C,B),k2_zfmisc_1(B,B),B) ) ) ) ).
fof(d7_realset1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m2_realset1(C,A,B)
=> k3_realset1(A,B,C) = k1_realset1(C,B) ) ) ).
fof(t10_realset1,axiom,
! [A,B] :
( m1_subset_1(B,A)
=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A)
& ! [D] :
( r2_hidden(D,k2_zfmisc_1(k4_xboole_0(A,k1_tarski(B)),k4_xboole_0(A,k1_tarski(B))))
=> r2_hidden(k1_funct_1(C,D),k4_xboole_0(A,k1_tarski(B))) ) ) ) ).
fof(d8_realset1,axiom,
! [A,B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ( m3_realset1(C,A,B)
<=> ! [D] :
( r2_hidden(D,k2_zfmisc_1(k4_xboole_0(A,k1_tarski(B)),k4_xboole_0(A,k1_tarski(B))))
=> r2_hidden(k1_funct_1(C,D),k4_xboole_0(A,k1_tarski(B))) ) ) ) ) ).
fof(t11_realset1,axiom,
! [A] :
( ~ v1_realset1(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m3_realset1(C,A,B)
=> ( v1_funct_1(k1_realset1(C,k4_xboole_0(A,k1_tarski(B))))
& v1_funct_2(k1_realset1(C,k4_xboole_0(A,k1_tarski(B))),k2_zfmisc_1(k4_xboole_0(A,k1_tarski(B)),k4_xboole_0(A,k1_tarski(B))),k4_xboole_0(A,k1_tarski(B)))
& m2_relset_1(k1_realset1(C,k4_xboole_0(A,k1_tarski(B))),k2_zfmisc_1(k4_xboole_0(A,k1_tarski(B)),k4_xboole_0(A,k1_tarski(B))),k4_xboole_0(A,k1_tarski(B))) ) ) ) ) ).
fof(d9_realset1,axiom,
! [A] :
( ~ v1_realset1(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m3_realset1(C,A,B)
=> k4_realset1(A,B,C) = k1_realset1(C,k4_xboole_0(A,k1_tarski(B))) ) ) ) ).
fof(d10_realset1,axiom,
! [A] :
( ~ v1_realset1(A)
=> ! [B] :
( m4_realset1(B,A)
<=> ? [C] :
( m1_subset_1(C,A)
& B = k1_tarski(C) ) ) ) ).
fof(t12_realset1,axiom,
! [A] :
( ~ v1_realset1(A)
=> ! [B] :
( m4_realset1(B,A)
=> ~ v1_xboole_0(k4_xboole_0(A,B)) ) ) ).
fof(t13_realset1,axiom,
! [A] :
( v1_finset_1(A)
=> ( v1_realset1(A)
<=> ~ r1_xreal_0(np__2,k4_card_1(A)) ) ) ).
fof(t14_realset1,axiom,
! [A] :
( ~ ( ~ v1_realset1(A)
& ! [B,C] :
~ ( r2_hidden(B,A)
& r2_hidden(C,A)
& B != C ) )
& ~ ( ? [B,C] :
( r2_hidden(B,A)
& r2_hidden(C,A)
& B != C )
& v1_realset1(A) ) ) ).
fof(t15_realset1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( ~ ( ~ v1_realset1(B)
& ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ~ ( r2_hidden(C,B)
& r2_hidden(D,B)
& C != D ) ) ) )
& ~ ( ? [C] :
( m1_subset_1(C,A)
& ? [D] :
( m1_subset_1(D,A)
& r2_hidden(C,B)
& r2_hidden(D,B)
& C != D ) )
& v1_realset1(B) ) ) ) ).
fof(dt_m1_realset1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ! [C] :
( m1_realset1(C,A,B)
=> m1_subset_1(C,k1_zfmisc_1(A)) ) ) ).
fof(existence_m1_realset1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ? [C] : m1_realset1(C,A,B) ) ).
fof(dt_m2_realset1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m2_realset1(C,A,B)
=> ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) ) ) ) ).
fof(existence_m2_realset1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ? [C] : m2_realset1(C,A,B) ) ).
fof(dt_m3_realset1,axiom,
! [A,B] :
( m1_subset_1(B,A)
=> ! [C] :
( m3_realset1(C,A,B)
=> ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) ) ) ) ).
fof(existence_m3_realset1,axiom,
! [A,B] :
( m1_subset_1(B,A)
=> ? [C] : m3_realset1(C,A,B) ) ).
fof(dt_m4_realset1,axiom,
$true ).
fof(existence_m4_realset1,axiom,
! [A] :
( ~ v1_realset1(A)
=> ? [B] : m4_realset1(B,A) ) ).
fof(dt_k1_realset1,axiom,
$true ).
fof(dt_k2_realset1,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A)
& m1_realset1(C,A,B) )
=> ( v1_funct_1(k2_realset1(A,B,C))
& v1_funct_2(k2_realset1(A,B,C),k2_zfmisc_1(C,C),C)
& m2_relset_1(k2_realset1(A,B,C),k2_zfmisc_1(C,C),C) ) ) ).
fof(redefinition_k2_realset1,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A)
& m1_realset1(C,A,B) )
=> k2_realset1(A,B,C) = k1_realset1(B,C) ) ).
fof(dt_k3_realset1,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k1_zfmisc_1(A))
& m2_realset1(C,A,B) )
=> ( v1_funct_1(k3_realset1(A,B,C))
& v1_funct_2(k3_realset1(A,B,C),k2_zfmisc_1(B,B),B)
& m2_relset_1(k3_realset1(A,B,C),k2_zfmisc_1(B,B),B) ) ) ).
fof(dt_k4_realset1,axiom,
! [A,B,C] :
( ( ~ v1_realset1(A)
& m1_subset_1(B,A)
& m3_realset1(C,A,B) )
=> ( v1_funct_1(k4_realset1(A,B,C))
& v1_funct_2(k4_realset1(A,B,C),k2_zfmisc_1(k4_xboole_0(A,k1_tarski(B)),k4_xboole_0(A,k1_tarski(B))),k4_xboole_0(A,k1_tarski(B)))
& m2_relset_1(k4_realset1(A,B,C),k2_zfmisc_1(k4_xboole_0(A,k1_tarski(B)),k4_xboole_0(A,k1_tarski(B))),k4_xboole_0(A,k1_tarski(B))) ) ) ).
fof(dt_k5_realset1,axiom,
! [A,B] :
( ( ~ v1_realset1(A)
& m1_subset_1(B,A) )
=> m4_realset1(k5_realset1(A,B),A) ) ).
fof(redefinition_k5_realset1,axiom,
! [A,B] :
( ( ~ v1_realset1(A)
& m1_subset_1(B,A) )
=> k5_realset1(A,B) = k1_tarski(B) ) ).
%------------------------------------------------------------------------------