SET007 Axioms: SET007+119.ax
%------------------------------------------------------------------------------
% File : SET007+119 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Infimum and Supremum of the Set of Real Numbers
% 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 : supinf_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 179 ( 65 unt; 0 def)
% Number of atoms : 619 ( 71 equ)
% Maximal formula atoms : 15 ( 3 avg)
% Number of connectives : 539 ( 99 ~; 5 |; 129 &)
% ( 27 <=>; 279 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 5 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 20 ( 18 usr; 1 prp; 0-2 aty)
% Number of functors : 22 ( 22 usr; 7 con; 0-3 aty)
% Number of variables : 244 ( 223 !; 21 ?)
% 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_supinf_1,axiom,
? [A] : v1_supinf_1(A) ).
fof(rc2_supinf_1,axiom,
? [A] : v2_supinf_1(A) ).
fof(rc3_supinf_1,axiom,
? [A] : v3_supinf_1(A) ).
fof(cc1_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> v3_supinf_1(A) ) ).
fof(cc2_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( v1_xcmplx_0(A)
& v1_xreal_0(A)
& v3_supinf_1(A) ) ) ).
fof(fc1_supinf_1,axiom,
~ v1_xboole_0(k3_supinf_1) ).
fof(fc2_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_supinf_1))
=> ~ v1_xboole_0(k7_supinf_1(A)) ) ).
fof(fc3_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_supinf_1))
=> ~ v1_xboole_0(k8_supinf_1(A)) ) ).
fof(fc4_supinf_1,axiom,
! [A] :
( m3_supinf_1(A,k3_supinf_1)
=> ~ v1_xboole_0(k13_supinf_1(A)) ) ).
fof(fc5_supinf_1,axiom,
! [A] :
( m3_supinf_1(A,k3_supinf_1)
=> ~ v1_xboole_0(k14_supinf_1(A)) ) ).
fof(d1_supinf_1,axiom,
k1_supinf_1 = k1_numbers ).
fof(t1_supinf_1,axiom,
$true ).
fof(t2_supinf_1,axiom,
~ r2_hidden(k1_supinf_1,k1_numbers) ).
fof(d2_supinf_1,axiom,
! [A] :
( v1_supinf_1(A)
<=> A = k1_supinf_1 ) ).
fof(t3_supinf_1,axiom,
$true ).
fof(t4_supinf_1,axiom,
v1_supinf_1(k1_supinf_1) ).
fof(d3_supinf_1,axiom,
k2_supinf_1 = k1_tarski(k1_numbers) ).
fof(t5_supinf_1,axiom,
$true ).
fof(t6_supinf_1,axiom,
~ r2_hidden(k2_supinf_1,k1_numbers) ).
fof(d4_supinf_1,axiom,
! [A] :
( v2_supinf_1(A)
<=> A = k2_supinf_1 ) ).
fof(t7_supinf_1,axiom,
$true ).
fof(t8_supinf_1,axiom,
v2_supinf_1(k2_supinf_1) ).
fof(d5_supinf_1,axiom,
! [A] :
( v3_supinf_1(A)
<=> r2_hidden(A,k2_xboole_0(k1_numbers,k2_tarski(k2_supinf_1,k1_supinf_1))) ) ).
fof(d6_supinf_1,axiom,
k3_supinf_1 = k2_xboole_0(k1_numbers,k2_tarski(k2_supinf_1,k1_supinf_1)) ).
fof(t9_supinf_1,axiom,
$true ).
fof(t10_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> m1_subset_1(A,k3_supinf_1) ) ).
fof(t11_supinf_1,axiom,
! [A] :
( ( A = k2_supinf_1
| A = k1_supinf_1 )
=> m1_subset_1(A,k3_supinf_1) ) ).
fof(t12_supinf_1,axiom,
$true ).
fof(t13_supinf_1,axiom,
$true ).
fof(t14_supinf_1,axiom,
k5_supinf_1 != k4_supinf_1 ).
fof(d7_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( ( ( r2_hidden(A,k1_numbers)
& r2_hidden(B,k1_numbers) )
=> ( r1_supinf_1(A,B)
<=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ? [D] :
( m1_subset_1(D,k1_numbers)
& C = A
& D = B
& r1_xreal_0(C,D) ) ) ) )
& ( ~ ( r2_hidden(A,k1_numbers)
& r2_hidden(B,k1_numbers) )
=> ( r1_supinf_1(A,B)
<=> ( A = k5_supinf_1
| B = k4_supinf_1 ) ) ) ) ) ) ).
fof(t15_supinf_1,axiom,
$true ).
fof(t16_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( ( m1_subset_1(A,k1_numbers)
& m1_subset_1(B,k1_numbers) )
=> ( r1_supinf_1(A,B)
<=> ? [C] :
( m1_subset_1(C,k1_numbers)
& ? [D] :
( m1_subset_1(D,k1_numbers)
& C = A
& D = B
& r1_xreal_0(C,D) ) ) ) ) ) ) ).
fof(t17_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ~ ( r2_hidden(A,k1_numbers)
& r1_supinf_1(A,k5_supinf_1) ) ) ).
fof(t18_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ~ ( r2_hidden(A,k1_numbers)
& r1_supinf_1(k4_supinf_1,A) ) ) ).
fof(t19_supinf_1,axiom,
~ r1_supinf_1(k4_supinf_1,k5_supinf_1) ).
fof(t20_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> r1_supinf_1(A,k4_supinf_1) ) ).
fof(t21_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> r1_supinf_1(k5_supinf_1,A) ) ).
fof(t22_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( ( r1_supinf_1(A,B)
& r1_supinf_1(B,A) )
=> A = B ) ) ) ).
fof(t23_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ( r1_supinf_1(A,k5_supinf_1)
=> A = k5_supinf_1 ) ) ).
fof(t24_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ( r1_supinf_1(k4_supinf_1,A)
=> A = k4_supinf_1 ) ) ).
fof(t25_supinf_1,axiom,
$true ).
fof(t26_supinf_1,axiom,
~ v1_xboole_0(k3_supinf_1) ).
fof(t27_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
<=> r2_hidden(A,k3_supinf_1) ) ).
fof(t28_supinf_1,axiom,
$true ).
fof(t29_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( ( r1_supinf_1(A,B)
& r1_supinf_1(B,C) )
=> r1_supinf_1(A,C) ) ) ) ) ).
fof(t30_supinf_1,axiom,
$true ).
fof(t31_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ( r2_hidden(A,k1_numbers)
=> ( ~ r1_supinf_1(A,k5_supinf_1)
& ~ r1_supinf_1(k4_supinf_1,A) ) ) ) ).
fof(d8_supinf_1,axiom,
$true ).
fof(d9_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_supinf_1))
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( m1_supinf_1(B,A)
<=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( r2_hidden(C,A)
=> r1_supinf_1(C,B) ) ) ) ) ) ).
fof(t32_supinf_1,axiom,
$true ).
fof(t33_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_supinf_1))
=> m1_supinf_1(k4_supinf_1,A) ) ).
fof(t34_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_supinf_1))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k3_supinf_1))
=> ( r1_tarski(A,B)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( m1_supinf_1(C,B)
=> m1_supinf_1(C,A) ) ) ) ) ) ).
fof(d10_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_supinf_1))
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( m2_supinf_1(B,A)
<=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( r2_hidden(C,A)
=> r1_supinf_1(B,C) ) ) ) ) ) ).
fof(t35_supinf_1,axiom,
$true ).
fof(t36_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_supinf_1))
=> m2_supinf_1(k5_supinf_1,A) ) ).
fof(t37_supinf_1,axiom,
$true ).
fof(t38_supinf_1,axiom,
$true ).
fof(t39_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_supinf_1))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k3_supinf_1))
=> ( r1_tarski(A,B)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( m2_supinf_1(C,B)
=> m2_supinf_1(C,A) ) ) ) ) ) ).
fof(t40_supinf_1,axiom,
$true ).
fof(t41_supinf_1,axiom,
m1_supinf_1(k4_supinf_1,k6_supinf_1) ).
fof(t42_supinf_1,axiom,
m2_supinf_1(k5_supinf_1,k6_supinf_1) ).
fof(d11_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_supinf_1))
=> ( v4_supinf_1(A)
<=> ? [B] :
( m1_supinf_1(B,A)
& r2_hidden(B,k6_supinf_1) ) ) ) ).
fof(t43_supinf_1,axiom,
$true ).
fof(t44_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_supinf_1))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k3_supinf_1))
=> ( ( r1_tarski(A,B)
& v4_supinf_1(B) )
=> v4_supinf_1(A) ) ) ) ).
fof(t45_supinf_1,axiom,
~ v4_supinf_1(k6_supinf_1) ).
fof(d12_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_supinf_1))
=> ( v5_supinf_1(A)
<=> ? [B] :
( m2_supinf_1(B,A)
& r2_hidden(B,k6_supinf_1) ) ) ) ).
fof(t46_supinf_1,axiom,
$true ).
fof(t47_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_supinf_1))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k3_supinf_1))
=> ( ( r1_tarski(A,B)
& v5_supinf_1(B) )
=> v5_supinf_1(A) ) ) ) ).
fof(t48_supinf_1,axiom,
~ v5_supinf_1(k6_supinf_1) ).
fof(d13_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_supinf_1))
=> ( v6_supinf_1(A)
<=> ( v4_supinf_1(A)
& v5_supinf_1(A) ) ) ) ).
fof(t49_supinf_1,axiom,
$true ).
fof(t50_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_supinf_1))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k3_supinf_1))
=> ( ( r1_tarski(A,B)
& v6_supinf_1(B) )
=> v6_supinf_1(A) ) ) ) ).
fof(t51_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_supinf_1))
=> ? [B] :
( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k3_supinf_1))
& ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( r2_hidden(C,B)
<=> m1_supinf_1(C,A) ) ) ) ) ).
fof(d14_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_supinf_1))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k3_supinf_1))
=> ( B = k7_supinf_1(A)
<=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( r2_hidden(C,B)
<=> m1_supinf_1(C,A) ) ) ) ) ) ).
fof(t52_supinf_1,axiom,
$true ).
fof(t53_supinf_1,axiom,
$true ).
fof(t54_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_supinf_1))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k3_supinf_1))
=> ( r1_tarski(A,B)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( r2_hidden(C,k7_supinf_1(B))
=> r2_hidden(C,k7_supinf_1(A)) ) ) ) ) ) ).
fof(t55_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_supinf_1))
=> ? [B] :
( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k3_supinf_1))
& ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( r2_hidden(C,B)
<=> m2_supinf_1(C,A) ) ) ) ) ).
fof(d15_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_supinf_1))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k3_supinf_1))
=> ( B = k8_supinf_1(A)
<=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( r2_hidden(C,B)
<=> m2_supinf_1(C,A) ) ) ) ) ) ).
fof(t56_supinf_1,axiom,
$true ).
fof(t57_supinf_1,axiom,
$true ).
fof(t58_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_supinf_1))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k3_supinf_1))
=> ( r1_tarski(A,B)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( r2_hidden(C,k8_supinf_1(B))
=> r2_hidden(C,k8_supinf_1(A)) ) ) ) ) ) ).
fof(t59_supinf_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ~ ( v4_supinf_1(A)
& A != k1_tarski(k5_supinf_1)
& ! [B] :
( m1_subset_1(B,k1_numbers)
=> ~ ( r2_hidden(B,A)
& B != k5_supinf_1 ) ) ) ) ).
fof(t60_supinf_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ~ ( v5_supinf_1(A)
& A != k1_tarski(k4_supinf_1)
& ! [B] :
( m1_subset_1(B,k1_numbers)
=> ~ ( r2_hidden(B,A)
& B != k4_supinf_1 ) ) ) ) ).
fof(t61_supinf_1,axiom,
$true ).
fof(t62_supinf_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ~ ( v4_supinf_1(A)
& A != k1_tarski(k5_supinf_1)
& ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ~ ( m1_supinf_1(B,A)
& ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( m1_supinf_1(C,A)
=> r1_supinf_1(B,C) ) ) ) ) ) ) ).
fof(t63_supinf_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ~ ( v5_supinf_1(A)
& A != k1_tarski(k4_supinf_1)
& ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ~ ( m2_supinf_1(B,A)
& ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( m2_supinf_1(C,A)
=> r1_supinf_1(C,B) ) ) ) ) ) ) ).
fof(t64_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_supinf_1))
=> ( A = k1_tarski(k5_supinf_1)
=> v4_supinf_1(A) ) ) ).
fof(t65_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_supinf_1))
=> ( A = k1_tarski(k4_supinf_1)
=> v5_supinf_1(A) ) ) ).
fof(t66_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_supinf_1))
=> ~ ( A = k1_tarski(k5_supinf_1)
& ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ~ ( m1_supinf_1(B,A)
& ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( m1_supinf_1(C,A)
=> r1_supinf_1(B,C) ) ) ) ) ) ) ).
fof(t67_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_supinf_1))
=> ~ ( A = k1_tarski(k4_supinf_1)
& ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ~ ( m2_supinf_1(B,A)
& ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( m2_supinf_1(C,A)
=> r1_supinf_1(C,B) ) ) ) ) ) ) ).
fof(t68_supinf_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ~ ( v4_supinf_1(A)
& ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ~ ( m1_supinf_1(B,A)
& ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( m1_supinf_1(C,A)
=> r1_supinf_1(B,C) ) ) ) ) ) ) ).
fof(t69_supinf_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ~ ( v5_supinf_1(A)
& ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ~ ( m2_supinf_1(B,A)
& ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( m2_supinf_1(C,A)
=> r1_supinf_1(C,B) ) ) ) ) ) ) ).
fof(t70_supinf_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ( ~ v4_supinf_1(A)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( m1_supinf_1(B,A)
=> B = k4_supinf_1 ) ) ) ) ).
fof(t71_supinf_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ( ~ v5_supinf_1(A)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( m2_supinf_1(B,A)
=> B = k5_supinf_1 ) ) ) ) ).
fof(t72_supinf_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ? [B] :
( m1_subset_1(B,k3_supinf_1)
& m1_supinf_1(B,A)
& ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( m1_supinf_1(C,A)
=> r1_supinf_1(B,C) ) ) ) ) ).
fof(t73_supinf_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ? [B] :
( m1_subset_1(B,k3_supinf_1)
& m2_supinf_1(B,A)
& ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( m2_supinf_1(C,A)
=> r1_supinf_1(C,B) ) ) ) ) ).
fof(d16_supinf_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( B = k9_supinf_1(A)
<=> ( m1_supinf_1(B,A)
& ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( m1_supinf_1(C,A)
=> r1_supinf_1(B,C) ) ) ) ) ) ) ).
fof(t74_supinf_1,axiom,
$true ).
fof(t75_supinf_1,axiom,
$true ).
fof(t76_supinf_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( r2_hidden(B,A)
=> r1_supinf_1(B,k9_supinf_1(A)) ) ) ) ).
fof(d17_supinf_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( B = k10_supinf_1(A)
<=> ( m2_supinf_1(B,A)
& ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( m2_supinf_1(C,A)
=> r1_supinf_1(C,B) ) ) ) ) ) ) ).
fof(t77_supinf_1,axiom,
$true ).
fof(t78_supinf_1,axiom,
$true ).
fof(t79_supinf_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( r2_hidden(B,A)
=> r1_supinf_1(k10_supinf_1(A),B) ) ) ) ).
fof(t80_supinf_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ! [B] :
( m1_supinf_1(B,A)
=> ( r2_hidden(B,A)
=> B = k9_supinf_1(A) ) ) ) ).
fof(t81_supinf_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ! [B] :
( m2_supinf_1(B,A)
=> ( r2_hidden(B,A)
=> B = k10_supinf_1(A) ) ) ) ).
fof(t82_supinf_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ( k9_supinf_1(A) = k10_supinf_1(k7_supinf_1(A))
& k10_supinf_1(A) = k9_supinf_1(k8_supinf_1(A)) ) ) ).
fof(t83_supinf_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ( v4_supinf_1(A)
=> ( A = k1_tarski(k5_supinf_1)
| r2_hidden(k9_supinf_1(A),k6_supinf_1) ) ) ) ).
fof(t84_supinf_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ( v5_supinf_1(A)
=> ( A = k1_tarski(k4_supinf_1)
| r2_hidden(k10_supinf_1(A),k6_supinf_1) ) ) ) ).
fof(t85_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> k9_supinf_1(k11_supinf_1(A)) = A ) ).
fof(t86_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> k10_supinf_1(k11_supinf_1(A)) = A ) ).
fof(t87_supinf_1,axiom,
$true ).
fof(t88_supinf_1,axiom,
$true ).
fof(t89_supinf_1,axiom,
$true ).
fof(t90_supinf_1,axiom,
$true ).
fof(t91_supinf_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k3_supinf_1)) )
=> ( r1_tarski(A,B)
=> r1_supinf_1(k9_supinf_1(A),k9_supinf_1(B)) ) ) ) ).
fof(t92_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( ( r1_supinf_1(A,C)
& r1_supinf_1(B,C) )
=> r1_supinf_1(k9_supinf_1(k12_supinf_1(A,B)),C) ) ) ) ) ).
fof(t93_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( ( r1_supinf_1(A,B)
=> k9_supinf_1(k12_supinf_1(A,B)) = B )
& ( r1_supinf_1(B,A)
=> k9_supinf_1(k12_supinf_1(A,B)) = A ) ) ) ) ).
fof(t94_supinf_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k3_supinf_1)) )
=> ( r1_tarski(A,B)
=> r1_supinf_1(k10_supinf_1(B),k10_supinf_1(A)) ) ) ) ).
fof(t95_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( ( r1_supinf_1(C,A)
& r1_supinf_1(C,B) )
=> r1_supinf_1(C,k10_supinf_1(k12_supinf_1(A,B))) ) ) ) ) ).
fof(t96_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( ( r1_supinf_1(A,B)
=> k10_supinf_1(k12_supinf_1(A,B)) = A )
& ( r1_supinf_1(B,A)
=> k10_supinf_1(k12_supinf_1(A,B)) = B ) ) ) ) ).
fof(t97_supinf_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( ? [C] :
( m1_subset_1(C,k3_supinf_1)
& r2_hidden(C,A)
& r1_supinf_1(B,C) )
=> r1_supinf_1(B,k9_supinf_1(A)) ) ) ) ).
fof(t98_supinf_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( ? [C] :
( m1_subset_1(C,k3_supinf_1)
& r2_hidden(C,A)
& r1_supinf_1(C,B) )
=> r1_supinf_1(k10_supinf_1(A),B) ) ) ) ).
fof(t99_supinf_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k3_supinf_1)) )
=> ( ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ~ ( r2_hidden(C,A)
& ! [D] :
( m1_subset_1(D,k3_supinf_1)
=> ~ ( r2_hidden(D,B)
& r1_supinf_1(C,D) ) ) ) )
=> r1_supinf_1(k9_supinf_1(A),k9_supinf_1(B)) ) ) ) ).
fof(t100_supinf_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k3_supinf_1)) )
=> ( ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ~ ( r2_hidden(C,B)
& ! [D] :
( m1_subset_1(D,k3_supinf_1)
=> ~ ( r2_hidden(D,A)
& r1_supinf_1(D,C) ) ) ) )
=> r1_supinf_1(k10_supinf_1(A),k10_supinf_1(B)) ) ) ) ).
fof(t101_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_supinf_1))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k3_supinf_1))
=> ! [C] :
( m1_supinf_1(C,A)
=> ! [D] :
( m1_supinf_1(D,B)
=> m1_supinf_1(k9_supinf_1(k12_supinf_1(C,D)),k4_subset_1(k3_supinf_1,A,B)) ) ) ) ) ).
fof(t102_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_supinf_1))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k3_supinf_1))
=> ! [C] :
( m2_supinf_1(C,A)
=> ! [D] :
( m2_supinf_1(D,B)
=> m2_supinf_1(k10_supinf_1(k12_supinf_1(C,D)),k4_subset_1(k3_supinf_1,A,B)) ) ) ) ) ).
fof(t103_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_supinf_1))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k3_supinf_1))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k3_supinf_1))
=> ! [D] :
( m1_supinf_1(D,A)
=> ! [E] :
( m1_supinf_1(E,B)
=> ( C = k5_subset_1(k3_supinf_1,A,B)
=> m1_supinf_1(k10_supinf_1(k12_supinf_1(D,E)),C) ) ) ) ) ) ) ).
fof(t104_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_supinf_1))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k3_supinf_1))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k3_supinf_1))
=> ! [D] :
( m2_supinf_1(D,A)
=> ! [E] :
( m2_supinf_1(E,B)
=> ( C = k5_subset_1(k3_supinf_1,A,B)
=> m2_supinf_1(k9_supinf_1(k12_supinf_1(D,E)),C) ) ) ) ) ) ) ).
fof(t105_supinf_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k3_supinf_1)) )
=> k9_supinf_1(k4_subset_1(k3_supinf_1,A,B)) = k9_supinf_1(k12_supinf_1(k9_supinf_1(A),k9_supinf_1(B))) ) ) ).
fof(t106_supinf_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k3_supinf_1)) )
=> k10_supinf_1(k4_subset_1(k3_supinf_1,A,B)) = k10_supinf_1(k12_supinf_1(k10_supinf_1(A),k10_supinf_1(B))) ) ) ).
fof(t107_supinf_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k3_supinf_1)) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k3_supinf_1)) )
=> ( C = k5_subset_1(k3_supinf_1,A,B)
=> r1_supinf_1(k9_supinf_1(C),k10_supinf_1(k12_supinf_1(k9_supinf_1(A),k9_supinf_1(B)))) ) ) ) ) ).
fof(t108_supinf_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k3_supinf_1)) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k3_supinf_1)) )
=> ( C = k5_subset_1(k3_supinf_1,A,B)
=> r1_supinf_1(k9_supinf_1(k12_supinf_1(k10_supinf_1(A),k10_supinf_1(B))),k10_supinf_1(C)) ) ) ) ) ).
fof(d18_supinf_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m3_supinf_1(B,A)
<=> ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
& ! [C] :
( r2_hidden(C,B)
=> ~ v1_xboole_0(C) ) ) ) ) ).
fof(d19_supinf_1,axiom,
! [A] :
( m3_supinf_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k3_supinf_1))
=> ( B = k13_supinf_1(A)
<=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( r2_hidden(C,B)
<=> ? [D] :
( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(k3_supinf_1))
& r2_hidden(D,A)
& C = k9_supinf_1(D) ) ) ) ) ) ) ).
fof(t109_supinf_1,axiom,
$true ).
fof(t110_supinf_1,axiom,
$true ).
fof(t111_supinf_1,axiom,
$true ).
fof(t112_supinf_1,axiom,
! [A] :
( m3_supinf_1(A,k3_supinf_1)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k3_supinf_1)) )
=> ( B = k3_tarski(A)
=> m1_supinf_1(k9_supinf_1(B),k13_supinf_1(A)) ) ) ) ).
fof(t113_supinf_1,axiom,
! [A] :
( m3_supinf_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k3_supinf_1))
=> ( B = k3_tarski(A)
=> m1_supinf_1(k9_supinf_1(k13_supinf_1(A)),B) ) ) ) ).
fof(t114_supinf_1,axiom,
! [A] :
( m3_supinf_1(A,k3_supinf_1)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k3_supinf_1)) )
=> ( B = k3_tarski(A)
=> k9_supinf_1(B) = k9_supinf_1(k13_supinf_1(A)) ) ) ) ).
fof(d20_supinf_1,axiom,
! [A] :
( m3_supinf_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k3_supinf_1))
=> ( B = k14_supinf_1(A)
<=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( r2_hidden(C,B)
<=> ? [D] :
( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(k3_supinf_1))
& r2_hidden(D,A)
& C = k10_supinf_1(D) ) ) ) ) ) ) ).
fof(t115_supinf_1,axiom,
$true ).
fof(t116_supinf_1,axiom,
$true ).
fof(t117_supinf_1,axiom,
! [A] :
( m3_supinf_1(A,k3_supinf_1)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k3_supinf_1)) )
=> ( B = k3_tarski(A)
=> m2_supinf_1(k10_supinf_1(B),k14_supinf_1(A)) ) ) ) ).
fof(t118_supinf_1,axiom,
! [A] :
( m3_supinf_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k3_supinf_1))
=> ( B = k3_tarski(A)
=> m2_supinf_1(k10_supinf_1(k14_supinf_1(A)),B) ) ) ) ).
fof(t119_supinf_1,axiom,
! [A] :
( m3_supinf_1(A,k3_supinf_1)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k3_supinf_1)) )
=> ( B = k3_tarski(A)
=> k10_supinf_1(B) = k10_supinf_1(k14_supinf_1(A)) ) ) ) ).
fof(t120_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( ( A = C
& B = D )
=> ( r1_xreal_0(C,D)
<=> r1_supinf_1(A,B) ) ) ) ) ) ) ).
fof(s1_supinf_1,axiom,
? [A] :
( m1_subset_1(A,k1_zfmisc_1(k2_xboole_0(k1_numbers,k2_tarski(k5_supinf_1,k4_supinf_1))))
& ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( r2_hidden(B,A)
<=> p1_s1_supinf_1(B) ) ) ) ).
fof(dt_m1_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_supinf_1))
=> ! [B] :
( m1_supinf_1(B,A)
=> m1_subset_1(B,k3_supinf_1) ) ) ).
fof(existence_m1_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_supinf_1))
=> ? [B] : m1_supinf_1(B,A) ) ).
fof(dt_m2_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_supinf_1))
=> ! [B] :
( m2_supinf_1(B,A)
=> m1_subset_1(B,k3_supinf_1) ) ) ).
fof(existence_m2_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_supinf_1))
=> ? [B] : m2_supinf_1(B,A) ) ).
fof(dt_m3_supinf_1,axiom,
$true ).
fof(existence_m3_supinf_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] : m3_supinf_1(B,A) ) ).
fof(reflexivity_r1_supinf_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k3_supinf_1)
& m1_subset_1(B,k3_supinf_1) )
=> r1_supinf_1(A,A) ) ).
fof(connectedness_r1_supinf_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k3_supinf_1)
& m1_subset_1(B,k3_supinf_1) )
=> ( r1_supinf_1(A,B)
| r1_supinf_1(B,A) ) ) ).
fof(dt_k1_supinf_1,axiom,
$true ).
fof(dt_k2_supinf_1,axiom,
$true ).
fof(dt_k3_supinf_1,axiom,
$true ).
fof(dt_k4_supinf_1,axiom,
m1_subset_1(k4_supinf_1,k3_supinf_1) ).
fof(redefinition_k4_supinf_1,axiom,
k4_supinf_1 = k1_supinf_1 ).
fof(dt_k5_supinf_1,axiom,
m1_subset_1(k5_supinf_1,k3_supinf_1) ).
fof(redefinition_k5_supinf_1,axiom,
k5_supinf_1 = k2_supinf_1 ).
fof(dt_k6_supinf_1,axiom,
( ~ v1_xboole_0(k6_supinf_1)
& m1_subset_1(k6_supinf_1,k1_zfmisc_1(k3_supinf_1)) ) ).
fof(redefinition_k6_supinf_1,axiom,
k6_supinf_1 = k1_numbers ).
fof(dt_k7_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_supinf_1))
=> m1_subset_1(k7_supinf_1(A),k1_zfmisc_1(k3_supinf_1)) ) ).
fof(dt_k8_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_supinf_1))
=> m1_subset_1(k8_supinf_1(A),k1_zfmisc_1(k3_supinf_1)) ) ).
fof(dt_k9_supinf_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> m1_subset_1(k9_supinf_1(A),k3_supinf_1) ) ).
fof(dt_k10_supinf_1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> m1_subset_1(k10_supinf_1(A),k3_supinf_1) ) ).
fof(dt_k11_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> m1_subset_1(k11_supinf_1(A),k1_zfmisc_1(k3_supinf_1)) ) ).
fof(redefinition_k11_supinf_1,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> k11_supinf_1(A) = k1_tarski(A) ) ).
fof(dt_k12_supinf_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k3_supinf_1)
& m1_subset_1(B,k3_supinf_1) )
=> m1_subset_1(k12_supinf_1(A,B),k1_zfmisc_1(k3_supinf_1)) ) ).
fof(commutativity_k12_supinf_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k3_supinf_1)
& m1_subset_1(B,k3_supinf_1) )
=> k12_supinf_1(A,B) = k12_supinf_1(B,A) ) ).
fof(redefinition_k12_supinf_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k3_supinf_1)
& m1_subset_1(B,k3_supinf_1) )
=> k12_supinf_1(A,B) = k2_tarski(A,B) ) ).
fof(dt_k13_supinf_1,axiom,
! [A] :
( m3_supinf_1(A,k3_supinf_1)
=> m1_subset_1(k13_supinf_1(A),k1_zfmisc_1(k3_supinf_1)) ) ).
fof(dt_k14_supinf_1,axiom,
! [A] :
( m3_supinf_1(A,k3_supinf_1)
=> m1_subset_1(k14_supinf_1(A),k1_zfmisc_1(k3_supinf_1)) ) ).
%------------------------------------------------------------------------------