SET007 Axioms: SET007+29.ax
%------------------------------------------------------------------------------
% File : SET007+29 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Non-Negative Real Numbers. Part I
% 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 : arytm_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 51 ( 11 unt; 0 def)
% Number of atoms : 248 ( 55 equ)
% Maximal formula atoms : 26 ( 4 avg)
% Number of connectives : 233 ( 36 ~; 3 |; 77 &)
% ( 17 <=>; 100 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 10 ( 8 usr; 1 prp; 0-3 aty)
% Number of functors : 24 ( 24 usr; 8 con; 0-2 aty)
% Number of variables : 113 ( 96 !; 17 ?)
% 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_arytm_2,axiom,
~ v1_xboole_0(k1_arytm_2) ).
fof(fc2_arytm_2,axiom,
~ v1_xboole_0(k2_arytm_2) ).
fof(t1_arytm_2,axiom,
r1_tarski(k6_arytm_3,k2_arytm_2) ).
fof(t2_arytm_2,axiom,
r1_tarski(k5_ordinal2,k2_arytm_2) ).
fof(t3_arytm_2,axiom,
! [A] : ~ r2_hidden(k4_tarski(k12_arytm_3,A),k2_arytm_2) ).
fof(d4_arytm_2,axiom,
! [A] :
( m2_subset_1(A,k1_zfmisc_1(k6_arytm_3),k1_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ( ( ? [C] :
( m1_subset_1(C,k6_arytm_3)
& ! [D] :
( m1_subset_1(D,k6_arytm_3)
=> ( r2_hidden(D,A)
<=> ~ r3_arytm_3(C,D) ) ) )
=> ( B = k4_arytm_2(A)
<=> ? [C] :
( m1_subset_1(C,k6_arytm_3)
& B = C
& ! [D] :
( m1_subset_1(D,k6_arytm_3)
=> ( r2_hidden(D,A)
<=> ~ r3_arytm_3(C,D) ) ) ) ) )
& ( ! [C] :
( m1_subset_1(C,k6_arytm_3)
=> ~ ! [D] :
( m1_subset_1(D,k6_arytm_3)
=> ( r2_hidden(D,A)
<=> ~ r3_arytm_3(C,D) ) ) )
=> ( B = k4_arytm_2(A)
<=> B = A ) ) ) ) ) ).
fof(d5_arytm_2,axiom,
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ( ( ( r2_hidden(A,k6_arytm_3)
& r2_hidden(B,k6_arytm_3) )
=> ( r1_arytm_2(A,B)
<=> ? [C] :
( m1_subset_1(C,k6_arytm_3)
& ? [D] :
( m1_subset_1(D,k6_arytm_3)
& A = C
& B = D
& r3_arytm_3(C,D) ) ) ) )
& ( r2_hidden(A,k6_arytm_3)
=> ( r2_hidden(B,k6_arytm_3)
| ( r1_arytm_2(A,B)
<=> r2_hidden(A,B) ) ) )
& ( r2_hidden(B,k6_arytm_3)
=> ( r2_hidden(A,k6_arytm_3)
| ( r1_arytm_2(A,B)
<=> ~ r2_hidden(B,A) ) ) )
& ~ ( ~ ( r2_hidden(A,k6_arytm_3)
& r2_hidden(B,k6_arytm_3) )
& ~ ( r2_hidden(A,k6_arytm_3)
& ~ r2_hidden(B,k6_arytm_3) )
& ~ ( ~ r2_hidden(A,k6_arytm_3)
& r2_hidden(B,k6_arytm_3) )
& ~ ( r1_arytm_2(A,B)
<=> r1_tarski(A,B) ) ) ) ) ) ).
fof(d8_arytm_2,axiom,
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ( ( B = k12_arytm_3
=> k7_arytm_2(A,B) = A )
& ( A = k12_arytm_3
=> k7_arytm_2(A,B) = B )
& ~ ( B != k12_arytm_3
& A != k12_arytm_3
& k7_arytm_2(A,B) != k4_arytm_2(k5_arytm_2(k3_arytm_2(A),k3_arytm_2(B))) ) ) ) ) ).
fof(d9_arytm_2,axiom,
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> k8_arytm_2(A,B) = k4_arytm_2(k6_arytm_2(k3_arytm_2(A),k3_arytm_2(B))) ) ) ).
fof(t4_arytm_2,axiom,
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ( A = k12_arytm_3
=> k8_arytm_2(A,B) = k12_arytm_3 ) ) ) ).
fof(t5_arytm_2,axiom,
$true ).
fof(t6_arytm_2,axiom,
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ( k7_arytm_2(A,B) = k12_arytm_3
=> A = k12_arytm_3 ) ) ) ).
fof(t7_arytm_2,axiom,
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ! [C] :
( m1_subset_1(C,k2_arytm_2)
=> k7_arytm_2(A,k7_arytm_2(B,C)) = k7_arytm_2(k7_arytm_2(A,B),C) ) ) ) ).
fof(t9_arytm_2,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k2_arytm_2))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k2_arytm_2))
=> ~ ( ? [C] :
( m1_subset_1(C,k2_arytm_2)
& r2_hidden(C,A) )
& ? [C] :
( m1_subset_1(C,k2_arytm_2)
& r2_hidden(C,B) )
& ! [C] :
( m1_subset_1(C,k2_arytm_2)
=> ! [D] :
( m1_subset_1(D,k2_arytm_2)
=> ( ( r2_hidden(C,A)
& r2_hidden(D,B) )
=> r1_arytm_2(C,D) ) ) )
& ! [C] :
( m1_subset_1(C,k2_arytm_2)
=> ? [D] :
( m1_subset_1(D,k2_arytm_2)
& ? [E] :
( m1_subset_1(E,k2_arytm_2)
& r2_hidden(D,A)
& r2_hidden(E,B)
& ~ ( r1_arytm_2(D,C)
& r1_arytm_2(C,E) ) ) ) ) ) ) ) ).
fof(t10_arytm_2,axiom,
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ~ ( r1_arytm_2(A,B)
& ! [C] :
( m1_subset_1(C,k2_arytm_2)
=> k7_arytm_2(A,C) != B ) ) ) ) ).
fof(t11_arytm_2,axiom,
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ~ ! [C] :
( m1_subset_1(C,k2_arytm_2)
=> ( k7_arytm_2(A,C) != B
& k7_arytm_2(B,C) != A ) ) ) ) ).
fof(t12_arytm_2,axiom,
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ! [C] :
( m1_subset_1(C,k2_arytm_2)
=> ( k7_arytm_2(A,B) = k7_arytm_2(A,C)
=> B = C ) ) ) ) ).
fof(t13_arytm_2,axiom,
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ! [C] :
( m1_subset_1(C,k2_arytm_2)
=> k8_arytm_2(A,k8_arytm_2(B,C)) = k8_arytm_2(k8_arytm_2(A,B),C) ) ) ) ).
fof(t14_arytm_2,axiom,
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ! [C] :
( m1_subset_1(C,k2_arytm_2)
=> k8_arytm_2(A,k7_arytm_2(B,C)) = k7_arytm_2(k8_arytm_2(A,B),k8_arytm_2(A,C)) ) ) ) ).
fof(t15_arytm_2,axiom,
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ~ ( A != k12_arytm_3
& ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> k8_arytm_2(A,B) != k13_arytm_3 ) ) ) ).
fof(t16_arytm_2,axiom,
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ( A = k13_arytm_3
=> k8_arytm_2(A,B) = B ) ) ) ).
fof(t17_arytm_2,axiom,
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ( ( r2_hidden(A,k5_ordinal2)
& r2_hidden(B,k5_ordinal2) )
=> r2_hidden(k7_arytm_2(B,A),k5_ordinal2) ) ) ) ).
fof(t18_arytm_2,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k2_arytm_2))
=> ( ( r2_hidden(k12_arytm_3,A)
& ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ! [C] :
( m1_subset_1(C,k2_arytm_2)
=> ( ( r2_hidden(B,A)
& C = k13_arytm_3 )
=> r2_hidden(k7_arytm_2(B,C),A) ) ) ) )
=> r1_tarski(k5_ordinal2,A) ) ) ).
fof(t19_arytm_2,axiom,
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ( r2_hidden(A,k5_ordinal2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ( r2_hidden(B,A)
<=> ( r2_hidden(B,k5_ordinal2)
& B != A
& r1_arytm_2(B,A) ) ) ) ) ) ).
fof(t20_arytm_2,axiom,
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ! [C] :
( m1_subset_1(C,k2_arytm_2)
=> ( A = k7_arytm_2(B,C)
=> r1_arytm_2(C,A) ) ) ) ) ).
fof(t21_arytm_2,axiom,
( r2_hidden(k12_arytm_3,k2_arytm_2)
& r2_hidden(k13_arytm_3,k2_arytm_2) ) ).
fof(t22_arytm_2,axiom,
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m1_subset_1(B,k2_arytm_2)
=> ~ ( r2_hidden(A,k6_arytm_3)
& r2_hidden(B,k6_arytm_3)
& ! [C] :
( m1_subset_1(C,k6_arytm_3)
=> ! [D] :
( m1_subset_1(D,k6_arytm_3)
=> ~ ( A = C
& B = D
& k8_arytm_2(A,B) = k11_arytm_3(C,D) ) ) ) ) ) ) ).
fof(connectedness_r1_arytm_2,axiom,
! [A,B] :
( ( m1_subset_1(A,k2_arytm_2)
& m1_subset_1(B,k2_arytm_2) )
=> ( r1_arytm_2(A,B)
| r1_arytm_2(B,A) ) ) ).
fof(dt_k1_arytm_2,axiom,
m1_subset_1(k1_arytm_2,k1_zfmisc_1(k1_zfmisc_1(k6_arytm_3))) ).
fof(dt_k2_arytm_2,axiom,
$true ).
fof(dt_k3_arytm_2,axiom,
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> m2_subset_1(k3_arytm_2(A),k1_zfmisc_1(k6_arytm_3),k1_arytm_2) ) ).
fof(dt_k4_arytm_2,axiom,
! [A] :
( m1_subset_1(A,k1_arytm_2)
=> m1_subset_1(k4_arytm_2(A),k2_arytm_2) ) ).
fof(dt_k5_arytm_2,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_arytm_2)
& m1_subset_1(B,k1_arytm_2) )
=> m2_subset_1(k5_arytm_2(A,B),k1_zfmisc_1(k6_arytm_3),k1_arytm_2) ) ).
fof(commutativity_k5_arytm_2,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_arytm_2)
& m1_subset_1(B,k1_arytm_2) )
=> k5_arytm_2(A,B) = k5_arytm_2(B,A) ) ).
fof(dt_k6_arytm_2,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_arytm_2)
& m1_subset_1(B,k1_arytm_2) )
=> m2_subset_1(k6_arytm_2(A,B),k1_zfmisc_1(k6_arytm_3),k1_arytm_2) ) ).
fof(commutativity_k6_arytm_2,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_arytm_2)
& m1_subset_1(B,k1_arytm_2) )
=> k6_arytm_2(A,B) = k6_arytm_2(B,A) ) ).
fof(dt_k7_arytm_2,axiom,
! [A,B] :
( ( m1_subset_1(A,k2_arytm_2)
& m1_subset_1(B,k2_arytm_2) )
=> m1_subset_1(k7_arytm_2(A,B),k2_arytm_2) ) ).
fof(commutativity_k7_arytm_2,axiom,
! [A,B] :
( ( m1_subset_1(A,k2_arytm_2)
& m1_subset_1(B,k2_arytm_2) )
=> k7_arytm_2(A,B) = k7_arytm_2(B,A) ) ).
fof(dt_k8_arytm_2,axiom,
! [A,B] :
( ( m1_subset_1(A,k2_arytm_2)
& m1_subset_1(B,k2_arytm_2) )
=> m1_subset_1(k8_arytm_2(A,B),k2_arytm_2) ) ).
fof(commutativity_k8_arytm_2,axiom,
! [A,B] :
( ( m1_subset_1(A,k2_arytm_2)
& m1_subset_1(B,k2_arytm_2) )
=> k8_arytm_2(A,B) = k8_arytm_2(B,A) ) ).
fof(d1_arytm_2,axiom,
k1_arytm_2 = k4_xboole_0(a_0_0_arytm_2,k1_tarski(k6_arytm_3)) ).
fof(d2_arytm_2,axiom,
k2_arytm_2 = k4_xboole_0(k2_xboole_0(k6_arytm_3,k1_arytm_2),a_0_1_arytm_2) ).
fof(d3_arytm_2,axiom,
! [A] :
( m1_subset_1(A,k2_arytm_2)
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(k6_arytm_3),k1_arytm_2)
=> ( ( r2_hidden(A,k6_arytm_3)
=> ( B = k3_arytm_2(A)
<=> ? [C] :
( m1_subset_1(C,k6_arytm_3)
& A = C
& B = a_1_0_arytm_2(C) ) ) )
& ( ~ r2_hidden(A,k6_arytm_3)
=> ( B = k3_arytm_2(A)
<=> B = A ) ) ) ) ) ).
fof(d6_arytm_2,axiom,
! [A] :
( m2_subset_1(A,k1_zfmisc_1(k6_arytm_3),k1_arytm_2)
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(k6_arytm_3),k1_arytm_2)
=> k5_arytm_2(A,B) = a_2_0_arytm_2(A,B) ) ) ).
fof(d7_arytm_2,axiom,
! [A] :
( m2_subset_1(A,k1_zfmisc_1(k6_arytm_3),k1_arytm_2)
=> ! [B] :
( m2_subset_1(B,k1_zfmisc_1(k6_arytm_3),k1_arytm_2)
=> k6_arytm_2(A,B) = a_2_1_arytm_2(A,B) ) ) ).
fof(t8_arytm_2,axiom,
v6_ordinal1(a_0_0_arytm_2) ).
fof(fraenkel_a_0_0_arytm_2,axiom,
! [A] :
( r2_hidden(A,a_0_0_arytm_2)
<=> ? [B] :
( m1_subset_1(B,k1_zfmisc_1(k6_arytm_3))
& A = B
& ! [C] :
( m1_subset_1(C,k6_arytm_3)
=> ( r2_hidden(C,B)
=> ( ! [D] :
( m1_subset_1(D,k6_arytm_3)
=> ( r3_arytm_3(D,C)
=> r2_hidden(D,B) ) )
& ? [D] :
( m1_subset_1(D,k6_arytm_3)
& r2_hidden(D,B)
& ~ r3_arytm_3(D,C) ) ) ) ) ) ) ).
fof(fraenkel_a_0_1_arytm_2,axiom,
! [A] :
( r2_hidden(A,a_0_1_arytm_2)
<=> ? [B] :
( m1_subset_1(B,k6_arytm_3)
& A = a_1_0_arytm_2(B)
& B != k12_arytm_3 ) ) ).
fof(fraenkel_a_1_0_arytm_2,axiom,
! [A,B] :
( m1_subset_1(B,k6_arytm_3)
=> ( r2_hidden(A,a_1_0_arytm_2(B))
<=> ? [C] :
( m1_subset_1(C,k6_arytm_3)
& A = C
& ~ r3_arytm_3(B,C) ) ) ) ).
fof(fraenkel_a_2_0_arytm_2,axiom,
! [A,B,C] :
( ( m2_subset_1(B,k1_zfmisc_1(k6_arytm_3),k1_arytm_2)
& m2_subset_1(C,k1_zfmisc_1(k6_arytm_3),k1_arytm_2) )
=> ( r2_hidden(A,a_2_0_arytm_2(B,C))
<=> ? [D,E] :
( m1_subset_1(D,k6_arytm_3)
& m1_subset_1(E,k6_arytm_3)
& A = k10_arytm_3(D,E)
& r2_hidden(D,B)
& r2_hidden(E,C) ) ) ) ).
fof(fraenkel_a_2_1_arytm_2,axiom,
! [A,B,C] :
( ( m2_subset_1(B,k1_zfmisc_1(k6_arytm_3),k1_arytm_2)
& m2_subset_1(C,k1_zfmisc_1(k6_arytm_3),k1_arytm_2) )
=> ( r2_hidden(A,a_2_1_arytm_2(B,C))
<=> ? [D,E] :
( m1_subset_1(D,k6_arytm_3)
& m1_subset_1(E,k6_arytm_3)
& A = k11_arytm_3(D,E)
& r2_hidden(D,B)
& r2_hidden(E,C) ) ) ) ).
%------------------------------------------------------------------------------