SET007 Axioms: SET007+906.ax
%------------------------------------------------------------------------------
% File : SET007+906 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Cardinal Numbers and Finite Sets
% 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 : card_fin [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 106 ( 0 unt; 0 def)
% Number of atoms : 788 ( 169 equ)
% Maximal formula atoms : 23 ( 7 avg)
% Number of connectives : 745 ( 63 ~; 14 |; 349 &)
% ( 20 <=>; 299 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 10 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of predicates : 28 ( 27 usr; 0 prp; 1-3 aty)
% Number of functors : 78 ( 78 usr; 8 con; 0-5 aty)
% Number of variables : 392 ( 375 !; 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(rc1_card_fin,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_relat_1(A)
& v1_funct_1(A)
& v1_card_fin(A) ) ).
fof(rc2_card_fin,axiom,
? [A] :
( v1_xboole_0(A)
& v1_relat_1(A)
& v1_funct_1(A)
& v2_funct_1(A)
& v1_membered(A)
& v2_membered(A)
& v3_membered(A)
& v4_membered(A)
& v5_membered(A)
& v1_finset_1(A)
& v1_card_fin(A) ) ).
fof(fc1_card_fin,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_card_fin(A) )
=> v1_finset_1(k1_funct_1(A,B)) ) ).
fof(fc2_card_fin,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_card_fin(A) )
=> ( v1_relat_1(k7_relat_1(A,B))
& v1_funct_1(k7_relat_1(A,B))
& v1_card_fin(k7_relat_1(A,B)) ) ) ).
fof(fc3_card_fin,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_card_fin(A)
& v1_relat_1(B)
& v1_funct_1(B) )
=> ( v1_relat_1(k5_relat_1(B,A))
& v1_funct_1(k5_relat_1(B,A))
& v1_card_fin(k5_relat_1(B,A)) ) ) ).
fof(fc4_card_fin,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_card_fin(A)
& v1_relat_1(B)
& v1_funct_1(B) )
=> ( v1_relat_1(k1_yellow20(A,B))
& v1_funct_1(k1_yellow20(A,B))
& v1_card_fin(k1_yellow20(A,B)) ) ) ).
fof(t1_card_fin,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( v1_finset_1(B)
=> ( ( r1_tarski(A,B)
& k4_card_1(A) = k4_card_1(B) )
=> A = B ) ) ) ).
fof(t4_card_fin,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( v1_finset_1(B)
=> ( ( A = k1_xboole_0
=> B = k1_xboole_0 )
=> k4_card_1(k1_funct_2(B,A)) = k3_newton(k4_card_1(A),k4_card_1(B)) ) ) ) ).
fof(t6_card_fin,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> m2_subset_1(k12_binop_2(k11_newton(A),k11_newton(k5_binarith(A,B))),k1_numbers,k5_numbers) ) ) ).
fof(d1_card_fin,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C,D,E] :
( m1_subset_1(E,k1_zfmisc_1(k1_funct_2(A,k2_tarski(C,D))))
=> ( E = k1_card_fin(A,B,C,D)
<=> ! [F] :
( r2_hidden(F,E)
<=> ? [G] :
( v1_funct_1(G)
& v1_funct_2(G,A,k2_tarski(C,D))
& m2_relset_1(G,A,k2_tarski(C,D))
& G = F
& k1_card_1(k3_funct_2(A,k2_tarski(C,D),G,k1_tarski(C))) = B ) ) ) ) ) ) ).
fof(t9_card_fin,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( k4_card_1(B) != C
=> v1_xboole_0(k1_card_fin(B,C,A,A)) ) ) ) ).
fof(t10_card_fin,axiom,
! [A,B,C] :
( v1_finset_1(C)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(D,k4_card_1(C))
=> v1_xboole_0(k1_card_fin(C,D,A,B)) ) ) ) ).
fof(t11_card_fin,axiom,
! [A,B,C] :
( v1_finset_1(C)
=> ( A != B
=> k4_card_1(k1_card_fin(C,np__0,A,B)) = np__1 ) ) ).
fof(t12_card_fin,axiom,
! [A,B,C] :
( v1_finset_1(C)
=> k4_card_1(k1_card_fin(C,k4_card_1(C),A,B)) = np__1 ) ).
fof(t13_card_fin,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( k1_funct_1(C,A) = B
& r2_hidden(A,k1_relat_1(C)) )
=> k2_xboole_0(k1_tarski(A),k10_relat_1(k7_relat_1(C,k4_xboole_0(k1_relat_1(C),k1_tarski(A))),k1_tarski(B))) = k10_relat_1(C,k1_tarski(B)) ) ) ).
fof(t15_card_fin,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( k1_funct_1(C,A) != B
=> k10_relat_1(k7_relat_1(C,k4_xboole_0(k1_relat_1(C),k1_tarski(A))),k1_tarski(B)) = k10_relat_1(C,k1_tarski(B)) ) ) ).
fof(t17_card_fin,axiom,
! [A,B,C,D] :
( v1_finset_1(D)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ ( A != B
& ~ r2_hidden(C,D)
& k4_card_1(k1_card_fin(k2_xboole_0(D,k1_tarski(C)),k23_binop_2(E,np__1),A,B)) != k23_binop_2(k4_card_1(k1_card_fin(D,k23_binop_2(E,np__1),A,B)),k4_card_1(k1_card_fin(D,E,A,B))) ) ) ) ).
fof(t18_card_fin,axiom,
! [A,B,C] :
( v1_finset_1(C)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( A != B
=> k4_card_1(k1_card_fin(C,D,A,B)) = k8_newton(D,k4_card_1(C)) ) ) ) ).
fof(t19_card_fin,axiom,
! [A,B,C] :
( v1_finset_1(C)
=> ! [D] :
( v1_finset_1(D)
=> ( A != B
=> r2_hidden(k1_funct_4(k2_funcop_1(C,B),k2_funcop_1(D,A)),k1_card_fin(k2_xboole_0(D,C),k4_card_1(D),A,B)) ) ) ) ).
fof(t20_card_fin,axiom,
! [A,B,C] :
( v1_finset_1(C)
=> ! [D] :
( v1_finset_1(D)
=> ( r1_xboole_0(C,D)
=> ( A = B
| r2_hidden(k1_funct_4(k2_funcop_1(C,A),k2_funcop_1(D,B)),k1_card_fin(k2_xboole_0(C,D),k4_card_1(C),A,B)) ) ) ) ) ).
fof(d2_card_fin,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C,D] :
( m1_subset_1(D,k1_zfmisc_1(k3_tarski(k2_relat_1(A))))
=> ( D = k2_card_fin(A,B,C)
<=> ! [E] :
( r2_hidden(E,D)
<=> ( r2_hidden(E,k3_tarski(k2_relat_1(A)))
& ! [F] :
( ( r2_hidden(F,k1_relat_1(B))
& k1_funct_1(B,F) = C )
=> r2_hidden(E,k1_funct_1(A,F)) ) ) ) ) ) ) ) ).
fof(t21_card_fin,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( ~ v1_xboole_0(k3_xboole_0(k1_relat_1(C),k10_relat_1(D,k1_tarski(A))))
=> ( r2_hidden(B,k2_card_fin(C,D,A))
<=> ! [E] :
( ( r2_hidden(E,k1_relat_1(D))
& k1_funct_1(D,E) = A )
=> r2_hidden(B,k1_funct_1(C,E)) ) ) ) ) ) ).
fof(t22_card_fin,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ~ v1_xboole_0(k2_card_fin(B,C,A))
=> r1_tarski(k10_relat_1(C,k1_tarski(A)),k1_relat_1(B)) ) ) ) ).
fof(t23_card_fin,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ~ v1_xboole_0(k2_card_fin(B,C,A))
=> ! [D,E] :
~ ( r2_hidden(D,k10_relat_1(C,k1_tarski(A)))
& r2_hidden(E,k10_relat_1(C,k1_tarski(A)))
& r1_xboole_0(k1_funct_1(B,D),k1_funct_1(B,E)) ) ) ) ) ).
fof(t24_card_fin,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ~ ( r2_hidden(A,k2_card_fin(C,D,B))
& r2_hidden(B,k2_relat_1(D))
& ! [E] :
~ ( r2_hidden(E,k1_relat_1(D))
& k1_funct_1(D,E) = B
& r2_hidden(A,k1_funct_1(C,E)) ) ) ) ) ).
fof(t25_card_fin,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( v1_xboole_0(B)
| v1_xboole_0(k3_tarski(k2_relat_1(B))) )
=> k2_card_fin(B,C,A) = k3_tarski(k2_relat_1(B)) ) ) ) ).
fof(t26_card_fin,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( k7_relat_1(B,k10_relat_1(C,k1_tarski(A))) = k2_funcop_1(k10_relat_1(C,k1_tarski(A)),k3_tarski(k2_relat_1(B)))
=> k2_card_fin(B,C,A) = k3_tarski(k2_relat_1(B)) ) ) ) ).
fof(t27_card_fin,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( k2_card_fin(B,C,A) = k3_tarski(k2_relat_1(B))
=> ( v1_xboole_0(k3_tarski(k2_relat_1(B)))
| k7_relat_1(B,k10_relat_1(C,k1_tarski(A))) = k2_funcop_1(k10_relat_1(C,k1_tarski(A)),k3_tarski(k2_relat_1(B))) ) ) ) ) ).
fof(t28_card_fin,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> k2_card_fin(B,k1_xboole_0,A) = k3_tarski(k2_relat_1(B)) ) ).
fof(t29_card_fin,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> r1_tarski(k2_card_fin(C,D,A),k2_card_fin(C,k7_relat_1(D,B),A)) ) ) ).
fof(t30_card_fin,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( k10_relat_1(C,k1_tarski(A)) = k10_relat_1(k7_relat_1(C,B),k1_tarski(A))
=> k2_card_fin(D,C,A) = k2_card_fin(D,k7_relat_1(C,B),A) ) ) ) ).
fof(t31_card_fin,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> r1_tarski(k2_card_fin(k7_relat_1(C,A),D,B),k2_card_fin(C,D,B)) ) ) ).
fof(t32_card_fin,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( ( r2_hidden(A,k2_relat_1(C))
& r1_tarski(k10_relat_1(C,k1_tarski(A)),B) )
=> k2_card_fin(k7_relat_1(D,B),C,A) = k2_card_fin(D,C,A) ) ) ) ).
fof(t33_card_fin,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( r2_hidden(A,k10_relat_1(C,k1_tarski(B)))
=> r1_tarski(k2_card_fin(D,C,B),k1_funct_1(D,A)) ) ) ) ).
fof(t34_card_fin,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( r2_hidden(A,k10_relat_1(C,k1_tarski(B)))
=> k3_xboole_0(k2_card_fin(D,k7_relat_1(C,k4_xboole_0(k1_relat_1(C),k1_tarski(A))),B),k1_funct_1(D,A)) = k2_card_fin(D,C,B) ) ) ) ).
fof(t35_card_fin,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E) )
=> ( k10_relat_1(D,k1_tarski(A)) = k10_relat_1(E,k1_tarski(B))
=> k2_card_fin(C,D,A) = k2_card_fin(C,E,B) ) ) ) ) ).
fof(t36_card_fin,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( k10_relat_1(B,k1_tarski(A)) = k1_xboole_0
=> k2_card_fin(C,B,A) = k3_tarski(k2_relat_1(C)) ) ) ) ).
fof(t37_card_fin,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ( k1_tarski(A) = k10_relat_1(C,k1_tarski(B))
=> k2_card_fin(D,C,B) = k1_funct_1(D,A) ) ) ) ).
fof(t38_card_fin,axiom,
! [A,B,C,D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E) )
=> ( k2_tarski(A,B) = k10_relat_1(D,k1_tarski(C))
=> k2_card_fin(E,D,C) = k3_xboole_0(k1_funct_1(E,A),k1_funct_1(E,B)) ) ) ) ).
fof(t39_card_fin,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ~ v1_xboole_0(C)
=> ( r2_hidden(A,k2_card_fin(C,k2_funcop_1(k1_relat_1(C),B),B))
<=> ! [D] :
( r2_hidden(D,k1_relat_1(C))
=> r2_hidden(A,k1_funct_1(C,D)) ) ) ) ) ).
fof(d3_card_fin,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v1_card_fin(A)
<=> ! [B] : v1_finset_1(k1_funct_1(A,B)) ) ) ).
fof(t40_card_fin,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_card_fin(C) )
=> ( r2_hidden(A,k2_relat_1(B))
=> v1_finset_1(k2_card_fin(C,B,A)) ) ) ) ).
fof(t41_card_fin,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_card_fin(A) )
=> ( v1_finset_1(k1_relat_1(A))
=> v1_finset_1(k3_tarski(k2_relat_1(A))) ) ) ).
fof(t42_card_fin,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_finset_1(B)
& m1_ordinal1(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] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k2_afinsq_1(B))
=> ( ( ~ v1_setwiseo(C,A)
& D = np__0 )
| k1_binop_1(C,k7_stirl2_1(A,k4_card_fin(A,B,D),C),k1_funct_1(B,D)) = k7_stirl2_1(A,k4_card_fin(A,B,k23_binop_2(D,np__1)),C) ) ) ) ) ) ) ).
fof(t43_card_fin,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_finset_1(B)
& m1_ordinal1(B,A) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( k1_afinsq_1(B) = k23_binop_2(C,np__1)
=> B = k1_ordinal4(k4_card_fin(A,B,C),k6_afinsq_1(k1_funct_1(B,C))) ) ) ) ) ).
fof(t44_card_fin,axiom,
! [A] :
( ( v1_finset_1(A)
& m1_ordinal1(A,k5_numbers) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(B,k2_afinsq_1(A))
=> k23_binop_2(k10_stirl2_1(k4_card_fin(k5_numbers,A,B)),k11_stirl2_1(A,B)) = k10_stirl2_1(k4_card_fin(k5_numbers,A,k23_binop_2(B,np__1))) ) ) ) ).
fof(t45_card_fin,axiom,
! [A] :
( ( v1_finset_1(A)
& m1_ordinal1(A,k5_numbers) )
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_tarski(k2_relat_1(A),k2_stirl2_1(np__0,B))
=> k10_stirl2_1(A) = k24_binop_2(B,k4_card_1(k10_relat_1(A,k1_stirl2_1(B)))) ) ) ) ).
fof(t46_card_fin,axiom,
! [A,B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(A,k1_card_fin(B,C,np__1,np__0))
<=> ? [D] :
( v1_finset_1(D)
& m1_ordinal1(D,k5_numbers)
& D = A
& k2_afinsq_1(D) = B
& r1_tarski(k2_relat_1(D),k2_stirl2_1(np__0,np__1))
& k10_stirl2_1(D) = C ) ) ) ) ).
fof(t47_card_fin,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_finset_1(B)
& m1_ordinal1(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) )
=> ( ( v1_setwiseo(C,A)
| r1_xreal_0(np__1,k1_afinsq_1(B)) )
=> k7_stirl2_1(A,B,C) = k2_finsop_1(A,k2_prgcor_2(A,B),C) ) ) ) ) ).
fof(t48_card_fin,axiom,
! [A] :
( ~ v1_xboole_0(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] :
( ( v1_finset_1(C)
& m1_ordinal1(C,A) )
=> ! [D] :
( ( v1_finset_1(D)
& m1_ordinal1(D,A) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k2_afinsq_1(C),k2_afinsq_1(C))
& v3_funct_2(E,k2_afinsq_1(C),k2_afinsq_1(C))
& m2_relset_1(E,k2_afinsq_1(C),k2_afinsq_1(C)) )
=> ( ( v1_binop_1(B,A)
& v2_binop_1(B,A)
& D = k5_relat_1(E,C) )
=> ( ( ~ v1_setwiseo(B,A)
& ~ r1_xreal_0(np__1,k1_afinsq_1(C)) )
| k7_stirl2_1(A,C,B) = k7_stirl2_1(A,D,B) ) ) ) ) ) ) ) ).
fof(d4_card_fin,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_card_fin(B) )
=> ( v1_finset_1(k1_relat_1(B))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( C = k5_card_fin(A,B)
<=> ! [D,E,F] :
( v1_finset_1(F)
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,k4_card_1(k1_card_fin(F,A,D,E)),k1_card_fin(F,A,D,E))
& m2_relset_1(G,k4_card_1(k1_card_fin(F,A,D,E)),k1_card_fin(F,A,D,E)) )
=> ~ ( k1_relat_1(B) = F
& v2_funct_1(G)
& D != E
& ! [H] :
( ( v1_finset_1(H)
& m1_ordinal1(H,k5_numbers) )
=> ~ ( k2_afinsq_1(H) = k4_relset_1(k4_card_1(k1_card_fin(F,A,D,E)),k1_card_fin(F,A,D,E),G)
& ! [I,J] :
( ( v1_relat_1(J)
& v1_funct_1(J) )
=> ( ( r2_hidden(I,k2_afinsq_1(H))
& J = k1_funct_1(G,I) )
=> k11_stirl2_1(H,I) = k1_card_1(k2_card_fin(B,J,D)) ) )
& C = k10_stirl2_1(H) ) ) ) ) ) ) ) ) ) ) ).
fof(t49_card_fin,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_card_fin(B) )
=> ! [C,D,E] :
( v1_finset_1(E)
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k4_card_1(k1_card_fin(E,A,C,D)),k1_card_fin(E,A,C,D))
& m2_relset_1(F,k4_card_1(k1_card_fin(E,A,C,D)),k1_card_fin(E,A,C,D)) )
=> ( ( k1_relat_1(B) = E
& v2_funct_1(F) )
=> ( C = D
| ! [G] :
( ( v1_finset_1(G)
& m1_ordinal1(G,k5_numbers) )
=> ( ( k2_afinsq_1(G) = k4_relset_1(k4_card_1(k1_card_fin(E,A,C,D)),k1_card_fin(E,A,C,D),F)
& ! [H,I] :
( ( v1_relat_1(I)
& v1_funct_1(I) )
=> ( ( r2_hidden(H,k2_afinsq_1(G))
& I = k1_funct_1(F,H) )
=> k11_stirl2_1(G,H) = k1_card_1(k2_card_fin(B,I,C)) ) ) )
=> k5_card_fin(A,B) = k10_stirl2_1(G) ) ) ) ) ) ) ) ) ).
fof(t50_card_fin,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_card_fin(B) )
=> ( ( v1_finset_1(k1_relat_1(B))
& A = np__0 )
=> k5_card_fin(A,B) = k1_card_1(k3_tarski(k2_relat_1(B))) ) ) ) ).
fof(t51_card_fin,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_card_fin(C) )
=> ( k1_relat_1(C) = A
=> ( r1_xreal_0(B,k4_card_1(A))
| k5_card_fin(B,C) = np__0 ) ) ) ) ) ).
fof(t52_card_fin,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_card_fin(A) )
=> ! [B] :
( v1_finset_1(B)
=> ( k1_relat_1(A) = B
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k4_card_1(B),B)
& m2_relset_1(C,k4_card_1(B),B) )
=> ~ ( v2_funct_1(C)
& ! [D] :
( ( v1_finset_1(D)
& m1_ordinal1(D,k5_numbers) )
=> ~ ( k2_afinsq_1(D) = k4_card_1(B)
& ! [E] :
( r2_hidden(E,k2_afinsq_1(D))
=> k11_stirl2_1(D,E) = k4_card_1(k1_funct_1(k5_relat_1(C,A),E)) )
& k5_card_fin(np__1,A) = k10_stirl2_1(D) ) ) ) ) ) ) ) ).
fof(t53_card_fin,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_card_fin(C) )
=> ( k1_relat_1(C) = B
=> k5_card_fin(k4_card_1(B),C) = k1_card_1(k2_card_fin(C,k2_funcop_1(B,A),A)) ) ) ) ).
fof(t54_card_fin,axiom,
! [A,B] :
( v1_finset_1(B)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_card_fin(C) )
=> ( C = k3_cqc_lang(A,B)
=> k5_card_fin(np__1,C) = k4_card_1(B) ) ) ) ).
fof(t55_card_fin,axiom,
! [A,B,C] :
( v1_finset_1(C)
=> ! [D] :
( v1_finset_1(D)
=> ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E)
& v1_card_fin(E) )
=> ( E = k4_funct_4(A,B,C,D)
=> ( A = B
| ( k5_card_fin(np__1,E) = k23_binop_2(k4_card_1(C),k4_card_1(D))
& k5_card_fin(np__2,E) = k4_card_1(k3_xboole_0(C,D)) ) ) ) ) ) ) ).
fof(t56_card_fin,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_card_fin(A) )
=> ! [B] :
( ( v1_finset_1(k1_relat_1(A))
& r2_hidden(B,k1_relat_1(A)) )
=> k5_card_fin(np__1,A) = k23_binop_2(k5_card_fin(np__1,k7_relat_1(A,k4_xboole_0(k1_relat_1(A),k1_tarski(B)))),k4_card_1(k1_funct_1(A,B))) ) ) ).
fof(t57_card_fin,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( k1_relat_1(k1_yellow20(B,k2_funcop_1(k1_relat_1(B),A))) = k1_relat_1(B)
& ! [C] :
( r2_hidden(C,k1_relat_1(B))
=> k1_funct_1(k1_yellow20(B,k2_funcop_1(k1_relat_1(B),A)),C) = k3_xboole_0(k1_funct_1(B,C),A) ) ) ) ).
fof(t58_card_fin,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> k3_xboole_0(k3_tarski(k2_relat_1(B)),A) = k3_tarski(k2_relat_1(k1_yellow20(B,k2_funcop_1(k1_relat_1(B),A)))) ) ).
fof(t59_card_fin,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> k3_xboole_0(k2_card_fin(C,D,A),B) = k2_card_fin(k1_yellow20(C,k2_funcop_1(k1_relat_1(C),B)),D,A) ) ) ).
fof(t60_card_fin,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) )
=> ( ( v2_funct_1(A)
& v2_funct_1(B)
& r1_xboole_0(k2_relat_1(A),k2_relat_1(B)) )
=> v2_funct_1(k1_ordinal4(A,B)) ) ) ) ).
fof(t61_card_fin,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_card_fin(B) )
=> ! [C] :
( v1_finset_1(C)
=> ! [D,E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ( k1_relat_1(B) = C
& r2_hidden(D,k1_relat_1(B)) )
=> ( r1_xreal_0(A,np__0)
| k5_card_fin(k23_binop_2(A,np__1),B) = k23_binop_2(k5_card_fin(k23_binop_2(A,np__1),k7_relat_1(B,k4_xboole_0(k1_relat_1(B),k1_tarski(D)))),k5_card_fin(A,k1_yellow20(k7_relat_1(B,k4_xboole_0(k1_relat_1(B),k1_tarski(D))),k2_funcop_1(k4_xboole_0(k1_relat_1(B),k1_tarski(D)),k1_funct_1(B,D))))) ) ) ) ) ) ) ).
fof(t62_card_fin,axiom,
! [A] :
( ~ v1_xboole_0(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] :
( ( v1_finset_1(C)
& m1_ordinal1(C,A) )
=> ! [D] :
( ( v1_finset_1(D)
& m1_ordinal1(D,A) )
=> ! [E] :
( ( v1_finset_1(E)
& m1_ordinal1(E,A) )
=> ( ( v1_binop_1(B,A)
& v2_binop_1(B,A)
& k1_afinsq_1(C) = k1_afinsq_1(D)
& k1_afinsq_1(C) = k1_afinsq_1(E)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( r2_hidden(F,k2_afinsq_1(E))
=> k1_funct_1(E,F) = k1_binop_1(B,k1_funct_1(C,F),k1_funct_1(D,F)) ) ) )
=> ( ( ~ v1_setwiseo(B,A)
& ~ r1_xreal_0(np__1,k1_afinsq_1(C)) )
| k7_stirl2_1(A,k5_afinsq_1(A,C,D),B) = k7_stirl2_1(A,E,B) ) ) ) ) ) ) ) ).
fof(d5_card_fin,axiom,
! [A] :
( ( v1_finset_1(A)
& m1_ordinal1(A,k4_numbers) )
=> k6_card_fin(A) = k7_stirl2_1(k4_numbers,A,k44_binop_2) ) ).
fof(t63_card_fin,axiom,
! [A] :
( ( v1_finset_1(A)
& m1_ordinal1(A,k5_numbers) )
=> ! [B] :
( ( v1_finset_1(B)
& m1_ordinal1(B,k4_numbers) )
=> ( B = A
=> k6_card_fin(B) = k10_stirl2_1(A) ) ) ) ).
fof(t64_card_fin,axiom,
! [A] :
( ( v1_finset_1(A)
& m1_ordinal1(A,k4_numbers) )
=> ! [B] :
( ( v1_finset_1(B)
& m1_ordinal1(B,k4_numbers) )
=> ! [C] :
( v1_int_1(C)
=> ( ( k2_afinsq_1(A) = k2_afinsq_1(B)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k2_afinsq_1(A))
=> k3_xcmplx_0(C,k7_card_fin(A,D)) = k7_card_fin(B,D) ) ) )
=> k3_xcmplx_0(C,k6_card_fin(A)) = k6_card_fin(B) ) ) ) ) ).
fof(t65_card_fin,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r2_hidden(A,k1_relat_1(B))
=> k3_tarski(k2_relat_1(B)) = k2_xboole_0(k3_tarski(k2_relat_1(k7_relat_1(B,k4_xboole_0(k1_relat_1(B),k1_tarski(A))))),k1_funct_1(B,A)) ) ) ).
fof(t66_card_fin,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_card_fin(A) )
=> ! [B] :
( v1_finset_1(B)
=> ? [C] :
( v1_finset_1(C)
& m1_ordinal1(C,k4_numbers)
& k2_afinsq_1(C) = k4_card_1(B)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k2_afinsq_1(C))
=> k7_card_fin(C,D) = k11_binop_2(k3_newton(k7_binop_2(np__1),D),k5_card_fin(k23_binop_2(D,np__1),A)) ) ) ) ) ) ).
fof(t67_card_fin,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_card_fin(A) )
=> ! [B] :
( v1_finset_1(B)
=> ( k1_relat_1(A) = B
=> ! [C] :
( ( v1_finset_1(C)
& m1_ordinal1(C,k4_numbers) )
=> ( ( k2_afinsq_1(C) = k4_card_1(B)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k2_afinsq_1(C))
=> k7_card_fin(C,D) = k11_binop_2(k3_newton(k7_binop_2(np__1),D),k5_card_fin(k23_binop_2(D,np__1),A)) ) ) )
=> k1_card_1(k3_tarski(k2_relat_1(A))) = k6_card_fin(C) ) ) ) ) ) ).
fof(t68_card_fin,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_card_fin(A) )
=> ! [B] :
( v1_finset_1(B)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( k1_relat_1(A) = B
=> ( ! [E,F] :
~ ( E != F
& ! [G] :
( ( v1_relat_1(G)
& v1_funct_1(G) )
=> ( r2_hidden(G,k1_card_fin(B,D,E,F))
=> k1_card_1(k2_card_fin(A,G,E)) = C ) ) )
| k5_card_fin(D,A) = k11_binop_2(C,k8_newton(D,k4_card_1(B))) ) ) ) ) ) ) ).
fof(t69_card_fin,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_card_fin(A) )
=> ! [B] :
( v1_finset_1(B)
=> ( k1_relat_1(A) = B
=> ! [C] :
( ( v1_finset_1(C)
& m1_ordinal1(C,k5_numbers) )
=> ~ ( k2_afinsq_1(C) = k4_card_1(B)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(D,k2_afinsq_1(C))
& ! [E,F] :
~ ( E != F
& ! [G] :
( ( v1_relat_1(G)
& v1_funct_1(G) )
=> ( r2_hidden(G,k1_card_fin(B,k23_binop_2(D,np__1),E,F))
=> k1_card_1(k2_card_fin(A,G,E)) = k11_stirl2_1(C,D) ) ) ) ) )
& ! [D] :
( ( v1_finset_1(D)
& m1_ordinal1(D,k4_numbers) )
=> ~ ( k2_afinsq_1(D) = k4_card_1(B)
& k1_card_1(k3_tarski(k2_relat_1(A))) = k6_card_fin(D)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k2_afinsq_1(D))
=> k7_card_fin(D,E) = k11_binop_2(k11_binop_2(k3_newton(k7_binop_2(np__1),E),k11_stirl2_1(C,E)),k8_newton(k23_binop_2(E,np__1),k4_card_1(B))) ) ) ) ) ) ) ) ) ) ).
fof(t71_card_fin,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(B,A)
& ! [C] :
( ( v1_finset_1(C)
& m1_ordinal1(C,k4_numbers) )
=> ~ ( k6_stirl2_1(A,B) = k3_xcmplx_0(k12_binop_2(np__1,k11_newton(B)),k6_card_fin(C))
& k2_afinsq_1(C) = k23_binop_2(B,np__1)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k2_afinsq_1(C))
=> k7_card_fin(C,D) = k11_binop_2(k11_binop_2(k3_newton(k7_binop_2(np__1),D),k8_newton(D,B)),k3_newton(k10_binop_2(B,D),A)) ) ) ) ) ) ) ) ).
fof(dt_k1_card_fin,axiom,
! [A,B,C,D] :
( ( v1_finset_1(A)
& m1_subset_1(B,k5_numbers) )
=> m1_subset_1(k1_card_fin(A,B,C,D),k1_zfmisc_1(k1_funct_2(A,k2_tarski(C,D)))) ) ).
fof(dt_k2_card_fin,axiom,
! [A,B,C] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_relat_1(B)
& v1_funct_1(B) )
=> m1_subset_1(k2_card_fin(A,B,C),k1_zfmisc_1(k3_tarski(k2_relat_1(A)))) ) ).
fof(dt_k3_card_fin,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_finset_1(A)
& m1_subset_1(B,k5_numbers) )
=> ( v1_relat_1(k3_card_fin(A,B))
& v1_funct_1(k3_card_fin(A,B))
& v5_ordinal1(k3_card_fin(A,B))
& v1_finset_1(k3_card_fin(A,B)) ) ) ).
fof(redefinition_k3_card_fin,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_finset_1(A)
& m1_subset_1(B,k5_numbers) )
=> k3_card_fin(A,B) = k7_relat_1(A,B) ) ).
fof(dt_k4_card_fin,axiom,
! [A,B,C] :
( ( v1_finset_1(B)
& m1_ordinal1(B,A)
& m1_subset_1(C,k5_numbers) )
=> ( v1_finset_1(k4_card_fin(A,B,C))
& m1_ordinal1(k4_card_fin(A,B,C),A) ) ) ).
fof(redefinition_k4_card_fin,axiom,
! [A,B,C] :
( ( v1_finset_1(B)
& m1_ordinal1(B,A)
& m1_subset_1(C,k5_numbers) )
=> k4_card_fin(A,B,C) = k7_relat_1(B,C) ) ).
fof(dt_k5_card_fin,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_card_fin(B) )
=> m2_subset_1(k5_card_fin(A,B),k1_numbers,k5_numbers) ) ).
fof(dt_k6_card_fin,axiom,
! [A] :
( ( v1_finset_1(A)
& m1_ordinal1(A,k4_numbers) )
=> v1_int_1(k6_card_fin(A)) ) ).
fof(dt_k7_card_fin,axiom,
! [A,B] :
( ( v1_finset_1(A)
& m1_ordinal1(A,k4_numbers) )
=> v1_int_1(k7_card_fin(A,B)) ) ).
fof(redefinition_k7_card_fin,axiom,
! [A,B] :
( ( v1_finset_1(A)
& m1_ordinal1(A,k4_numbers) )
=> k7_card_fin(A,B) = k1_funct_1(A,B) ) ).
fof(t2_card_fin,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( v1_finset_1(B)
=> ! [C,D] :
~ ( ( B = k1_xboole_0
=> A = k1_xboole_0 )
& ~ r2_hidden(C,A)
& k4_card_1(k1_funct_2(A,B)) != k1_card_1(a_4_0_card_fin(A,B,C,D)) ) ) ) ).
fof(t3_card_fin,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( v1_finset_1(B)
=> ! [C,D] :
( r2_hidden(D,B)
=> ( r2_hidden(C,A)
| k4_card_1(k1_funct_2(A,B)) = k1_card_1(a_4_1_card_fin(A,B,C,D)) ) ) ) ) ).
fof(t5_card_fin,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( v1_finset_1(B)
=> ! [C,D] :
~ ( ( v1_xboole_0(B)
=> v1_xboole_0(A) )
& ~ r2_hidden(C,A)
& ~ r2_hidden(D,B)
& k1_card_1(a_2_0_card_fin(A,B)) != k1_card_1(a_4_2_card_fin(A,B,C,D)) ) ) ) ).
fof(t7_card_fin,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( v1_finset_1(B)
=> ( r1_xreal_0(k4_card_1(A),k4_card_1(B))
=> k1_card_1(a_2_0_card_fin(A,B)) = k12_binop_2(k11_newton(k4_card_1(B)),k11_newton(k5_binarith(k4_card_1(B),k4_card_1(A)))) ) ) ) ).
fof(t8_card_fin,axiom,
! [A] :
( v1_finset_1(A)
=> k1_card_1(a_1_0_card_fin(A)) = k11_newton(k4_card_1(A)) ) ).
fof(t14_card_fin,axiom,
! [A,B,C,D] :
( v1_finset_1(D)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( ~ r2_hidden(A,D)
=> k4_card_1(k1_card_fin(D,E,B,C)) = k1_card_1(a_5_0_card_fin(A,B,C,D,E)) ) ) ) ).
fof(t16_card_fin,axiom,
! [A,B,C,D] :
( v1_finset_1(D)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ ( ~ r2_hidden(A,D)
& B != C
& k4_card_1(k1_card_fin(D,E,B,C)) != k1_card_1(a_5_1_card_fin(A,B,C,D,E)) ) ) ) ).
fof(t70_card_fin,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( v1_finset_1(B)
=> ~ ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ! [C] :
( ( v1_finset_1(C)
& m1_ordinal1(C,k4_numbers) )
=> ~ ( k2_afinsq_1(C) = k23_binop_2(k4_card_1(B),np__1)
& k6_card_fin(C) = k1_card_1(a_2_1_card_fin(A,B))
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k2_afinsq_1(C))
=> k7_card_fin(C,D) = k11_binop_2(k11_binop_2(k3_newton(k7_binop_2(np__1),D),k8_newton(D,k4_card_1(B))),k3_newton(k10_binop_2(k4_card_1(B),D),k4_card_1(A))) ) ) ) ) ) ) ) ).
fof(t72_card_fin,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( v1_finset_1(B)
=> ! [C] :
( v1_finset_1(C)
=> ( r1_tarski(C,A)
=> ( ( v1_xboole_0(B)
& ~ v1_xboole_0(A) )
| ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,B)
& m2_relset_1(D,A,B) )
=> ( ( v2_funct_1(D)
& k4_card_1(A) = k4_card_1(B) )
=> k11_newton(k5_binarith(k4_card_1(A),k4_card_1(C))) = k1_card_1(a_4_3_card_fin(A,B,C,D)) ) ) ) ) ) ) ) ).
fof(t73_card_fin,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ~ ( k1_relat_1(B) = A
& v2_funct_1(B)
& ! [C] :
( ( v1_finset_1(C)
& m1_ordinal1(C,k4_numbers) )
=> ~ ( k6_card_fin(C) = k1_card_1(a_2_2_card_fin(A,B))
& k2_afinsq_1(C) = k23_binop_2(k4_card_1(A),np__1)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k2_afinsq_1(C))
=> k7_card_fin(C,D) = k12_binop_2(k11_binop_2(k3_newton(k7_binop_2(np__1),D),k11_newton(k4_card_1(A))),k11_newton(D)) ) ) ) ) ) ) ) ).
fof(t74_card_fin,axiom,
! [A] :
( v1_finset_1(A)
=> ? [B] :
( v1_finset_1(B)
& m1_ordinal1(B,k4_numbers)
& k6_card_fin(B) = k1_card_1(a_1_1_card_fin(A))
& k2_afinsq_1(B) = k23_binop_2(k4_card_1(A),np__1)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k2_afinsq_1(B))
=> k7_card_fin(B,C) = k12_binop_2(k11_binop_2(k3_newton(k7_binop_2(np__1),C),k11_newton(k4_card_1(A))),k11_newton(C)) ) ) ) ) ).
fof(fraenkel_a_4_0_card_fin,axiom,
! [A,B,C,D,E] :
( ( v1_finset_1(B)
& v1_finset_1(C) )
=> ( r2_hidden(A,a_4_0_card_fin(B,C,D,E))
<=> ? [F] :
( v1_funct_1(F)
& v1_funct_2(F,k2_xboole_0(B,k1_tarski(D)),k2_xboole_0(C,k1_tarski(E)))
& m2_relset_1(F,k2_xboole_0(B,k1_tarski(D)),k2_xboole_0(C,k1_tarski(E)))
& A = F
& r1_tarski(k5_relset_1(k2_xboole_0(B,k1_tarski(D)),k2_xboole_0(C,k1_tarski(E)),k2_partfun1(k2_xboole_0(B,k1_tarski(D)),k2_xboole_0(C,k1_tarski(E)),F,B)),C)
& k1_funct_1(F,D) = E ) ) ) ).
fof(fraenkel_a_4_1_card_fin,axiom,
! [A,B,C,D,E] :
( ( v1_finset_1(B)
& v1_finset_1(C) )
=> ( r2_hidden(A,a_4_1_card_fin(B,C,D,E))
<=> ? [F] :
( v1_funct_1(F)
& v1_funct_2(F,k2_xboole_0(B,k1_tarski(D)),C)
& m2_relset_1(F,k2_xboole_0(B,k1_tarski(D)),C)
& A = F
& k1_funct_1(F,D) = E ) ) ) ).
fof(fraenkel_a_2_0_card_fin,axiom,
! [A,B,C] :
( ( v1_finset_1(B)
& v1_finset_1(C) )
=> ( r2_hidden(A,a_2_0_card_fin(B,C))
<=> ? [D] :
( v1_funct_1(D)
& v1_funct_2(D,B,C)
& m2_relset_1(D,B,C)
& A = D
& v2_funct_1(D) ) ) ) ).
fof(fraenkel_a_4_2_card_fin,axiom,
! [A,B,C,D,E] :
( ( v1_finset_1(B)
& v1_finset_1(C) )
=> ( r2_hidden(A,a_4_2_card_fin(B,C,D,E))
<=> ? [F] :
( v1_funct_1(F)
& v1_funct_2(F,k2_xboole_0(B,k1_tarski(D)),k2_xboole_0(C,k1_tarski(E)))
& m2_relset_1(F,k2_xboole_0(B,k1_tarski(D)),k2_xboole_0(C,k1_tarski(E)))
& A = F
& v2_funct_1(F)
& k1_funct_1(F,D) = E ) ) ) ).
fof(fraenkel_a_1_0_card_fin,axiom,
! [A,B] :
( v1_finset_1(B)
=> ( r2_hidden(A,a_1_0_card_fin(B))
<=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,B,B)
& m2_relset_1(C,B,B)
& A = C
& v1_funct_1(C)
& v1_funct_2(C,B,B)
& v3_funct_2(C,B,B)
& m2_relset_1(C,B,B) ) ) ) ).
fof(fraenkel_a_5_0_card_fin,axiom,
! [A,B,C,D,E,F] :
( ( v1_finset_1(E)
& m2_subset_1(F,k1_numbers,k5_numbers) )
=> ( r2_hidden(A,a_5_0_card_fin(B,C,D,E,F))
<=> ? [G] :
( v1_funct_1(G)
& v1_funct_2(G,k2_xboole_0(E,k1_tarski(B)),k2_tarski(C,D))
& m2_relset_1(G,k2_xboole_0(E,k1_tarski(B)),k2_tarski(C,D))
& A = G
& k1_card_1(k3_funct_2(k2_xboole_0(E,k1_tarski(B)),k2_tarski(C,D),G,k1_tarski(C))) = k23_binop_2(F,np__1)
& k1_funct_1(G,B) = C ) ) ) ).
fof(fraenkel_a_5_1_card_fin,axiom,
! [A,B,C,D,E,F] :
( ( v1_finset_1(E)
& m2_subset_1(F,k1_numbers,k5_numbers) )
=> ( r2_hidden(A,a_5_1_card_fin(B,C,D,E,F))
<=> ? [G] :
( v1_funct_1(G)
& v1_funct_2(G,k2_xboole_0(E,k1_tarski(B)),k2_tarski(C,D))
& m2_relset_1(G,k2_xboole_0(E,k1_tarski(B)),k2_tarski(C,D))
& A = G
& k1_card_1(k3_funct_2(k2_xboole_0(E,k1_tarski(B)),k2_tarski(C,D),G,k1_tarski(C))) = F
& k1_funct_1(G,B) = D ) ) ) ).
fof(fraenkel_a_2_1_card_fin,axiom,
! [A,B,C] :
( ( v1_finset_1(B)
& v1_finset_1(C) )
=> ( r2_hidden(A,a_2_1_card_fin(B,C))
<=> ? [D] :
( v1_funct_1(D)
& v1_funct_2(D,B,C)
& m2_relset_1(D,B,C)
& A = D
& v2_funct_2(D,B,C) ) ) ) ).
fof(fraenkel_a_4_3_card_fin,axiom,
! [A,B,C,D,E] :
( ( v1_finset_1(B)
& v1_finset_1(C)
& v1_finset_1(D)
& v1_funct_1(E)
& v1_funct_2(E,B,C)
& m2_relset_1(E,B,C) )
=> ( r2_hidden(A,a_4_3_card_fin(B,C,D,E))
<=> ? [F] :
( v1_funct_1(F)
& v1_funct_2(F,B,C)
& m2_relset_1(F,B,C)
& A = F
& v2_funct_1(F)
& r1_tarski(k5_relset_1(B,C,k2_partfun1(B,C,F,k4_xboole_0(B,D))),k2_funct_2(B,C,E,k4_xboole_0(B,D)))
& ! [G] :
( r2_hidden(G,D)
=> k1_funct_1(F,G) = k1_funct_1(E,G) ) ) ) ) ).
fof(fraenkel_a_2_2_card_fin,axiom,
! [A,B,C] :
( ( v1_finset_1(B)
& v1_relat_1(C)
& v1_funct_1(C) )
=> ( r2_hidden(A,a_2_2_card_fin(B,C))
<=> ? [D] :
( v1_funct_1(D)
& v1_funct_2(D,B,k2_relat_1(C))
& m2_relset_1(D,B,k2_relat_1(C))
& A = D
& v2_funct_1(D)
& ! [E] :
~ ( r2_hidden(E,B)
& k1_funct_1(D,E) = k1_funct_1(C,E) ) ) ) ) ).
fof(fraenkel_a_1_1_card_fin,axiom,
! [A,B] :
( v1_finset_1(B)
=> ( r2_hidden(A,a_1_1_card_fin(B))
<=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,B,B)
& m2_relset_1(C,B,B)
& A = C
& v2_funct_1(C)
& ! [D] :
~ ( r2_hidden(D,B)
& k1_funct_1(C,D) = D ) ) ) ) ).
%------------------------------------------------------------------------------