SET007 Axioms: SET007+171.ax
%------------------------------------------------------------------------------
% File : SET007+171 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Mahlo and Inaccessible Cardinals
% 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_lar [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 74 ( 1 unt; 0 def)
% Number of atoms : 501 ( 19 equ)
% Maximal formula atoms : 14 ( 6 avg)
% Number of connectives : 556 ( 129 ~; 1 |; 248 &)
% ( 18 <=>; 160 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 8 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 35 ( 33 usr; 1 prp; 0-3 aty)
% Number of functors : 27 ( 27 usr; 2 con; 0-3 aty)
% Number of variables : 180 ( 170 !; 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(cc1_card_lar,axiom,
! [A] :
( ( v3_ordinal1(A)
& ~ v1_finset_1(A)
& v1_card_1(A) )
=> ( v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A)
& v4_ordinal1(A) ) ) ).
fof(cc2_card_lar,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v3_ordinal1(A)
& v4_ordinal1(A) )
=> ( ~ v1_xboole_0(A)
& v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A)
& ~ v1_finset_1(A) ) ) ).
fof(cc3_card_lar,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A)
& ~ v2_card_1(A) )
=> ( ~ v1_xboole_0(A)
& v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A)
& v4_ordinal1(A)
& ~ v1_finset_1(A)
& v1_card_1(A)
& ~ v1_card_4(A) ) ) ).
fof(rc1_card_lar,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A)
& v4_ordinal1(A)
& ~ v1_finset_1(A)
& v1_card_1(A)
& v1_card_5(A)
& ~ v1_card_4(A) ) ).
fof(rc2_card_lar,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A)
& ~ v1_card_4(A) )
=> ? [B] :
( m1_subset_1(B,A)
& ~ v1_xboole_0(B)
& v1_ordinal1(B)
& v2_ordinal1(B)
& v3_ordinal1(B)
& v4_ordinal1(B)
& ~ v1_finset_1(B)
& v1_card_1(B) ) ) ).
fof(fc1_card_lar,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v3_ordinal1(A) )
=> ( ~ v1_xboole_0(k4_classes1(A))
& v1_ordinal1(k4_classes1(A)) ) ) ).
fof(d1_card_lar,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal1(A)
& ~ v1_finset_1(A) )
=> ! [B] :
( r1_card_lar(A,B)
<=> ( r1_tarski(B,A)
& k7_ordinal2(B) = A ) ) ) ).
fof(d2_card_lar,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal1(A)
& ~ v1_finset_1(A) )
=> ! [B] :
( r2_card_lar(A,B)
<=> ( r1_tarski(B,A)
& ! [C] :
( ( v3_ordinal1(C)
& v4_ordinal1(C)
& ~ v1_finset_1(C) )
=> ( ( r2_hidden(C,A)
& k7_ordinal2(k3_xboole_0(B,C)) = C )
=> r2_hidden(C,B) ) ) ) ) ) ).
fof(d3_card_lar,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal1(A)
& ~ v1_finset_1(A) )
=> ! [B] :
( r3_card_lar(A,B)
<=> ( r2_card_lar(A,B)
& r1_card_lar(A,B) ) ) ) ).
fof(d4_card_lar,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal1(A)
& ~ v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( v1_card_lar(B,A)
<=> k7_ordinal2(B) = A ) ) ) ).
fof(d5_card_lar,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal1(A)
& ~ v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( v2_card_lar(B,A)
<=> ! [C] :
( ( v3_ordinal1(C)
& v4_ordinal1(C)
& ~ v1_finset_1(C) )
=> ( ( r2_hidden(C,A)
& k7_ordinal2(k3_xboole_0(B,C)) = C )
=> r2_hidden(C,B) ) ) ) ) ) ).
fof(t1_card_lar,axiom,
$true ).
fof(t2_card_lar,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal1(A)
& ~ v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( r3_card_lar(A,B)
<=> ( v2_card_lar(B,A)
& v1_card_lar(B,A) ) ) ) ) ).
fof(t3_card_lar,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal1(A)
& ~ v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> r1_tarski(B,k7_ordinal2(B)) ) ) ).
fof(t4_card_lar,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal1(A)
& ~ v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( ! [C] :
( v3_ordinal1(C)
=> ~ ( r2_hidden(C,B)
& ! [D] :
( v3_ordinal1(D)
=> ~ ( r2_hidden(D,B)
& r2_hidden(C,D) ) ) ) )
=> ( v1_xboole_0(B)
| ( v3_ordinal1(k7_ordinal2(B))
& v4_ordinal1(k7_ordinal2(B))
& ~ v1_finset_1(k7_ordinal2(B)) ) ) ) ) ) ).
fof(t5_card_lar,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal1(A)
& ~ v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( ~ ( ~ v1_card_lar(B,A)
& ! [C] :
( v3_ordinal1(C)
=> ~ ( r2_hidden(C,A)
& r1_tarski(B,C) ) ) )
& ~ ( ? [C] :
( v3_ordinal1(C)
& r2_hidden(C,A)
& r1_tarski(B,C) )
& v1_card_lar(B,A) ) ) ) ) ).
fof(t6_card_lar,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal1(A)
& ~ v1_finset_1(A) )
=> ! [B] :
( ( v3_ordinal1(B)
& v4_ordinal1(B)
& ~ v1_finset_1(B) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ~ ( k7_ordinal2(k3_xboole_0(C,B)) != B
& ! [D] :
( v3_ordinal1(D)
=> ~ ( r2_hidden(D,B)
& r1_tarski(k3_xboole_0(C,B),D) ) ) ) ) ) ) ).
fof(t7_card_lar,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal1(A)
& ~ v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( v1_card_lar(B,A)
<=> ! [C] :
( v3_ordinal1(C)
=> ~ ( r2_hidden(C,A)
& ! [D] :
( v3_ordinal1(D)
=> ~ ( r2_hidden(D,B)
& r1_ordinal1(C,D) ) ) ) ) ) ) ) ).
fof(t8_card_lar,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal1(A)
& ~ v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ~ ( v1_card_lar(B,A)
& v1_xboole_0(B) ) ) ) ).
fof(t10_card_lar,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal1(A)
& ~ v1_finset_1(A) )
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( ( v1_card_lar(C,A)
& r2_hidden(B,A) )
=> ( r2_hidden(k2_card_lar(A,C,B),C)
& r2_hidden(B,k2_card_lar(A,C,B)) ) ) ) ) ) ).
fof(t11_card_lar,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal1(A)
& ~ v1_finset_1(A) )
=> ( v2_card_lar(k2_subset_1(A),A)
& v1_card_lar(k2_subset_1(A),A) ) ) ).
fof(t12_card_lar,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal1(A)
& ~ v1_finset_1(A) )
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( ( r2_hidden(B,A)
& v2_card_lar(C,A)
& v1_card_lar(C,A) )
=> ( v2_card_lar(k3_card_lar(A,C,B),A)
& v1_card_lar(k3_card_lar(A,C,B),A) ) ) ) ) ) ).
fof(t13_card_lar,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal1(A)
& ~ v1_finset_1(A) )
=> ! [B] :
( v3_ordinal1(B)
=> ( r2_hidden(B,A)
=> ( v2_card_lar(k1_card_fil(A,B),A)
& v1_card_lar(k1_card_fil(A,B),A) ) ) ) ) ).
fof(d7_card_lar,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal1(A)
& ~ v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( v3_card_lar(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ~ ( v2_card_lar(C,A)
& v1_card_lar(C,A)
& v1_xboole_0(k1_card_lar(A,B,C)) ) ) ) ) ) ).
fof(t14_card_lar,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal1(A)
& ~ v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( ( v3_card_lar(B,A)
& r1_tarski(B,C) )
=> v3_card_lar(C,A) ) ) ) ) ).
fof(d8_card_lar,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal1(A)
& ~ v1_finset_1(A) )
=> ! [B] :
( r4_card_lar(A,B)
<=> ( r1_tarski(B,A)
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ~ ( v2_card_lar(C,A)
& v1_card_lar(C,A)
& v1_xboole_0(k1_card_lar(A,B,C)) ) ) ) ) ) ).
fof(t15_card_lar,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal1(A)
& ~ v1_finset_1(A) )
=> ! [B,C] :
( ( r4_card_lar(A,B)
& r1_tarski(B,C)
& r1_tarski(C,A) )
=> r4_card_lar(A,C) ) ) ).
fof(t16_card_lar,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal1(A)
& ~ v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( v3_card_lar(B,A)
=> v1_card_lar(B,A) ) ) ) ).
fof(t17_card_lar,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal1(A)
& ~ v1_finset_1(A) )
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( r1_tarski(k3_xboole_0(D,B),C)
=> r1_tarski(k1_card_lar(A,B,k4_card_lar(A,D)),k1_ordinal1(C)) ) ) ) ) ) ).
fof(t18_card_lar,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal1(A)
& ~ v1_finset_1(A) )
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r1_tarski(C,B)
=> r1_tarski(k4_card_lar(A,C),k1_ordinal1(B)) ) ) ) ) ).
fof(t19_card_lar,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal1(A)
& ~ v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> v2_card_lar(k4_card_lar(A,B),A) ) ) ).
fof(t21_card_lar,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( v1_card_lar(B,A)
=> r1_tarski(k1_card_5(A),k1_card_1(B)) ) ) ) ).
fof(t22_card_lar,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A)
& ~ v1_card_4(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( ! [C] :
( m1_card_lar(C,A,B)
=> v2_card_lar(C,A) )
=> v2_card_lar(k6_setfam_1(A,B),A) ) ) ) ).
fof(t23_card_lar,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A)
& ~ v1_card_4(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( r2_hidden(k3_card_1(np__0),k1_card_5(A))
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,B)
& m2_relset_1(C,k5_numbers,B) )
=> r2_hidden(k7_ordinal2(k2_relat_1(C)),A) ) ) ) ) ).
fof(t24_card_lar,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A)
& ~ v1_card_4(A) )
=> ( r2_hidden(k3_card_1(np__0),k1_card_5(A))
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ( ( r2_hidden(k1_card_1(B),k1_card_5(A))
& ! [C] :
( m1_card_lar(C,A,B)
=> ( v2_card_lar(C,A)
& v1_card_lar(C,A) ) ) )
=> ( v2_card_lar(k6_setfam_1(A,B),A)
& v1_card_lar(k6_setfam_1(A,B),A) ) ) ) ) ) ).
fof(t25_card_lar,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A)
& ~ v1_card_4(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( ( r2_hidden(k3_card_1(np__0),k1_card_5(A))
& v1_card_lar(B,A) )
=> ! [C] :
( v3_ordinal1(C)
=> ~ ( r2_hidden(C,A)
& ! [D] :
( ( v3_ordinal1(D)
& v4_ordinal1(D)
& ~ v1_finset_1(D) )
=> ~ ( r2_hidden(D,A)
& r2_hidden(C,D)
& r2_hidden(D,k4_card_lar(A,B)) ) ) ) ) ) ) ) ).
fof(t26_card_lar,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A)
& ~ v1_card_4(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( ( r2_hidden(k3_card_1(np__0),k1_card_5(A))
& v1_card_lar(B,A) )
=> v1_card_lar(k4_card_lar(A,B),A) ) ) ) ).
fof(t27_card_lar,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A)
& ~ v1_card_4(A) )
=> ( v5_card_lar(A)
=> v4_card_lar(A) ) ) ).
fof(t28_card_lar,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A)
& ~ v1_card_4(A) )
=> ( v4_card_lar(A)
=> v1_card_5(A) ) ) ).
fof(t29_card_lar,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A)
& ~ v1_card_4(A) )
=> ( v4_card_lar(A)
=> v2_card_1(A) ) ) ).
fof(t30_card_lar,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A)
& ~ v1_card_4(A) )
=> ( v4_card_lar(A)
=> v4_card_fil(A) ) ) ).
fof(t31_card_lar,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A)
& ~ v1_card_4(A) )
=> ( v5_card_lar(A)
=> v5_card_fil(A) ) ) ).
fof(t32_card_lar,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A)
& ~ v1_card_4(A) )
=> ( v5_card_lar(A)
=> v6_card_fil(A) ) ) ).
fof(t33_card_lar,axiom,
! [A] :
( ! [B] :
~ ( r2_hidden(B,A)
& ! [C] :
~ ( r2_hidden(C,A)
& r1_tarski(B,C)
& v1_card_1(C) ) )
=> v1_card_1(k3_tarski(A)) ) ).
fof(t34_card_lar,axiom,
! [A,B] :
( ( ~ v1_finset_1(B)
& v1_card_1(B) )
=> ( ( r2_hidden(k1_card_1(A),k1_card_5(B))
& ! [C] :
( r2_hidden(C,A)
=> r2_hidden(k1_card_1(C),B) ) )
=> r2_hidden(k1_card_1(k3_tarski(A)),B) ) ) ).
fof(t35_card_lar,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A)
& ~ v1_card_4(A) )
=> ! [B] :
( v3_ordinal1(B)
=> ( ( v6_card_fil(A)
& r2_hidden(B,A) )
=> r2_hidden(k1_card_1(k4_classes1(B)),A) ) ) ) ).
fof(t36_card_lar,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A)
& ~ v1_card_4(A) )
=> ( v6_card_fil(A)
=> k1_card_1(k4_classes1(A)) = A ) ) ).
fof(t37_card_lar,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A)
& ~ v1_card_4(A) )
=> ( v6_card_fil(A)
=> v2_classes1(k4_classes1(A)) ) ) ).
fof(t38_card_lar,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v3_ordinal1(A) )
=> ~ v1_xboole_0(k4_classes1(A)) ) ).
fof(t39_card_lar,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A)
& ~ v1_card_4(A) )
=> ( v6_card_fil(A)
=> ( ~ v1_xboole_0(k4_classes1(A))
& v1_classes2(k4_classes1(A)) ) ) ) ).
fof(t40_card_lar,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A)
& ~ v1_card_4(A) )
=> ( v6_card_fil(A)
=> v1_zf_model(k4_classes1(A)) ) ) ).
fof(dt_m1_card_lar,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ! [C] :
( m1_card_lar(C,A,B)
=> m1_subset_1(C,k1_zfmisc_1(A)) ) ) ).
fof(existence_m1_card_lar,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ? [C] : m1_card_lar(C,A,B) ) ).
fof(redefinition_m1_card_lar,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ! [C] :
( m1_card_lar(C,A,B)
<=> m1_subset_1(C,B) ) ) ).
fof(dt_k1_card_lar,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> m1_subset_1(k1_card_lar(A,B,C),k1_zfmisc_1(A)) ) ).
fof(commutativity_k1_card_lar,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> k1_card_lar(A,B,C) = k1_card_lar(A,C,B) ) ).
fof(idempotence_k1_card_lar,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> k1_card_lar(A,B,B) = B ) ).
fof(redefinition_k1_card_lar,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> k1_card_lar(A,B,C) = k3_xboole_0(B,C) ) ).
fof(dt_k2_card_lar,axiom,
! [A,B,C] :
( ( v3_ordinal1(A)
& v4_ordinal1(A)
& ~ v1_finset_1(A)
& m1_subset_1(B,k1_zfmisc_1(A))
& v3_ordinal1(C) )
=> m1_subset_1(k2_card_lar(A,B,C),B) ) ).
fof(dt_k3_card_lar,axiom,
! [A,B,C] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> m1_subset_1(k3_card_lar(A,B,C),k1_zfmisc_1(A)) ) ).
fof(redefinition_k3_card_lar,axiom,
! [A,B,C] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> k3_card_lar(A,B,C) = k4_xboole_0(B,C) ) ).
fof(dt_k4_card_lar,axiom,
! [A,B] :
( ( v3_ordinal1(A)
& v4_ordinal1(A)
& ~ v1_finset_1(A)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> m1_subset_1(k4_card_lar(A,B),k1_zfmisc_1(A)) ) ).
fof(t9_card_lar,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal1(A)
& ~ v1_finset_1(A) )
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ~ ( v1_card_lar(C,A)
& r2_hidden(B,A)
& ! [D] :
( m1_subset_1(D,A)
=> ~ r2_hidden(D,a_3_0_card_lar(A,B,C)) ) ) ) ) ) ).
fof(d6_card_lar,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal1(A)
& ~ v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( v3_ordinal1(C)
=> ( ( v1_card_lar(B,A)
& r2_hidden(C,A) )
=> k2_card_lar(A,B,C) = k6_ordinal2(a_3_1_card_lar(A,B,C)) ) ) ) ) ).
fof(d9_card_lar,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal1(A)
& ~ v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> k4_card_lar(A,B) = a_2_0_card_lar(A,B) ) ) ).
fof(t20_card_lar,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal1(A)
& ~ v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ~ ( v1_card_lar(B,A)
& ~ v1_card_lar(k4_card_lar(A,B),A)
& ! [C] :
( v3_ordinal1(C)
=> ~ ( r2_hidden(C,A)
& r3_card_lar(A,a_3_2_card_lar(A,B,C)) ) ) ) ) ) ).
fof(d10_card_lar,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A)
& ~ v1_card_4(A) )
=> ( v4_card_lar(A)
<=> r4_card_lar(A,a_1_0_card_lar(A)) ) ) ).
fof(d11_card_lar,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A)
& ~ v1_card_4(A) )
=> ( v5_card_lar(A)
<=> r4_card_lar(A,a_1_1_card_lar(A)) ) ) ).
fof(fraenkel_a_3_0_card_lar,axiom,
! [A,B,C,D] :
( ( v3_ordinal1(B)
& v4_ordinal1(B)
& ~ v1_finset_1(B)
& v3_ordinal1(C)
& m1_subset_1(D,k1_zfmisc_1(B)) )
=> ( r2_hidden(A,a_3_0_card_lar(B,C,D))
<=> ? [E] :
( m1_subset_1(E,B)
& A = E
& r2_hidden(E,D)
& r2_hidden(C,E) ) ) ) ).
fof(fraenkel_a_3_1_card_lar,axiom,
! [A,B,C,D] :
( ( v3_ordinal1(B)
& v4_ordinal1(B)
& ~ v1_finset_1(B)
& m1_subset_1(C,k1_zfmisc_1(B))
& v3_ordinal1(D) )
=> ( r2_hidden(A,a_3_1_card_lar(B,C,D))
<=> ? [E] :
( m1_subset_1(E,B)
& A = E
& r2_hidden(E,C)
& r2_hidden(D,E) ) ) ) ).
fof(fraenkel_a_2_0_card_lar,axiom,
! [A,B,C] :
( ( v3_ordinal1(B)
& v4_ordinal1(B)
& ~ v1_finset_1(B)
& m1_subset_1(C,k1_zfmisc_1(B)) )
=> ( r2_hidden(A,a_2_0_card_lar(B,C))
<=> ? [D] :
( m1_subset_1(D,B)
& A = D
& ~ v1_finset_1(D)
& v4_ordinal1(D)
& k7_ordinal2(k3_xboole_0(C,D)) = D ) ) ) ).
fof(fraenkel_a_3_2_card_lar,axiom,
! [A,B,C,D] :
( ( v3_ordinal1(B)
& v4_ordinal1(B)
& ~ v1_finset_1(B)
& m1_subset_1(C,k1_zfmisc_1(B))
& v3_ordinal1(D) )
=> ( r2_hidden(A,a_3_2_card_lar(B,C,D))
<=> ? [E] :
( m1_subset_1(E,B)
& A = k1_ordinal1(E)
& r2_hidden(E,C)
& r2_hidden(D,k1_ordinal1(E)) ) ) ) ).
fof(fraenkel_a_1_0_card_lar,axiom,
! [A,B] :
( ( ~ v1_finset_1(B)
& v1_card_1(B)
& ~ v1_card_4(B) )
=> ( r2_hidden(A,a_1_0_card_lar(B))
<=> ? [C] :
( ~ v1_finset_1(C)
& v1_card_1(C)
& m1_subset_1(C,B)
& A = C
& v1_card_5(C) ) ) ) ).
fof(fraenkel_a_1_1_card_lar,axiom,
! [A,B] :
( ( ~ v1_finset_1(B)
& v1_card_1(B)
& ~ v1_card_4(B) )
=> ( r2_hidden(A,a_1_1_card_lar(B))
<=> ? [C] :
( ~ v1_finset_1(C)
& v1_card_1(C)
& m1_subset_1(C,B)
& A = C
& v6_card_fil(C) ) ) ) ).
%------------------------------------------------------------------------------