SET007 Axioms: SET007+192.ax
%------------------------------------------------------------------------------
% File : SET007+192 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Zero-Based Finite Sequences
% 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 : afinsq_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 94 ( 12 unt; 0 def)
% Number of atoms : 637 ( 89 equ)
% Maximal formula atoms : 23 ( 6 avg)
% Number of connectives : 574 ( 31 ~; 4 |; 387 &)
% ( 14 <=>; 138 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 32 ( 30 usr; 1 prp; 0-3 aty)
% Number of functors : 38 ( 38 usr; 10 con; 0-4 aty)
% Number of variables : 201 ( 189 !; 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_afinsq_1,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_finset_1(A) ) ).
fof(cc1_afinsq_1,axiom,
! [A] :
( v4_ordinal2(A)
=> v1_finset_1(A) ) ).
fof(fc1_afinsq_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_finset_1(A) )
=> ( v1_ordinal1(k1_relat_1(A))
& v2_ordinal1(k1_relat_1(A))
& v3_ordinal1(k1_relat_1(A))
& v4_ordinal2(k1_relat_1(A))
& v1_xcmplx_0(k1_relat_1(A))
& v1_xreal_0(k1_relat_1(A)) ) ) ).
fof(rc2_afinsq_1,axiom,
! [A] :
? [B] :
( m1_ordinal1(B,A)
& v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_finset_1(B) ) ).
fof(fc2_afinsq_1,axiom,
( v1_xboole_0(k1_xboole_0)
& v1_relat_1(k1_xboole_0)
& v3_relat_1(k1_xboole_0)
& v1_funct_1(k1_xboole_0)
& v2_funct_1(k1_xboole_0)
& v1_ordinal1(k1_xboole_0)
& v2_ordinal1(k1_xboole_0)
& v3_ordinal1(k1_xboole_0)
& v5_ordinal1(k1_xboole_0)
& v4_ordinal2(k1_xboole_0)
& v1_xcmplx_0(k1_xboole_0)
& v1_finset_1(k1_xboole_0)
& v1_finseq_1(k1_xboole_0)
& v1_funcop_1(k1_xboole_0)
& v1_funct_7(k1_xboole_0)
& v1_xreal_0(k1_xboole_0)
& ~ v2_xreal_0(k1_xboole_0)
& ~ v3_xreal_0(k1_xboole_0) ) ).
fof(rc3_afinsq_1,axiom,
! [A] :
? [B] :
( m1_relset_1(B,k5_numbers,A)
& v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_finset_1(B) ) ).
fof(rc4_afinsq_1,axiom,
? [A] :
( v1_xboole_0(A)
& v1_relat_1(A)
& v1_funct_1(A)
& v2_funct_1(A)
& v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A)
& v5_ordinal1(A)
& v4_ordinal2(A)
& v1_xcmplx_0(A)
& v1_finset_1(A)
& v1_xreal_0(A)
& ~ v2_xreal_0(A)
& ~ v3_xreal_0(A) ) ).
fof(rc5_afinsq_1,axiom,
! [A] :
? [B] :
( m1_ordinal1(B,A)
& v1_xboole_0(B)
& v1_relat_1(B)
& v1_funct_1(B)
& v2_funct_1(B)
& v1_ordinal1(B)
& v2_ordinal1(B)
& v3_ordinal1(B)
& v5_ordinal1(B)
& v4_ordinal2(B)
& v1_xcmplx_0(B)
& v1_finset_1(B)
& v1_xreal_0(B)
& ~ v2_xreal_0(B)
& ~ v3_xreal_0(B) ) ).
fof(fc3_afinsq_1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_finset_1(A)
& v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_finset_1(B) )
=> ( v1_relat_1(k1_ordinal4(A,B))
& v1_funct_1(k1_ordinal4(A,B))
& v5_ordinal1(k1_ordinal4(A,B))
& v1_finset_1(k1_ordinal4(A,B)) ) ) ).
fof(fc4_afinsq_1,axiom,
! [A] :
( v1_relat_1(k3_afinsq_1(A))
& v1_funct_1(k3_afinsq_1(A)) ) ).
fof(fc5_afinsq_1,axiom,
! [A] :
( v1_relat_1(k3_afinsq_1(A))
& v1_funct_1(k3_afinsq_1(A))
& v5_ordinal1(k3_afinsq_1(A))
& v1_finset_1(k3_afinsq_1(A)) ) ).
fof(fc6_afinsq_1,axiom,
! [A,B] :
( v1_relat_1(k7_afinsq_1(A,B))
& v1_funct_1(k7_afinsq_1(A,B)) ) ).
fof(fc7_afinsq_1,axiom,
! [A,B,C] :
( v1_relat_1(k8_afinsq_1(A,B,C))
& v1_funct_1(k8_afinsq_1(A,B,C)) ) ).
fof(fc8_afinsq_1,axiom,
! [A,B] :
( v1_relat_1(k7_afinsq_1(A,B))
& v1_funct_1(k7_afinsq_1(A,B))
& v5_ordinal1(k7_afinsq_1(A,B))
& v1_finset_1(k7_afinsq_1(A,B)) ) ).
fof(fc9_afinsq_1,axiom,
! [A,B,C] :
( v1_relat_1(k8_afinsq_1(A,B,C))
& v1_funct_1(k8_afinsq_1(A,B,C))
& v5_ordinal1(k8_afinsq_1(A,B,C))
& v1_finset_1(k8_afinsq_1(A,B,C)) ) ).
fof(fc10_afinsq_1,axiom,
! [A] : ~ v1_xboole_0(k10_afinsq_1(A)) ).
fof(fc11_afinsq_1,axiom,
! [A,B,C] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_finset_1(A) )
=> ( v1_relat_1(k2_funct_7(A,B,C))
& v1_funct_1(k2_funct_7(A,B,C))
& v5_ordinal1(k2_funct_7(A,B,C))
& v1_finset_1(k2_funct_7(A,B,C)) ) ) ).
fof(t1_afinsq_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> r2_hidden(A,k1_nat_1(A,np__1)) ) ).
fof(t2_afinsq_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)
=> A = k3_xboole_0(A,B) ) ) ) ).
fof(t3_afinsq_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( A = k3_xboole_0(A,B)
=> r1_xreal_0(A,B) ) ) ) ).
fof(t4_afinsq_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k2_xboole_0(A,k1_tarski(A)) = k1_nat_1(A,np__1) ) ).
fof(t5_afinsq_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> r1_tarski(k2_finseq_1(A),k1_nat_1(A,np__1)) ) ).
fof(t6_afinsq_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k1_nat_1(A,np__1) = k2_xboole_0(k1_tarski(np__0),k2_finseq_1(A)) ) ).
fof(t7_afinsq_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( ( v1_finset_1(A)
& v5_ordinal1(A) )
<=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& k1_relat_1(A) = B ) ) ) ).
fof(d1_afinsq_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_finset_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( B = k1_afinsq_1(A)
<=> B = k1_relat_1(A) ) ) ) ).
fof(t8_afinsq_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ~ ( ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& r1_tarski(k1_relat_1(A),B) )
& ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_finset_1(B) )
=> ~ r1_tarski(A,B) ) ) ) ).
fof(t9_afinsq_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_finset_1(B) )
=> ~ ( r2_hidden(A,B)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(C,k2_afinsq_1(B))
& A = k4_tarski(C,k1_funct_1(B,C)) ) ) ) ) ).
fof(t10_afinsq_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_finset_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_finset_1(B) )
=> ( ( k2_afinsq_1(A) = k2_afinsq_1(B)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k2_afinsq_1(A))
=> k1_funct_1(A,C) = k1_funct_1(B,C) ) ) )
=> A = B ) ) ) ).
fof(t11_afinsq_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_finset_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_finset_1(B) )
=> ( ( k1_afinsq_1(A) = k1_afinsq_1(B)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(k1_afinsq_1(A),C)
=> k1_funct_1(A,C) = k1_funct_1(B,C) ) ) )
=> A = B ) ) ) ).
fof(t12_afinsq_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_finset_1(B) )
=> ( v1_relat_1(k2_ordinal1(B,A))
& v1_funct_1(k2_ordinal1(B,A))
& v5_ordinal1(k2_ordinal1(B,A))
& v1_finset_1(k2_ordinal1(B,A)) ) ) ) ).
fof(t13_afinsq_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_finset_1(B) )
=> ( r1_tarski(k2_relat_1(B),k1_relat_1(A))
=> ( v1_relat_1(k5_relat_1(B,A))
& v1_funct_1(k5_relat_1(B,A))
& v5_ordinal1(k5_relat_1(B,A))
& v1_finset_1(k5_relat_1(B,A)) ) ) ) ) ).
fof(t14_afinsq_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_finset_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& v1_finset_1(C) )
=> ( C = k2_ordinal1(B,A)
=> ( r1_xreal_0(k1_afinsq_1(B),A)
| ( k1_afinsq_1(C) = A
& k2_afinsq_1(C) = A ) ) ) ) ) ) ).
fof(t15_afinsq_1,axiom,
! [A,B] :
( ( v1_finset_1(B)
& m1_ordinal1(B,A) )
=> ( v1_funct_1(B)
& m2_relset_1(B,k5_numbers,A) ) ) ).
fof(t16_afinsq_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B,C] :
( ( v1_finset_1(C)
& m1_ordinal1(C,B) )
=> ( v1_finset_1(k2_ordinal1(C,A))
& m1_ordinal1(k2_ordinal1(C,A),B) ) ) ) ).
fof(t17_afinsq_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ? [C] :
( v1_finset_1(C)
& m1_ordinal1(C,B)
& k1_afinsq_1(C) = A ) ) ) ).
fof(t18_afinsq_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_finset_1(A) )
=> ( k1_afinsq_1(A) = np__0
<=> A = k1_xboole_0 ) ) ).
fof(t19_afinsq_1,axiom,
! [A] :
( v1_finset_1(k1_xboole_0)
& m1_ordinal1(k1_xboole_0,A) ) ).
fof(d2_afinsq_1,axiom,
! [A] : k3_afinsq_1(A) = k1_tarski(k4_tarski(np__0,A)) ).
fof(d3_afinsq_1,axiom,
! [A] : k4_afinsq_1(A) = k1_xboole_0 ).
fof(d4_afinsq_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_finset_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_finset_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C) )
=> ( C = k1_ordinal4(A,B)
<=> ( k1_relat_1(C) = k1_nat_1(k1_afinsq_1(A),k1_afinsq_1(B))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k2_afinsq_1(A))
=> k1_funct_1(C,D) = k1_funct_1(A,D) ) )
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k2_afinsq_1(B))
=> k1_funct_1(C,k1_nat_1(k1_afinsq_1(A),D)) = k1_funct_1(B,D) ) ) ) ) ) ) ) ).
fof(t20_afinsq_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_finset_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_finset_1(B) )
=> k1_afinsq_1(k1_ordinal4(A,B)) = k1_nat_1(k1_afinsq_1(A),k1_afinsq_1(B)) ) ) ).
fof(t21_afinsq_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_finset_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& v1_finset_1(C) )
=> ( r1_xreal_0(k1_afinsq_1(B),A)
=> ( r1_xreal_0(k1_nat_1(k1_afinsq_1(B),k1_afinsq_1(C)),A)
| k1_funct_1(k1_ordinal4(B,C),A) = k1_funct_1(C,k5_real_1(A,k1_afinsq_1(B))) ) ) ) ) ) ).
fof(t22_afinsq_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_finset_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& v1_finset_1(C) )
=> ( r1_xreal_0(k1_afinsq_1(B),A)
=> ( r1_xreal_0(k1_afinsq_1(k1_ordinal4(B,C)),A)
| k1_funct_1(k1_ordinal4(B,C),A) = k1_funct_1(C,k5_real_1(A,k1_afinsq_1(B))) ) ) ) ) ) ).
fof(t23_afinsq_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_finset_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& v1_finset_1(C) )
=> ~ ( r2_hidden(A,k2_afinsq_1(k1_ordinal4(B,C)))
& ~ r2_hidden(A,k2_afinsq_1(B))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(D,k2_afinsq_1(C))
& A = k1_nat_1(k1_afinsq_1(B),D) ) ) ) ) ) ) ).
fof(t24_afinsq_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B) )
=> r1_ordinal1(k1_relat_1(A),k1_relat_1(k1_ordinal4(A,B))) ) ) ).
fof(t25_afinsq_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_finset_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& v1_finset_1(C) )
=> ~ ( r2_hidden(A,k2_afinsq_1(B))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( D = A
& r2_hidden(k1_nat_1(k1_afinsq_1(C),D),k2_afinsq_1(k1_ordinal4(C,B))) ) ) ) ) ) ).
fof(t26_afinsq_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_finset_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& v1_finset_1(C) )
=> ( r2_hidden(A,k2_afinsq_1(B))
=> r2_hidden(k1_nat_1(k1_afinsq_1(C),A),k2_afinsq_1(k1_ordinal4(C,B))) ) ) ) ) ).
fof(t27_afinsq_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_finset_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_finset_1(B) )
=> r1_tarski(k2_relat_1(A),k2_relat_1(k1_ordinal4(A,B))) ) ) ).
fof(t28_afinsq_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_finset_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_finset_1(B) )
=> r1_tarski(k2_relat_1(A),k2_relat_1(k1_ordinal4(B,A))) ) ) ).
fof(t29_afinsq_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_finset_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_finset_1(B) )
=> k2_relat_1(k1_ordinal4(A,B)) = k2_xboole_0(k2_relat_1(A),k2_relat_1(B)) ) ) ).
fof(t30_afinsq_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_finset_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_finset_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& v1_finset_1(C) )
=> k1_ordinal4(k1_ordinal4(A,B),C) = k1_ordinal4(A,k1_ordinal4(B,C)) ) ) ) ).
fof(t31_afinsq_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_finset_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_finset_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& v1_finset_1(C) )
=> ( ( k1_ordinal4(A,B) = k1_ordinal4(C,B)
| k1_ordinal4(B,A) = k1_ordinal4(B,C) )
=> A = C ) ) ) ) ).
fof(t32_afinsq_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_finset_1(A) )
=> ( k1_ordinal4(A,k1_xboole_0) = A
& k1_ordinal4(k1_xboole_0,A) = A ) ) ).
fof(t33_afinsq_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_finset_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_finset_1(B) )
=> ( k1_ordinal4(A,B) = k1_xboole_0
=> ( A = k1_xboole_0
& B = k1_xboole_0 ) ) ) ) ).
fof(d5_afinsq_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( B = k6_afinsq_1(A)
<=> ( k1_relat_1(B) = np__1
& k1_funct_1(B,np__0) = A ) ) ) ).
fof(t34_afinsq_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_finset_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_finset_1(B) )
=> ! [C] :
( ( v1_finset_1(k1_ordinal4(A,B))
& m1_ordinal1(k1_ordinal4(A,B),C) )
=> ( v1_finset_1(A)
& m1_ordinal1(A,C)
& v1_finset_1(B)
& m1_ordinal1(B,C) ) ) ) ) ).
fof(d6_afinsq_1,axiom,
! [A,B] : k7_afinsq_1(A,B) = k1_ordinal4(k6_afinsq_1(A),k6_afinsq_1(B)) ).
fof(d7_afinsq_1,axiom,
! [A,B,C] : k8_afinsq_1(A,B,C) = k1_ordinal4(k1_ordinal4(k6_afinsq_1(A),k6_afinsq_1(B)),k6_afinsq_1(C)) ).
fof(t35_afinsq_1,axiom,
! [A] : k6_afinsq_1(A) = k1_tarski(k4_tarski(np__0,A)) ).
fof(t36_afinsq_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_finset_1(B) )
=> ( B = k6_afinsq_1(A)
<=> ( k2_afinsq_1(B) = np__1
& k2_relat_1(B) = k1_tarski(A) ) ) ) ).
fof(t37_afinsq_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_finset_1(B) )
=> ( B = k6_afinsq_1(A)
<=> ( k1_afinsq_1(B) = np__1
& k2_relat_1(B) = k1_tarski(A) ) ) ) ).
fof(t38_afinsq_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_finset_1(B) )
=> ( B = k6_afinsq_1(A)
<=> ( k1_afinsq_1(B) = np__1
& k1_funct_1(B,np__0) = A ) ) ) ).
fof(t39_afinsq_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_finset_1(B) )
=> k1_funct_1(k1_ordinal4(k6_afinsq_1(A),B),np__0) = A ) ).
fof(t40_afinsq_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_finset_1(B) )
=> k1_funct_1(k1_ordinal4(B,k6_afinsq_1(A)),k1_afinsq_1(B)) = A ) ).
fof(t41_afinsq_1,axiom,
! [A,B,C] :
( k8_afinsq_1(A,B,C) = k1_ordinal4(k6_afinsq_1(A),k7_afinsq_1(B,C))
& k8_afinsq_1(A,B,C) = k1_ordinal4(k7_afinsq_1(A,B),k6_afinsq_1(C)) ) ).
fof(t42_afinsq_1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& v1_finset_1(C) )
=> ( C = k7_afinsq_1(A,B)
<=> ( k1_afinsq_1(C) = np__2
& k1_funct_1(C,np__0) = A
& k1_funct_1(C,np__1) = B ) ) ) ).
fof(t43_afinsq_1,axiom,
! [A,B,C,D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v5_ordinal1(D)
& v1_finset_1(D) )
=> ( D = k8_afinsq_1(A,B,C)
<=> ( k1_afinsq_1(D) = np__3
& k1_funct_1(D,np__0) = A
& k1_funct_1(D,np__1) = B
& k1_funct_1(D,np__2) = C ) ) ) ).
fof(t44_afinsq_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_finset_1(A) )
=> ~ ( A != k1_xboole_0
& ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_finset_1(B) )
=> ! [C] : A != k1_ordinal4(B,k6_afinsq_1(C)) ) ) ) ).
fof(t45_afinsq_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_finset_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_finset_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& v1_finset_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D)
& v5_ordinal1(D)
& v1_finset_1(D) )
=> ~ ( k1_ordinal4(A,B) = k1_ordinal4(C,D)
& r1_xreal_0(k1_afinsq_1(A),k1_afinsq_1(C))
& ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E)
& v5_ordinal1(E)
& v1_finset_1(E) )
=> k1_ordinal4(A,E) != C ) ) ) ) ) ) ).
fof(d8_afinsq_1,axiom,
! [A,B] :
( B = k10_afinsq_1(A)
<=> ! [C] :
( r2_hidden(C,B)
<=> ( v1_finset_1(C)
& m1_ordinal1(C,A) ) ) ) ).
fof(t46_afinsq_1,axiom,
! [A,B] :
( r2_hidden(A,k10_afinsq_1(B))
<=> ( v1_finset_1(A)
& m1_ordinal1(A,B) ) ) ).
fof(t47_afinsq_1,axiom,
! [A] : r2_hidden(k1_xboole_0,k10_afinsq_1(A)) ).
fof(t48_afinsq_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_finset_1(A) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( k1_afinsq_1(k2_funct_7(A,B,C)) = k1_afinsq_1(A)
& ( ~ r1_xreal_0(k1_afinsq_1(A),B)
=> k1_funct_1(k2_funct_7(A,B,C),B) = C )
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( D != B
=> k1_funct_1(k2_funct_7(A,B,C),D) = k1_funct_1(A,D) ) ) ) ) ) ).
fof(s1_afinsq_1,axiom,
( ( ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B,C] :
( ( r2_hidden(A,f1_s1_afinsq_1)
& p1_s1_afinsq_1(A,B)
& p1_s1_afinsq_1(A,C) )
=> B = C ) )
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(A,f1_s1_afinsq_1)
& ! [B] : ~ p1_s1_afinsq_1(A,B) ) ) )
=> ? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_finset_1(A)
& k2_afinsq_1(A) = f1_s1_afinsq_1
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(B,f1_s1_afinsq_1)
=> p1_s1_afinsq_1(B,k1_funct_1(A,B)) ) ) ) ) ).
fof(s2_afinsq_1,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_finset_1(A)
& k1_afinsq_1(A) = f1_s2_afinsq_1
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(B,f1_s2_afinsq_1)
=> k1_funct_1(A,B) = f2_s2_afinsq_1(B) ) ) ) ).
fof(s3_afinsq_1,axiom,
( ( p1_s3_afinsq_1(k1_xboole_0)
& ! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_finset_1(A) )
=> ! [B] :
( p1_s3_afinsq_1(A)
=> p1_s3_afinsq_1(k1_ordinal4(A,k6_afinsq_1(B))) ) ) )
=> ! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_finset_1(A) )
=> p1_s3_afinsq_1(A) ) ) ).
fof(s4_afinsq_1,axiom,
? [A] :
! [B] :
( r2_hidden(B,A)
<=> ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& v1_finset_1(C)
& r2_hidden(C,k10_afinsq_1(f1_s4_afinsq_1))
& p1_s4_afinsq_1(C)
& B = C ) ) ).
fof(dt_k1_afinsq_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_finset_1(A) )
=> m2_subset_1(k1_afinsq_1(A),k1_numbers,k5_numbers) ) ).
fof(redefinition_k1_afinsq_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_finset_1(A) )
=> k1_afinsq_1(A) = k1_card_1(A) ) ).
fof(dt_k2_afinsq_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_finset_1(A) )
=> m1_subset_1(k2_afinsq_1(A),k1_zfmisc_1(k5_numbers)) ) ).
fof(redefinition_k2_afinsq_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_finset_1(A) )
=> k2_afinsq_1(A) = k1_relat_1(A) ) ).
fof(dt_k3_afinsq_1,axiom,
$true ).
fof(dt_k4_afinsq_1,axiom,
! [A] :
( v1_xboole_0(k4_afinsq_1(A))
& v1_finset_1(k4_afinsq_1(A))
& m1_ordinal1(k4_afinsq_1(A),A) ) ).
fof(dt_k5_afinsq_1,axiom,
! [A,B,C] :
( ( v1_finset_1(B)
& m1_ordinal1(B,A)
& v1_finset_1(C)
& m1_ordinal1(C,A) )
=> m1_ordinal1(k5_afinsq_1(A,B,C),A) ) ).
fof(redefinition_k5_afinsq_1,axiom,
! [A,B,C] :
( ( v1_finset_1(B)
& m1_ordinal1(B,A)
& v1_finset_1(C)
& m1_ordinal1(C,A) )
=> k5_afinsq_1(A,B,C) = k1_ordinal4(B,C) ) ).
fof(dt_k6_afinsq_1,axiom,
! [A] :
( v1_relat_1(k6_afinsq_1(A))
& v1_funct_1(k6_afinsq_1(A)) ) ).
fof(redefinition_k6_afinsq_1,axiom,
! [A] : k6_afinsq_1(A) = k3_afinsq_1(A) ).
fof(dt_k7_afinsq_1,axiom,
$true ).
fof(dt_k8_afinsq_1,axiom,
$true ).
fof(dt_k9_afinsq_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A) )
=> ( v1_finset_1(k9_afinsq_1(A,B))
& m1_ordinal1(k9_afinsq_1(A,B),A) ) ) ).
fof(redefinition_k9_afinsq_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A) )
=> k9_afinsq_1(A,B) = k3_afinsq_1(B) ) ).
fof(dt_k10_afinsq_1,axiom,
$true ).
fof(dt_k11_afinsq_1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(B)
& m1_ordinal1(B,A)
& m1_subset_1(C,k5_numbers)
& m1_subset_1(D,A) )
=> ( v1_finset_1(k11_afinsq_1(A,B,C,D))
& m1_ordinal1(k11_afinsq_1(A,B,C,D),A) ) ) ).
fof(redefinition_k11_afinsq_1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(B)
& m1_ordinal1(B,A)
& m1_subset_1(C,k5_numbers)
& m1_subset_1(D,A) )
=> k11_afinsq_1(A,B,C,D) = k2_funct_7(B,C,D) ) ).
%------------------------------------------------------------------------------