SET007 Axioms: SET007+148.ax
%------------------------------------------------------------------------------
% File : SET007+148 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Countable Sets and Hessenberg's Theorem
% 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_4 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 97 ( 10 unt; 0 def)
% Number of atoms : 414 ( 57 equ)
% Maximal formula atoms : 14 ( 4 avg)
% Number of connectives : 390 ( 73 ~; 16 |; 124 &)
% ( 20 <=>; 157 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 6 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 26 ( 24 usr; 1 prp; 0-3 aty)
% Number of functors : 42 ( 42 usr; 11 con; 0-4 aty)
% Number of variables : 169 ( 159 !; 10 ?)
% 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_card_4,axiom,
( ~ v1_xboole_0(k5_ordinal2)
& v1_ordinal1(k5_ordinal2)
& v2_ordinal1(k5_ordinal2)
& v3_ordinal1(k5_ordinal2)
& ~ v1_finset_1(k5_ordinal2)
& v1_membered(k5_ordinal2)
& v2_membered(k5_ordinal2)
& v3_membered(k5_ordinal2)
& v4_membered(k5_ordinal2)
& v5_membered(k5_ordinal2) ) ).
fof(t1_card_4,axiom,
! [A] :
( v1_finset_1(A)
<=> v1_finset_1(k1_card_1(A)) ) ).
fof(t2_card_4,axiom,
! [A] :
( v1_finset_1(A)
<=> r2_hidden(k1_card_1(A),k3_card_1(np__0)) ) ).
fof(t3_card_4,axiom,
! [A] :
( v1_finset_1(A)
=> ( r2_hidden(k1_card_1(A),k3_card_1(np__0))
& r2_hidden(k1_card_1(A),k5_ordinal2) ) ) ).
fof(t4_card_4,axiom,
! [A] :
( v1_finset_1(A)
<=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& k1_card_1(A) = k1_card_1(B) ) ) ).
fof(t5_card_4,axiom,
! [A] :
( v3_ordinal1(A)
=> k4_xboole_0(k1_ordinal1(A),k1_tarski(A)) = A ) ).
fof(t6_card_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v3_ordinal1(B)
=> ( r2_wellord2(B,A)
=> B = A ) ) ) ).
fof(t7_card_4,axiom,
! [A] :
( v3_ordinal1(A)
=> ( v1_finset_1(A)
<=> r2_hidden(A,k5_ordinal2) ) ) ).
fof(t8_card_4,axiom,
! [A] :
( v3_ordinal1(A)
=> ( ~ v1_finset_1(A)
<=> r1_ordinal1(k5_ordinal2,A) ) ) ).
fof(t9_card_4,axiom,
! [A] :
( v1_card_1(A)
=> ( v1_finset_1(A)
<=> r2_hidden(A,k3_card_1(np__0)) ) ) ).
fof(t10_card_4,axiom,
$true ).
fof(t11_card_4,axiom,
! [A] :
( v1_card_1(A)
=> ( ~ v1_finset_1(A)
<=> r1_ordinal1(k3_card_1(np__0),A) ) ) ).
fof(t12_card_4,axiom,
$true ).
fof(t13_card_4,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( v1_card_1(B)
=> ( v1_finset_1(A)
=> ( v1_finset_1(B)
| ( r2_hidden(A,B)
& r1_tarski(A,B) ) ) ) ) ) ).
fof(t14_card_4,axiom,
! [A] :
( ~ ( ~ v1_finset_1(A)
& ! [B] :
~ ( r1_tarski(B,A)
& k1_card_1(B) = k3_card_1(np__0) ) )
& ~ ( ? [B] :
( r1_tarski(B,A)
& k1_card_1(B) = k3_card_1(np__0) )
& v1_finset_1(A) ) ) ).
fof(t15_card_4,axiom,
~ v1_finset_1(k5_numbers) ).
fof(t16_card_4,axiom,
$true ).
fof(t17_card_4,axiom,
! [A] :
( A = k1_xboole_0
<=> k1_card_1(A) = np__0 ) ).
fof(t18_card_4,axiom,
$true ).
fof(t19_card_4,axiom,
! [A] :
( v1_card_1(A)
=> r1_tarski(np__0,A) ) ).
fof(t20_card_4,axiom,
! [A,B] :
( k1_card_1(A) = k1_card_1(B)
<=> k2_card_1(A) = k2_card_1(B) ) ).
fof(t21_card_4,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( v1_card_1(B)
=> ( A = B
<=> k2_card_1(B) = k2_card_1(A) ) ) ) ).
fof(t22_card_4,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( v1_card_1(B)
=> ( r2_hidden(A,B)
<=> r1_tarski(k2_card_1(A),B) ) ) ) ).
fof(t23_card_4,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( v1_card_1(B)
=> ( r2_hidden(A,k2_card_1(B))
<=> r1_tarski(A,B) ) ) ) ).
fof(t24_card_4,axiom,
! [A] :
( v1_card_1(A)
=> ( r2_hidden(np__0,A)
<=> r1_tarski(np__1,A) ) ) ).
fof(t25_card_4,axiom,
! [A] :
( v1_card_1(A)
=> ( r2_hidden(np__1,A)
<=> r1_tarski(np__2,A) ) ) ).
fof(t26_card_4,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( v1_card_1(B)
=> ( v1_finset_1(A)
=> ( ( ~ r1_tarski(B,A)
& ~ r2_hidden(B,A) )
| v1_finset_1(B) ) ) ) ) ).
fof(t27_card_4,axiom,
! [A] :
( v3_ordinal1(A)
=> ( v4_ordinal1(A)
<=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(B,A)
=> r2_hidden(k14_ordinal2(B,C),A) ) ) ) ) ) ).
fof(t28_card_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( v3_ordinal1(B)
=> ( k14_ordinal2(B,k1_ordinal1(A)) = k14_ordinal2(k1_ordinal1(B),A)
& k14_ordinal2(B,k1_nat_1(A,np__1)) = k14_ordinal2(k1_ordinal1(B),A) ) ) ) ).
fof(t29_card_4,axiom,
! [A] :
( v3_ordinal1(A)
=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& k15_ordinal2(A,k1_ordinal1(k4_ordinal2)) = k14_ordinal2(A,B) ) ) ).
fof(t30_card_4,axiom,
! [A] :
( v3_ordinal1(A)
=> ( v4_ordinal1(A)
=> k15_ordinal2(A,k1_ordinal1(k4_ordinal2)) = A ) ) ).
fof(t31_card_4,axiom,
! [A] :
( v3_ordinal1(A)
=> ( r1_ordinal1(k5_ordinal2,A)
=> k14_ordinal2(k4_ordinal2,A) = A ) ) ).
fof(t32_card_4,axiom,
! [A] :
( v1_card_1(A)
=> ( ~ v1_finset_1(A)
=> v4_ordinal1(A) ) ) ).
fof(t33_card_4,axiom,
! [A] :
( v1_card_1(A)
=> ( ~ v1_finset_1(A)
=> k1_card_2(A,A) = A ) ) ).
fof(t34_card_4,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( v1_card_1(B)
=> ~ ( ~ v1_finset_1(A)
& ( r1_tarski(B,A)
| r2_hidden(B,A) )
& ~ ( k1_card_2(A,B) = A
& k1_card_2(B,A) = A ) ) ) ) ).
fof(t35_card_4,axiom,
! [A,B] :
~ ( ~ v1_finset_1(A)
& ( r2_wellord2(A,B)
| r2_wellord2(B,A) )
& ~ ( r2_wellord2(k2_xboole_0(A,B),A)
& k1_card_1(k2_xboole_0(A,B)) = k1_card_1(A) ) ) ).
fof(t36_card_4,axiom,
! [A,B] :
( v1_finset_1(B)
=> ( v1_finset_1(A)
| ( r2_wellord2(k2_xboole_0(A,B),A)
& k1_card_1(k2_xboole_0(A,B)) = k1_card_1(A) ) ) ) ).
fof(t37_card_4,axiom,
! [A,B] :
~ ( ~ v1_finset_1(A)
& ( r2_hidden(k1_card_1(B),k1_card_1(A))
| r1_tarski(k1_card_1(B),k1_card_1(A)) )
& ~ ( r2_wellord2(k2_xboole_0(A,B),A)
& k1_card_1(k2_xboole_0(A,B)) = k1_card_1(A) ) ) ).
fof(t38_card_4,axiom,
! [A] :
( ( v1_finset_1(A)
& v1_card_1(A) )
=> ! [B] :
( ( v1_finset_1(B)
& v1_card_1(B) )
=> v1_finset_1(k1_card_2(A,B)) ) ) ).
fof(t39_card_4,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( v1_card_1(B)
=> ( ~ v1_finset_1(A)
=> ( ~ v1_finset_1(k1_card_2(A,B))
& ~ v1_finset_1(k1_card_2(B,A)) ) ) ) ) ).
fof(t40_card_4,axiom,
! [A] :
( ( v1_finset_1(A)
& v1_card_1(A) )
=> ! [B] :
( ( v1_finset_1(B)
& v1_card_1(B) )
=> v1_finset_1(k2_card_2(A,B)) ) ) ).
fof(t41_card_4,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( v1_card_1(B)
=> ! [C] :
( v1_card_1(C)
=> ! [D] :
( v1_card_1(D)
=> ( ~ ( ~ ( r2_hidden(A,B)
& r2_hidden(C,D) )
& ~ ( r1_tarski(A,B)
& r2_hidden(C,D) )
& ~ ( r2_hidden(A,B)
& r1_tarski(C,D) )
& ~ ( r1_tarski(A,B)
& r1_tarski(C,D) ) )
=> ( r1_tarski(k1_card_2(A,C),k1_card_2(B,D))
& r1_tarski(k1_card_2(C,A),k1_card_2(B,D)) ) ) ) ) ) ) ).
fof(t42_card_4,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( v1_card_1(B)
=> ! [C] :
( v1_card_1(C)
=> ( ( r2_hidden(A,B)
| r1_tarski(A,B) )
=> ( r1_tarski(k1_card_2(C,A),k1_card_2(C,B))
& r1_tarski(k1_card_2(C,A),k1_card_2(B,C))
& r1_tarski(k1_card_2(A,C),k1_card_2(C,B))
& r1_tarski(k1_card_2(A,C),k1_card_2(B,C)) ) ) ) ) ) ).
fof(d1_card_4,axiom,
! [A] :
( v1_card_4(A)
<=> r1_tarski(k1_card_1(A),k3_card_1(np__0)) ) ).
fof(t43_card_4,axiom,
! [A] :
( v1_finset_1(A)
=> v1_card_4(A) ) ).
fof(t44_card_4,axiom,
( v1_card_4(k5_ordinal2)
& v1_card_4(k5_numbers) ) ).
fof(t45_card_4,axiom,
! [A] :
( v1_card_4(A)
<=> ? [B] :
( v1_relat_1(B)
& v1_funct_1(B)
& k1_relat_1(B) = k5_numbers
& r1_tarski(A,k2_relat_1(B)) ) ) ).
fof(t46_card_4,axiom,
! [A,B] :
( ( r1_tarski(A,B)
& v1_card_4(B) )
=> v1_card_4(A) ) ).
fof(t47_card_4,axiom,
! [A,B] :
( ( v1_card_4(A)
& v1_card_4(B) )
=> v1_card_4(k2_xboole_0(A,B)) ) ).
fof(t48_card_4,axiom,
! [A,B] :
( v1_card_4(A)
=> ( v1_card_4(k3_xboole_0(A,B))
& v1_card_4(k3_xboole_0(B,A)) ) ) ).
fof(t49_card_4,axiom,
! [A,B] :
( v1_card_4(A)
=> v1_card_4(k4_xboole_0(A,B)) ) ).
fof(t50_card_4,axiom,
! [A,B] :
( ( v1_card_4(A)
& v1_card_4(B) )
=> v1_card_4(k5_xboole_0(A,B)) ) ).
fof(t51_card_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( ~ ( ~ ( B = np__0
& A != np__0 )
& k3_newton(B,A) = np__0 )
& ~ ( k3_newton(B,A) != np__0
& B = np__0
& A != np__0 ) ) ) ) ).
fof(t52_card_4,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)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( k2_nat_1(k1_card_4(np__2,A),k1_nat_1(k2_nat_1(np__2,B),np__1)) = k2_nat_1(k1_card_4(np__2,C),k1_nat_1(k2_nat_1(np__2,D),np__1))
=> ( A = C
& B = D ) ) ) ) ) ) ).
fof(t53_card_4,axiom,
( r2_wellord2(k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k5_numbers)
& k1_card_1(k5_numbers) = k1_card_1(k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers)) ) ).
fof(t54_card_4,axiom,
k2_card_2(k3_card_1(np__0),k3_card_1(np__0)) = k3_card_1(np__0) ).
fof(t55_card_4,axiom,
! [A,B] :
( ( v1_card_4(A)
& v1_card_4(B) )
=> v1_card_4(k2_zfmisc_1(A,B)) ) ).
fof(t56_card_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( r2_wellord2(k4_finseq_2(np__1,A),A)
& k1_card_1(k4_finseq_2(np__1,A)) = k1_card_1(A) ) ) ).
fof(t57_card_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_wellord2(k2_zfmisc_1(k4_finseq_2(B,A),k4_finseq_2(C,A)),k4_finseq_2(k1_nat_1(B,C),A))
& k1_card_1(k2_zfmisc_1(k4_finseq_2(B,A),k4_finseq_2(C,A))) = k1_card_1(k4_finseq_2(k1_nat_1(B,C),A)) ) ) ) ) ).
fof(t58_card_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( v1_card_4(A)
=> v1_card_4(k4_finseq_2(B,A)) ) ) ) ).
fof(t59_card_4,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( v1_card_1(B)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( r1_tarski(k1_card_1(k1_relat_1(C)),A)
& ! [D] :
( r2_hidden(D,k1_relat_1(C))
=> r1_tarski(k1_card_1(k1_funct_1(C,D)),B) ) )
=> r1_tarski(k1_card_1(k3_card_3(C)),k2_card_2(A,B)) ) ) ) ) ).
fof(t60_card_4,axiom,
! [A,B] :
( v1_card_1(B)
=> ! [C] :
( v1_card_1(C)
=> ( ( r1_tarski(k1_card_1(A),B)
& ! [D] :
( r2_hidden(D,A)
=> r1_tarski(k1_card_1(D),C) ) )
=> r1_tarski(k1_card_1(k3_tarski(A)),k2_card_2(B,C)) ) ) ) ).
fof(t61_card_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( ( v1_card_4(k1_relat_1(A))
& ! [B] :
( r2_hidden(B,k1_relat_1(A))
=> v1_card_4(k1_funct_1(A,B)) ) )
=> v1_card_4(k3_card_3(A)) ) ) ).
fof(t62_card_4,axiom,
! [A] :
( ( v1_card_4(A)
& ! [B] :
( r2_hidden(B,A)
=> v1_card_4(B) ) )
=> v1_card_4(k3_tarski(A)) ) ).
fof(t63_card_4,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( ( v1_finset_1(k1_relat_1(A))
& ! [B] :
( r2_hidden(B,k1_relat_1(A))
=> v1_finset_1(k1_funct_1(A,B)) ) )
=> v1_finset_1(k3_card_3(A)) ) ) ).
fof(t64_card_4,axiom,
$true ).
fof(t65_card_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_card_4(A)
=> v1_card_4(k3_finseq_2(A)) ) ) ).
fof(t66_card_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> r1_tarski(k3_card_1(np__0),k1_card_1(k3_finseq_2(A))) ) ).
fof(t67_card_4,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_tarski(k2_card_2(k3_card_1(np__0),k1_card_1(A)),k3_card_1(np__0))
& r1_tarski(k2_card_2(k1_card_1(A),k3_card_1(np__0)),k3_card_1(np__0)) ) ) ).
fof(t68_card_4,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( v1_card_1(B)
=> ! [C] :
( v1_card_1(C)
=> ! [D] :
( v1_card_1(D)
=> ( ~ ( ~ ( r2_hidden(A,B)
& r2_hidden(C,D) )
& ~ ( r1_tarski(A,B)
& r2_hidden(C,D) )
& ~ ( r2_hidden(A,B)
& r1_tarski(C,D) )
& ~ ( r1_tarski(A,B)
& r1_tarski(C,D) ) )
=> ( r1_tarski(k2_card_2(A,C),k2_card_2(B,D))
& r1_tarski(k2_card_2(C,A),k2_card_2(B,D)) ) ) ) ) ) ) ).
fof(t69_card_4,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( v1_card_1(B)
=> ! [C] :
( v1_card_1(C)
=> ( ( r2_hidden(A,B)
| r1_tarski(A,B) )
=> ( r1_tarski(k2_card_2(C,A),k2_card_2(C,B))
& r1_tarski(k2_card_2(C,A),k2_card_2(B,C))
& r1_tarski(k2_card_2(A,C),k2_card_2(C,B))
& r1_tarski(k2_card_2(A,C),k2_card_2(B,C)) ) ) ) ) ) ).
fof(t70_card_4,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( v1_card_1(B)
=> ! [C] :
( v1_card_1(C)
=> ! [D] :
( v1_card_1(D)
=> ~ ( ~ ( ~ ( r2_hidden(A,B)
& r2_hidden(C,D) )
& ~ ( r1_tarski(A,B)
& r2_hidden(C,D) )
& ~ ( r2_hidden(A,B)
& r1_tarski(C,D) )
& ~ ( r1_tarski(A,B)
& r1_tarski(C,D) ) )
& A != np__0
& ~ r1_tarski(k3_card_2(A,C),k3_card_2(B,D)) ) ) ) ) ) ).
fof(t71_card_4,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( v1_card_1(B)
=> ! [C] :
( v1_card_1(C)
=> ~ ( ( r2_hidden(A,B)
| r1_tarski(A,B) )
& C != np__0
& ~ ( r1_tarski(k3_card_2(C,A),k3_card_2(C,B))
& r1_tarski(k3_card_2(A,C),k3_card_2(B,C)) ) ) ) ) ) ).
fof(t72_card_4,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( v1_card_1(B)
=> ( r1_tarski(A,k1_card_2(A,B))
& r1_tarski(B,k1_card_2(A,B)) ) ) ) ).
fof(t73_card_4,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( v1_card_1(B)
=> ( A != np__0
=> ( r1_tarski(B,k2_card_2(B,A))
& r1_tarski(B,k2_card_2(A,B)) ) ) ) ) ).
fof(t74_card_4,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( v1_card_1(B)
=> ! [C] :
( v1_card_1(C)
=> ! [D] :
( v1_card_1(D)
=> ( ( r2_hidden(A,B)
& r2_hidden(C,D) )
=> ( r2_hidden(k1_card_2(A,C),k1_card_2(B,D))
& r2_hidden(k1_card_2(C,A),k1_card_2(B,D)) ) ) ) ) ) ) ).
fof(t75_card_4,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( v1_card_1(B)
=> ! [C] :
( v1_card_1(C)
=> ( r2_hidden(k1_card_2(A,B),k1_card_2(A,C))
=> r2_hidden(B,C) ) ) ) ) ).
fof(t76_card_4,axiom,
! [A,B] :
( ( k1_card_2(k1_card_1(A),k1_card_1(B)) = k1_card_1(A)
& r2_hidden(k1_card_1(B),k1_card_1(A)) )
=> k1_card_1(k4_xboole_0(A,B)) = k1_card_1(A) ) ).
fof(t77_card_4,axiom,
! [A] :
( v1_card_1(A)
=> ( ~ v1_finset_1(A)
=> k2_card_2(A,A) = A ) ) ).
fof(t78_card_4,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( v1_card_1(B)
=> ( r2_hidden(np__0,B)
=> ( v1_finset_1(A)
| ( ~ r1_tarski(B,A)
& ~ r2_hidden(B,A) )
| ( k2_card_2(A,B) = A
& k2_card_2(B,A) = A ) ) ) ) ) ).
fof(t79_card_4,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( v1_card_1(B)
=> ~ ( ~ v1_finset_1(A)
& ( r1_tarski(B,A)
| r2_hidden(B,A) )
& ~ ( r1_tarski(k2_card_2(A,B),A)
& r1_tarski(k2_card_2(B,A),A) ) ) ) ) ).
fof(t80_card_4,axiom,
! [A] :
( ~ v1_finset_1(A)
=> ( r2_wellord2(k2_zfmisc_1(A,A),A)
& k1_card_1(k2_zfmisc_1(A,A)) = k1_card_1(A) ) ) ).
fof(t81_card_4,axiom,
! [A,B] :
( v1_finset_1(B)
=> ( v1_finset_1(A)
| B = k1_xboole_0
| ( r2_wellord2(k2_zfmisc_1(A,B),A)
& k1_card_1(k2_zfmisc_1(A,B)) = k1_card_1(A) ) ) ) ).
fof(t82_card_4,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( v1_card_1(B)
=> ! [C] :
( v1_card_1(C)
=> ! [D] :
( v1_card_1(D)
=> ( ( r2_hidden(A,B)
& r2_hidden(C,D) )
=> ( r2_hidden(k2_card_2(A,C),k2_card_2(B,D))
& r2_hidden(k2_card_2(C,A),k2_card_2(B,D)) ) ) ) ) ) ) ).
fof(t83_card_4,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( v1_card_1(B)
=> ! [C] :
( v1_card_1(C)
=> ( r2_hidden(k2_card_2(A,B),k2_card_2(A,C))
=> r2_hidden(B,C) ) ) ) ) ).
fof(t84_card_4,axiom,
! [A] :
( ~ v1_finset_1(A)
=> k1_card_1(A) = k2_card_2(k3_card_1(np__0),k1_card_1(A)) ) ).
fof(t85_card_4,axiom,
! [A,B] :
( v1_finset_1(A)
=> ( A = k1_xboole_0
| v1_finset_1(B)
| k2_card_2(k1_card_1(B),k1_card_1(A)) = k1_card_1(B) ) ) ).
fof(t86_card_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( ~ v1_finset_1(A)
& B != np__0
& ~ ( r2_wellord2(k4_finseq_2(B,A),A)
& k1_card_1(k4_finseq_2(B,A)) = k1_card_1(A) ) ) ) ) ).
fof(t87_card_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ~ v1_finset_1(A)
=> k1_card_1(A) = k1_card_1(k3_finseq_2(A)) ) ) ).
fof(dt_k1_card_4,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> m2_subset_1(k1_card_4(A,B),k1_numbers,k5_numbers) ) ).
fof(redefinition_k1_card_4,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> k1_card_4(A,B) = k2_newton(A,B) ) ).
fof(s1_card_4,axiom,
v1_card_4(a_0_0_card_4) ).
fof(s2_card_4,axiom,
v1_card_4(a_0_1_card_4) ).
fof(s3_card_4,axiom,
v1_card_4(a_0_2_card_4) ).
fof(fraenkel_a_0_0_card_4,axiom,
! [A] :
( r2_hidden(A,a_0_0_card_4)
<=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& A = f1_s1_card_4(B)
& p1_s1_card_4(B) ) ) ).
fof(fraenkel_a_0_1_card_4,axiom,
! [A] :
( r2_hidden(A,a_0_1_card_4)
<=> ? [B,C] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& m2_subset_1(C,k1_numbers,k5_numbers)
& A = f1_s2_card_4(B,C)
& p1_s2_card_4(B,C) ) ) ).
fof(fraenkel_a_0_2_card_4,axiom,
! [A] :
( r2_hidden(A,a_0_2_card_4)
<=> ? [B,C,D] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& m2_subset_1(C,k1_numbers,k5_numbers)
& m2_subset_1(D,k1_numbers,k5_numbers)
& A = f1_s3_card_4(B,C,D)
& p1_s3_card_4(B,C,D) ) ) ).
%------------------------------------------------------------------------------