SET007 Axioms: SET007+47.ax
%------------------------------------------------------------------------------
% File : SET007+47 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Fundamental Properties of Natural 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 : nat_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 116 ( 29 unt; 0 def)
% Number of atoms : 473 ( 80 equ)
% Maximal formula atoms : 19 ( 4 avg)
% Number of connectives : 420 ( 63 ~; 6 |; 126 &)
% ( 9 <=>; 216 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 6 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 22 ( 20 usr; 1 prp; 0-3 aty)
% Number of functors : 22 ( 22 usr; 9 con; 0-2 aty)
% Number of variables : 203 ( 190 !; 13 ?)
% 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_nat_1,axiom,
! [A,B] :
( ( v4_ordinal2(A)
& v4_ordinal2(B) )
=> ( v4_ordinal2(k2_xcmplx_0(A,B))
& v1_xcmplx_0(k2_xcmplx_0(A,B))
& v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ).
fof(fc2_nat_1,axiom,
! [A,B] :
( ( v4_ordinal2(A)
& v4_ordinal2(B) )
=> ( v4_ordinal2(k3_xcmplx_0(A,B))
& v1_xcmplx_0(k3_xcmplx_0(A,B))
& v1_xreal_0(k3_xcmplx_0(A,B)) ) ) ).
fof(rc1_nat_1,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v4_ordinal2(A)
& v1_xcmplx_0(A)
& v1_xreal_0(A) ) ).
fof(fc3_nat_1,axiom,
! [A,B] :
( ( v4_ordinal2(A)
& ~ v1_xboole_0(B)
& v4_ordinal2(B) )
=> ( ~ v1_xboole_0(k2_xcmplx_0(A,B))
& v4_ordinal2(k2_xcmplx_0(A,B))
& v1_xcmplx_0(k2_xcmplx_0(A,B))
& v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ).
fof(fc4_nat_1,axiom,
! [A,B] :
( ( v4_ordinal2(A)
& ~ v1_xboole_0(B)
& v4_ordinal2(B) )
=> ( ~ v1_xboole_0(k2_xcmplx_0(B,A))
& v4_ordinal2(k2_xcmplx_0(B,A))
& v1_xcmplx_0(k2_xcmplx_0(B,A))
& v1_xreal_0(k2_xcmplx_0(B,A)) ) ) ).
fof(cc1_nat_1,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) ) ) ).
fof(rc2_nat_1,axiom,
? [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
& ~ v1_xboole_0(A)
& v3_ordinal1(A) ) ).
fof(rc3_nat_1,axiom,
? [A] :
( m1_subset_1(A,k5_numbers)
& ~ v1_xboole_0(A)
& v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A)
& v4_ordinal2(A)
& v1_xcmplx_0(A)
& v1_xreal_0(A) ) ).
fof(cc2_nat_1,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) ) ) ).
fof(cc3_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ( v4_ordinal2(A)
& v1_xcmplx_0(A)
& v1_xreal_0(A)
& ~ v3_xreal_0(A) ) ) ).
fof(t1_nat_1,axiom,
$true ).
fof(t2_nat_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ( ( r2_hidden(np__0,A)
& ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r2_hidden(B,A)
=> r2_hidden(k3_real_1(B,np__1),A) ) ) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r2_hidden(B,A) ) ) ) ).
fof(t3_nat_1,axiom,
$true ).
fof(t4_nat_1,axiom,
$true ).
fof(t5_nat_1,axiom,
$true ).
fof(t6_nat_1,axiom,
$true ).
fof(t7_nat_1,axiom,
$true ).
fof(t8_nat_1,axiom,
$true ).
fof(t9_nat_1,axiom,
$true ).
fof(t10_nat_1,axiom,
$true ).
fof(t11_nat_1,axiom,
$true ).
fof(t12_nat_1,axiom,
$true ).
fof(t13_nat_1,axiom,
$true ).
fof(t14_nat_1,axiom,
$true ).
fof(t15_nat_1,axiom,
$true ).
fof(t16_nat_1,axiom,
$true ).
fof(t17_nat_1,axiom,
$true ).
fof(t18_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> r1_xreal_0(np__0,A) ) ).
fof(t19_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ~ ( np__0 != A
& r1_xreal_0(A,np__0) ) ) ).
fof(t20_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( v4_ordinal2(C)
=> ( r1_xreal_0(A,B)
=> r1_xreal_0(k3_xcmplx_0(A,C),k3_xcmplx_0(B,C)) ) ) ) ) ).
fof(t21_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ~ r1_xreal_0(k2_xcmplx_0(A,np__1),np__0) ) ).
fof(t22_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ~ ( A != np__0
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> A != k1_nat_1(B,np__1) ) ) ) ).
fof(t23_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( k2_xcmplx_0(A,B) = np__0
=> ( A = np__0
& B = np__0 ) ) ) ) ).
fof(t24_nat_1,axiom,
$true ).
fof(t25_nat_1,axiom,
$true ).
fof(t26_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ~ ( r1_xreal_0(A,k2_xcmplx_0(B,np__1))
& ~ r1_xreal_0(A,B)
& A != k2_xcmplx_0(B,np__1) ) ) ) ).
fof(t27_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ~ ( r1_xreal_0(A,B)
& r1_xreal_0(B,k2_xcmplx_0(A,np__1))
& A != B
& B != k2_xcmplx_0(A,np__1) ) ) ) ).
fof(t28_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ~ ( r1_xreal_0(A,B)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> B != k2_xcmplx_0(A,C) ) ) ) ) ).
fof(t29_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> r1_xreal_0(A,k2_xcmplx_0(A,B)) ) ) ).
fof(t30_nat_1,axiom,
$true ).
fof(t31_nat_1,axiom,
$true ).
fof(t32_nat_1,axiom,
$true ).
fof(t33_nat_1,axiom,
$true ).
fof(t34_nat_1,axiom,
$true ).
fof(t35_nat_1,axiom,
$true ).
fof(t36_nat_1,axiom,
$true ).
fof(t37_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( v4_ordinal2(C)
=> ( r1_xreal_0(A,B)
=> r1_xreal_0(A,k2_xcmplx_0(B,C)) ) ) ) ) ).
fof(t38_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( ~ r1_xreal_0(k2_xcmplx_0(B,np__1),A)
<=> r1_xreal_0(A,B) ) ) ) ).
fof(t39_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ( ~ r1_xreal_0(np__1,A)
=> A = np__0 ) ) ).
fof(t40_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( k3_xcmplx_0(A,B) = np__1
=> ( A = np__1
& B = np__1 ) ) ) ) ).
fof(t41_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ~ ( A != np__0
& r1_xreal_0(k2_xcmplx_0(B,A),B) ) ) ) ).
fof(t42_nat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(A,np__0)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& B = k1_nat_1(k2_nat_1(A,C),D)
& ~ r1_xreal_0(A,D) ) ) ) ) ) ).
fof(t43_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( v4_ordinal2(C)
=> ! [D] :
( v4_ordinal2(D)
=> ! [E] :
( v4_ordinal2(E)
=> ! [F] :
( v4_ordinal2(F)
=> ( ( A = k2_xcmplx_0(k3_xcmplx_0(B,C),E)
& A = k2_xcmplx_0(k3_xcmplx_0(B,D),F) )
=> ( r1_xreal_0(B,E)
| r1_xreal_0(B,F)
| ( C = D
& E = F ) ) ) ) ) ) ) ) ) ).
fof(d1_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( C = k3_nat_1(A,B)
<=> ~ ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( A = k2_xcmplx_0(k3_xcmplx_0(B,C),D)
& ~ r1_xreal_0(B,D) ) )
& ~ ( C = np__0
& B = np__0 ) ) ) ) ) ) ).
fof(d2_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( C = k4_nat_1(A,B)
<=> ~ ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( A = k2_xcmplx_0(k3_xcmplx_0(B,D),C)
& ~ r1_xreal_0(B,C) ) )
& ~ ( C = np__0
& B = np__0 ) ) ) ) ) ) ).
fof(t44_nat_1,axiom,
$true ).
fof(t45_nat_1,axiom,
$true ).
fof(t46_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ~ ( ~ r1_xreal_0(A,np__0)
& r1_xreal_0(A,k4_nat_1(B,A)) ) ) ) ).
fof(t47_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( ~ r1_xreal_0(A,np__0)
=> B = k2_xcmplx_0(k3_xcmplx_0(A,k3_nat_1(B,A)),k4_nat_1(B,A)) ) ) ) ).
fof(d3_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( r1_nat_1(A,B)
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& B = k3_xcmplx_0(A,C) ) ) ) ) ).
fof(t48_nat_1,axiom,
$true ).
fof(t49_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( r1_nat_1(A,B)
<=> B = k3_xcmplx_0(A,k3_nat_1(B,A)) ) ) ) ).
fof(t50_nat_1,axiom,
$true ).
fof(t51_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( v4_ordinal2(C)
=> ( ( r1_nat_1(A,B)
& r1_nat_1(B,C) )
=> r1_nat_1(A,C) ) ) ) ) ).
fof(t52_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( ( r1_nat_1(A,B)
& r1_nat_1(B,A) )
=> A = B ) ) ) ).
fof(t53_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ( r1_nat_1(A,np__0)
& r1_nat_1(np__1,A) ) ) ).
fof(t54_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( r1_nat_1(B,A)
=> ( r1_xreal_0(A,np__0)
| r1_xreal_0(B,A) ) ) ) ) ).
fof(t55_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( v4_ordinal2(C)
=> ( ( r1_nat_1(A,B)
& r1_nat_1(A,C) )
=> r1_nat_1(A,k2_xcmplx_0(B,C)) ) ) ) ) ).
fof(t56_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( v4_ordinal2(C)
=> ( r1_nat_1(A,B)
=> r1_nat_1(A,k3_xcmplx_0(B,C)) ) ) ) ) ).
fof(t57_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( v4_ordinal2(C)
=> ( ( r1_nat_1(A,B)
& r1_nat_1(A,k2_xcmplx_0(B,C)) )
=> r1_nat_1(A,C) ) ) ) ) ).
fof(t58_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( v4_ordinal2(C)
=> ( ( r1_nat_1(A,B)
& r1_nat_1(A,C) )
=> r1_nat_1(A,k4_nat_1(B,C)) ) ) ) ) ).
fof(d4_nat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( C = k5_nat_1(A,B)
<=> ( r1_nat_1(A,C)
& r1_nat_1(B,C)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r1_nat_1(A,D)
& r1_nat_1(B,D) )
=> r1_nat_1(C,D) ) ) ) ) ) ) ) ).
fof(d5_nat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( C = k6_nat_1(A,B)
<=> ( r1_nat_1(C,A)
& r1_nat_1(C,B)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r1_nat_1(D,A)
& r1_nat_1(D,B) )
=> r1_nat_1(D,C) ) ) ) ) ) ) ) ).
fof(t59_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( ~ r1_xreal_0(k2_xcmplx_0(A,B),A)
<=> r1_xreal_0(np__1,B) ) ) ) ).
fof(t60_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( ~ r1_xreal_0(B,A)
=> m2_subset_1(k6_xcmplx_0(B,np__1),k1_numbers,k5_numbers) ) ) ) ).
fof(t61_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( r1_xreal_0(A,B)
=> m2_subset_1(k6_xcmplx_0(B,A),k1_numbers,k5_numbers) ) ) ) ).
fof(t62_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ( k4_nat_1(A,np__2) = np__0
| k4_nat_1(A,np__2) = np__1 ) ) ).
fof(t63_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> k4_nat_1(k3_xcmplx_0(A,B),A) = np__0 ) ) ).
fof(t64_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( ~ r1_xreal_0(A,np__1)
=> k4_nat_1(np__1,A) = np__1 ) ) ) ).
fof(t65_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( v4_ordinal2(C)
=> ! [D] :
( v4_ordinal2(D)
=> ( ( k4_nat_1(A,C) = np__0
& B = k6_xcmplx_0(A,k3_xcmplx_0(D,C)) )
=> k4_nat_1(B,C) = np__0 ) ) ) ) ) ).
fof(t66_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( v4_ordinal2(C)
=> ( k4_nat_1(A,C) = np__0
=> ( C = np__0
| r1_xreal_0(C,B)
| k4_nat_1(k2_xcmplx_0(A,B),C) = B ) ) ) ) ) ).
fof(t67_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( v4_ordinal2(C)
=> ( k4_nat_1(A,C) = np__0
=> k4_nat_1(k2_xcmplx_0(A,B),C) = k4_nat_1(B,C) ) ) ) ) ).
fof(t68_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( A != np__0
=> k3_nat_1(k3_xcmplx_0(A,B),A) = B ) ) ) ).
fof(t69_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( v4_ordinal2(C)
=> ( k4_nat_1(A,B) = np__0
=> k3_nat_1(k2_xcmplx_0(A,C),B) = k1_nat_1(k3_nat_1(A,B),k3_nat_1(C,B)) ) ) ) ) ).
fof(t70_nat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(k1_nat_1(B,np__1),A)
& r1_xreal_0(B,A)
& A != B ) ) ) ).
fof(t71_nat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(np__2,A)
& A != np__0
& A != np__1 ) ) ).
fof(t72_nat_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( B != np__0
=> k3_nat_1(k2_nat_1(A,B),B) = A ) ) ) ).
fof(t73_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( v4_ordinal2(C)
=> k4_nat_1(A,B) = k4_nat_1(k2_xcmplx_0(k3_xcmplx_0(B,C),A),B) ) ) ) ).
fof(t74_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( v4_ordinal2(C)
=> k4_nat_1(k2_xcmplx_0(A,B),C) = k4_nat_1(k2_xcmplx_0(k4_nat_1(A,C),B),C) ) ) ) ).
fof(t75_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( v4_ordinal2(C)
=> k4_nat_1(k2_xcmplx_0(A,B),C) = k4_nat_1(k2_xcmplx_0(A,k4_nat_1(B,C)),C) ) ) ) ).
fof(t76_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( ~ r1_xreal_0(A,B)
=> k4_nat_1(B,A) = B ) ) ) ).
fof(t77_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> k4_nat_1(A,A) = np__0 ) ).
fof(t78_nat_1,axiom,
! [A] :
( v4_ordinal2(A)
=> np__0 = k4_nat_1(np__0,A) ) ).
fof(s1_nat_1,axiom,
( ( p1_s1_nat_1(np__0)
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( p1_s1_nat_1(A)
=> p1_s1_nat_1(k1_nat_1(A,np__1)) ) ) )
=> ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> p1_s1_nat_1(A) ) ) ).
fof(s2_nat_1,axiom,
( ( p1_s2_nat_1(np__0)
& ! [A] :
( v4_ordinal2(A)
=> ( p1_s2_nat_1(A)
=> p1_s2_nat_1(k2_xcmplx_0(A,np__1)) ) ) )
=> ! [A] :
( v4_ordinal2(A)
=> p1_s2_nat_1(A) ) ) ).
fof(s3_nat_1,axiom,
( ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( p1_s3_nat_1(A,B)
<=> ~ ( ~ ( A = np__0
& B = f1_s3_nat_1 )
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( A = k1_nat_1(C,np__1)
& p1_s3_nat_1(C,D)
& B = f2_s3_nat_1(A,D) ) ) ) ) ) ) )
=> ( ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& p1_s3_nat_1(A,B) ) )
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( p1_s3_nat_1(A,B)
& p1_s3_nat_1(A,C) )
=> B = C ) ) ) ) ) ) ).
fof(s4_nat_1,axiom,
( ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(A,B)
=> p1_s4_nat_1(B) ) )
=> p1_s4_nat_1(A) ) )
=> ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> p1_s4_nat_1(A) ) ) ).
fof(s5_nat_1,axiom,
( ? [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
& p1_s5_nat_1(A) )
=> ? [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
& p1_s5_nat_1(A)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( p1_s5_nat_1(B)
=> r1_xreal_0(A,B) ) ) ) ) ).
fof(s6_nat_1,axiom,
( ( ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( p1_s6_nat_1(A)
=> r1_xreal_0(A,f1_s6_nat_1) ) )
& ? [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
& p1_s6_nat_1(A) ) )
=> ? [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
& p1_s6_nat_1(A)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( p1_s6_nat_1(B)
=> r1_xreal_0(B,A) ) ) ) ) ).
fof(s7_nat_1,axiom,
( ( ? [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
& p1_s7_nat_1(A) )
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ( A != np__0
& p1_s7_nat_1(A)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(A,B)
& p1_s7_nat_1(B) ) ) ) ) )
=> p1_s7_nat_1(np__0) ) ).
fof(s8_nat_1,axiom,
( ( ~ r1_xreal_0(f3_s8_nat_1,np__0)
& ~ r1_xreal_0(f2_s8_nat_1,f3_s8_nat_1)
& f1_s8_nat_1(np__0) = f2_s8_nat_1
& f1_s8_nat_1(np__1) = f3_s8_nat_1
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> f1_s8_nat_1(k1_nat_1(A,np__2)) = k4_nat_1(f1_s8_nat_1(A),f1_s8_nat_1(k1_nat_1(A,np__1))) ) )
=> ? [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
& f1_s8_nat_1(A) = k6_nat_1(f2_s8_nat_1,f3_s8_nat_1)
& f1_s8_nat_1(k1_nat_1(A,np__1)) = np__0 ) ) ).
fof(reflexivity_r1_nat_1,axiom,
! [A,B] :
( ( v4_ordinal2(A)
& v4_ordinal2(B) )
=> r1_nat_1(A,A) ) ).
fof(dt_k1_nat_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> m2_subset_1(k1_nat_1(A,B),k1_numbers,k5_numbers) ) ).
fof(commutativity_k1_nat_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> k1_nat_1(A,B) = k1_nat_1(B,A) ) ).
fof(redefinition_k1_nat_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> k1_nat_1(A,B) = k2_xcmplx_0(A,B) ) ).
fof(dt_k2_nat_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> m2_subset_1(k2_nat_1(A,B),k1_numbers,k5_numbers) ) ).
fof(commutativity_k2_nat_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> k2_nat_1(A,B) = k2_nat_1(B,A) ) ).
fof(redefinition_k2_nat_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> k2_nat_1(A,B) = k3_xcmplx_0(A,B) ) ).
fof(dt_k3_nat_1,axiom,
! [A,B] :
( ( v4_ordinal2(A)
& v4_ordinal2(B) )
=> m2_subset_1(k3_nat_1(A,B),k1_numbers,k5_numbers) ) ).
fof(dt_k4_nat_1,axiom,
! [A,B] :
( ( v4_ordinal2(A)
& v4_ordinal2(B) )
=> m2_subset_1(k4_nat_1(A,B),k1_numbers,k5_numbers) ) ).
fof(dt_k5_nat_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> m2_subset_1(k5_nat_1(A,B),k1_numbers,k5_numbers) ) ).
fof(commutativity_k5_nat_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> k5_nat_1(A,B) = k5_nat_1(B,A) ) ).
fof(idempotence_k5_nat_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> k5_nat_1(A,A) = A ) ).
fof(dt_k6_nat_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> m2_subset_1(k6_nat_1(A,B),k1_numbers,k5_numbers) ) ).
fof(commutativity_k6_nat_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> k6_nat_1(A,B) = k6_nat_1(B,A) ) ).
fof(idempotence_k6_nat_1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> k6_nat_1(A,A) = A ) ).
%------------------------------------------------------------------------------