SET007 Axioms: SET007+170.ax
%------------------------------------------------------------------------------
% File : SET007+170 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Basic Facts about Inaccessible and Measurable 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_fil [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 95 ( 5 unt; 0 def)
% Number of atoms : 552 ( 29 equ)
% Maximal formula atoms : 21 ( 5 avg)
% Number of connectives : 612 ( 155 ~; 3 |; 227 &)
% ( 30 <=>; 197 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 8 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 32 ( 31 usr; 1 prp; 0-3 aty)
% Number of functors : 37 ( 37 usr; 6 con; 0-6 aty)
% Number of variables : 217 ( 198 !; 19 ?)
% 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_fil,axiom,
! [A] :
( ~ v1_finset_1(A)
=> ( v1_ordinal1(k1_card_1(A))
& v2_ordinal1(k1_card_1(A))
& v3_ordinal1(k1_card_1(A))
& ~ v1_xboole_0(k1_card_1(A))
& ~ v1_finset_1(k1_card_1(A))
& v1_card_1(k1_card_1(A)) ) ) ).
fof(fc2_card_fil,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ~ v1_xboole_0(k7_setfam_1(A,B)) ) ).
fof(rc1_card_fil,axiom,
! [A] :
( ~ v1_finset_1(A)
=> ? [B] :
( m1_card_fil(B,A)
& ~ v1_xboole_0(B)
& ~ v2_card_fil(B,A)
& v3_card_fil(B,A) ) ) ).
fof(cc1_card_fil,axiom,
! [A] :
( ~ v1_finset_1(A)
=> ! [B] :
( m1_card_fil(B,A)
=> ( ( v1_card_fil(B,A)
& v3_card_fil(B,A) )
=> ~ v2_card_fil(B,A) ) ) ) ).
fof(cc2_card_fil,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A)
& v4_card_fil(A) )
=> ( v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A)
& ~ v1_xboole_0(A)
& ~ v1_finset_1(A)
& v1_card_1(A)
& v2_card_1(A)
& v1_card_5(A) ) ) ).
fof(cc3_card_fil,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A)
& v5_card_fil(A) )
=> ( v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A)
& ~ v1_xboole_0(A)
& ~ v1_finset_1(A)
& v1_card_1(A)
& v2_card_1(A) ) ) ).
fof(cc4_card_fil,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A)
& v6_card_fil(A) )
=> ( v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A)
& ~ v1_xboole_0(A)
& ~ v1_finset_1(A)
& v1_card_1(A)
& v2_card_1(A)
& v1_card_5(A)
& v5_card_fil(A) ) ) ).
fof(cc5_card_fil,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A)
& v6_card_fil(A) )
=> ( v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A)
& ~ v1_xboole_0(A)
& ~ v1_finset_1(A)
& v1_card_1(A)
& v2_card_1(A)
& v1_card_5(A)
& v4_card_fil(A) ) ) ).
fof(fc3_card_fil,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A) )
=> ( v1_ordinal1(k2_card_1(A))
& v2_ordinal1(k2_card_1(A))
& v3_ordinal1(k2_card_1(A))
& ~ v1_xboole_0(k2_card_1(A))
& ~ v1_finset_1(k2_card_1(A))
& v1_card_1(k2_card_1(A))
& ~ v2_card_1(k2_card_1(A)) ) ) ).
fof(rc2_card_fil,axiom,
? [A] :
( v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A)
& ~ v1_xboole_0(A)
& ~ v1_finset_1(A)
& v1_card_1(A)
& ~ v2_card_1(A) ) ).
fof(cc6_card_fil,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A)
& ~ v2_card_1(A) )
=> ( v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A)
& ~ v1_xboole_0(A)
& ~ v1_finset_1(A)
& v1_card_1(A)
& v1_card_5(A) ) ) ).
fof(fc4_card_fil,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A)
& ~ v2_card_1(A) )
=> ( v1_ordinal1(k7_card_fil(A))
& v2_ordinal1(k7_card_fil(A))
& v3_ordinal1(k7_card_fil(A))
& ~ v1_xboole_0(k7_card_fil(A))
& ~ v1_finset_1(k7_card_fil(A))
& v1_card_1(k7_card_fil(A)) ) ) ).
fof(t1_card_fil,axiom,
! [A,B] :
( ~ v1_finset_1(B)
=> r2_hidden(k1_card_1(k1_tarski(A)),k1_card_1(B)) ) ).
fof(t2_card_fil,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ~ v1_xboole_0(k1_tarski(A))
& m1_subset_1(k1_tarski(A),k1_zfmisc_1(k1_zfmisc_1(A)))
& ~ r2_hidden(k1_xboole_0,k1_tarski(A))
& ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( ( ( r2_hidden(B,k1_tarski(A))
& r2_hidden(C,k1_tarski(A)) )
=> r2_hidden(k5_subset_1(A,B,C),k1_tarski(A)) )
& ( ( r2_hidden(B,k1_tarski(A))
& r1_tarski(B,C) )
=> r2_hidden(C,k1_tarski(A)) ) ) ) ) ) ) ).
fof(d1_card_fil,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ( m1_card_fil(B,A)
<=> ( ~ r2_hidden(k1_xboole_0,B)
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( ( ( r2_hidden(C,B)
& r2_hidden(D,B) )
=> r2_hidden(k5_subset_1(A,C,D),B) )
& ( ( r2_hidden(C,B)
& r1_tarski(C,D) )
=> r2_hidden(D,B) ) ) ) ) ) ) ) ) ).
fof(t3_card_fil,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_card_fil(B,A)
<=> ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
& ~ r2_hidden(k1_xboole_0,B)
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( ( ( r2_hidden(C,B)
& r2_hidden(D,B) )
=> r2_hidden(k5_subset_1(A,C,D),B) )
& ( ( r2_hidden(C,B)
& r1_tarski(C,D) )
=> r2_hidden(D,B) ) ) ) ) ) ) ) ).
fof(t4_card_fil,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> m1_card_fil(k1_tarski(A),A) ) ).
fof(t5_card_fil,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_card_fil(B,A)
=> r2_hidden(A,B) ) ) ).
fof(t6_card_fil,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_card_fil(C,A)
=> ~ ( r2_hidden(B,C)
& r2_hidden(k1_card_fil(A,B),C) ) ) ) ) ).
fof(t7_card_fil,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_card_fil(B,A)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ( ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( r2_hidden(D,C)
<=> r2_hidden(k3_subset_1(A,D),B) ) )
=> ( ~ r2_hidden(A,C)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
=> ( ( ( r2_hidden(D,C)
& r2_hidden(E,C) )
=> r2_hidden(k4_subset_1(A,D,E),C) )
& ( ( r2_hidden(D,C)
& r1_tarski(E,D) )
=> r2_hidden(E,C) ) ) ) ) ) ) ) ) ) ).
fof(d2_card_fil,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ( m2_card_fil(B,A)
<=> ( ~ r2_hidden(A,B)
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( ( ( r2_hidden(C,B)
& r2_hidden(D,B) )
=> r2_hidden(k4_subset_1(A,C,D),B) )
& ( ( r2_hidden(C,B)
& r1_tarski(D,C) )
=> r2_hidden(D,B) ) ) ) ) ) ) ) ) ).
fof(t10_card_fil,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_card_fil(B,A)
=> ! [C] :
( m2_card_fil(C,A)
=> ( ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ~ ( r2_hidden(D,B)
& r2_hidden(D,k2_card_fil(A,B)) ) )
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ~ ( r2_hidden(D,C)
& r2_hidden(D,k7_setfam_1(A,C)) ) ) ) ) ) ) ).
fof(t11_card_fil,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_card_fil(B,A)
=> r2_hidden(k1_xboole_0,B) ) ) ).
fof(d3_card_fil,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( v1_card_1(B)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ( r1_card_fil(A,B,C)
<=> ! [D] :
( ~ v1_xboole_0(D)
=> ( ( r1_tarski(D,C)
& r2_hidden(k1_card_1(D),B) )
=> r2_hidden(k1_setfam_1(D),C) ) ) ) ) ) ) ).
fof(d4_card_fil,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( v1_card_1(B)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ( r2_card_fil(A,B,C)
<=> ! [D] :
( ~ v1_xboole_0(D)
=> ( ( r1_tarski(D,C)
& r2_hidden(k1_card_1(D),B) )
=> r2_hidden(k3_tarski(D),C) ) ) ) ) ) ) ).
fof(t12_card_fil,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(B))) )
=> ( r1_card_fil(B,A,C)
=> r2_card_fil(B,A,k7_setfam_1(B,C)) ) ) ) ) ).
fof(d5_card_fil,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_card_fil(B,A)
=> ( v1_card_fil(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r2_hidden(C,B)
=> k1_card_1(C) = k1_card_1(A) ) ) ) ) ) ).
fof(d6_card_fil,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_card_fil(B,A)
=> ( v2_card_fil(B,A)
<=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
& r2_hidden(C,B)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( r2_hidden(D,B)
=> r1_tarski(C,D) ) ) ) ) ) ) ).
fof(d7_card_fil,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_card_fil(B,A)
=> ( v3_card_fil(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r2_hidden(C,B)
| r2_hidden(k1_card_fil(A,C),B) ) ) ) ) ) ).
fof(t13_card_fil,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_card_fil(C,A)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( r2_hidden(D,k3_card_fil(A,C,B))
<=> ? [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
& r2_hidden(E,C)
& r1_tarski(k5_subset_1(A,E,B),D) ) ) ) ) ) ) ).
fof(t14_card_fil,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_card_fil(C,A)
=> ( ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ~ ( r2_hidden(D,C)
& r1_xboole_0(D,B) ) )
=> ( r2_hidden(B,k3_card_fil(A,C,B))
& m1_card_fil(k3_card_fil(A,C,B),A)
& r1_tarski(C,k3_card_fil(A,C,B)) ) ) ) ) ) ).
fof(t15_card_fil,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( r2_hidden(B,k4_card_fil(A))
<=> m1_card_fil(B,A) ) ) ).
fof(t16_card_fil,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k4_card_fil(A))) )
=> ( v6_ordinal1(B)
=> m1_card_fil(k3_tarski(B),A) ) ) ) ).
fof(t17_card_fil,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_card_fil(B,A)
=> ? [C] :
( m1_card_fil(C,A)
& r1_tarski(B,C)
& v3_card_fil(C,A) ) ) ) ).
fof(d11_card_fil,axiom,
! [A] :
( ~ v1_finset_1(A)
=> k6_card_fil(A) = k2_card_fil(A,k5_card_fil(A)) ) ).
fof(t18_card_fil,axiom,
! [A] :
( ~ v1_finset_1(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( r2_hidden(B,k5_card_fil(A))
<=> r2_hidden(k1_card_1(k1_card_fil(A,B)),k1_card_1(A)) ) ) ) ).
fof(t19_card_fil,axiom,
! [A] :
( ~ v1_finset_1(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( r2_hidden(B,k6_card_fil(A))
<=> r2_hidden(k1_card_1(B),k1_card_1(A)) ) ) ) ).
fof(t20_card_fil,axiom,
! [A] :
( ~ v1_finset_1(A)
=> ! [B] :
( m1_card_fil(B,A)
=> ( r1_tarski(k5_card_fil(A),B)
=> v1_card_fil(B,A) ) ) ) ).
fof(t21_card_fil,axiom,
! [A] :
( ~ v1_finset_1(A)
=> ! [B] :
( m1_card_fil(B,A)
=> ( ( v1_card_fil(B,A)
& v3_card_fil(B,A) )
=> r1_tarski(k5_card_fil(A),B) ) ) ) ).
fof(t22_card_fil,axiom,
! [A] :
( ~ v1_finset_1(A)
=> ! [B] :
( ( v3_card_fil(B,A)
& m1_card_fil(B,A) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r2_hidden(C,B)
<=> ~ r2_hidden(C,k2_card_fil(A,B)) ) ) ) ) ).
fof(t23_card_fil,axiom,
! [A] :
( ~ v1_finset_1(A)
=> ! [B] :
( m1_card_fil(B,A)
=> ( ( v3_card_fil(B,A)
& r1_card_fil(A,k1_card_1(A),B) )
=> ( v2_card_fil(B,A)
| v1_card_fil(B,A) ) ) ) ) ).
fof(t24_card_fil,axiom,
! [A] :
( v1_card_1(A)
=> r1_tarski(k2_card_1(A),k3_card_2(np__2,A)) ) ).
fof(d12_card_fil,axiom,
( r3_card_fil
<=> ! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A) )
=> k2_card_1(A) = k3_card_2(np__2,A) ) ) ).
fof(d13_card_fil,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A) )
=> ( v4_card_fil(A)
<=> ( v1_card_5(A)
& v2_card_1(A) ) ) ) ).
fof(t25_card_fil,axiom,
v4_card_fil(k3_card_1(np__0)) ).
fof(d14_card_fil,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A) )
=> ( v5_card_fil(A)
<=> ! [B] :
( v1_card_1(B)
=> ( r2_hidden(B,A)
=> r2_hidden(k3_card_2(np__2,B),A) ) ) ) ) ).
fof(t26_card_fil,axiom,
v5_card_fil(k3_card_1(np__0)) ).
fof(t27_card_fil,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A) )
=> ( v5_card_fil(A)
=> v2_card_1(A) ) ) ).
fof(t28_card_fil,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A) )
=> ( ( r3_card_fil
& v2_card_1(A) )
=> v5_card_fil(A) ) ) ).
fof(d15_card_fil,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A) )
=> ( v6_card_fil(A)
<=> ( v1_card_5(A)
& v5_card_fil(A) ) ) ) ).
fof(t29_card_fil,axiom,
v6_card_fil(k3_card_1(np__0)) ).
fof(t30_card_fil,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A) )
=> ( v6_card_fil(A)
=> v4_card_fil(A) ) ) ).
fof(t31_card_fil,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A) )
=> ( ( r3_card_fil
& v4_card_fil(A) )
=> v6_card_fil(A) ) ) ).
fof(d16_card_fil,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A) )
=> ( v7_card_fil(A)
<=> ? [B] :
( m1_card_fil(B,A)
& r1_card_fil(A,A,B)
& ~ v2_card_fil(B,A)
& v3_card_fil(B,A) ) ) ) ).
fof(t32_card_fil,axiom,
! [A] :
( ( v3_ordinal1(A)
& v4_ordinal1(A) )
=> ! [B] :
( ( r1_tarski(B,A)
& k7_ordinal2(B) = A )
=> k3_tarski(B) = A ) ) ).
fof(t33_card_fil,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A) )
=> ( v7_card_fil(A)
=> v1_card_5(A) ) ) ).
fof(d17_card_fil,axiom,
! [A] :
( ( v1_card_1(A)
& ~ v2_card_1(A) )
=> ! [B] :
( v1_card_1(B)
=> ( B = k7_card_fil(A)
<=> A = k2_card_1(B) ) ) ) ).
fof(t34_card_fil,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A)
& ~ v2_card_1(A) )
=> ? [B] :
( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(k7_card_fil(A),A),k1_zfmisc_1(A))
& m2_relset_1(B,k2_zfmisc_1(k7_card_fil(A),A),k1_zfmisc_1(A))
& r4_card_fil(A,B) ) ) ).
fof(t35_card_fil,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A)
& ~ v2_card_1(A) )
=> ! [B] :
( m2_card_fil(B,A)
=> ~ ( r2_card_fil(A,A,B)
& r1_tarski(k6_card_fil(A),B)
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ~ ( k1_card_1(C) = A
& ! [D] :
~ ( r2_hidden(D,C)
& r2_hidden(D,B) )
& ! [D,E] :
( ( r2_hidden(D,C)
& r2_hidden(E,C) )
=> ( D = E
| r1_xboole_0(D,E) ) ) ) ) ) ) ) ).
fof(t36_card_fil,axiom,
! [A,B] :
( v1_card_1(B)
=> ~ ( r1_tarski(B,k1_card_1(A))
& ! [C] :
~ ( r1_tarski(C,A)
& k1_card_1(C) = B ) ) ) ).
fof(t37_card_fil,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A)
& ~ v2_card_1(A) )
=> ! [B] :
( m1_card_fil(B,A)
=> ~ ( v1_card_fil(B,A)
& v3_card_fil(B,A)
& r1_card_fil(A,A,B) ) ) ) ).
fof(t38_card_fil,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A) )
=> ( v7_card_fil(A)
=> v2_card_1(A) ) ) ).
fof(t39_card_fil,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A) )
=> ( v7_card_fil(A)
=> v4_card_fil(A) ) ) ).
fof(t40_card_fil,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A) )
=> ( v7_card_fil(A)
=> v5_card_fil(A) ) ) ).
fof(t41_card_fil,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A) )
=> ( v7_card_fil(A)
=> v6_card_fil(A) ) ) ).
fof(dt_m1_card_fil,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_card_fil(B,A)
=> ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) ) ) ).
fof(existence_m1_card_fil,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] : m1_card_fil(B,A) ) ).
fof(dt_m2_card_fil,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_card_fil(B,A)
=> ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) ) ) ).
fof(existence_m2_card_fil,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] : m2_card_fil(B,A) ) ).
fof(dt_k1_card_fil,axiom,
! [A,B] : m1_subset_1(k1_card_fil(A,B),k1_zfmisc_1(A)) ).
fof(redefinition_k1_card_fil,axiom,
! [A,B] : k1_card_fil(A,B) = k4_xboole_0(A,B) ).
fof(dt_k2_card_fil,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_card_fil(B,A) )
=> m2_card_fil(k2_card_fil(A,B),A) ) ).
fof(involutiveness_k2_card_fil,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_card_fil(B,A) )
=> k2_card_fil(A,k2_card_fil(A,B)) = B ) ).
fof(redefinition_k2_card_fil,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_card_fil(B,A) )
=> k2_card_fil(A,B) = k7_setfam_1(A,B) ) ).
fof(dt_k3_card_fil,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_card_fil(B,A)
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> ( ~ v1_xboole_0(k3_card_fil(A,B,C))
& m1_subset_1(k3_card_fil(A,B,C),k1_zfmisc_1(k1_zfmisc_1(A))) ) ) ).
fof(dt_k4_card_fil,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ~ v1_xboole_0(k4_card_fil(A))
& m1_subset_1(k4_card_fil(A),k1_zfmisc_1(k1_zfmisc_1(k1_zfmisc_1(A)))) ) ) ).
fof(dt_k5_card_fil,axiom,
! [A] :
( ~ v1_finset_1(A)
=> m1_card_fil(k5_card_fil(A),A) ) ).
fof(dt_k6_card_fil,axiom,
! [A] :
( ~ v1_finset_1(A)
=> m2_card_fil(k6_card_fil(A),A) ) ).
fof(dt_k7_card_fil,axiom,
! [A] :
( ( v1_card_1(A)
& ~ v2_card_1(A) )
=> v1_card_1(k7_card_fil(A)) ) ).
fof(t8_card_fil,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> k7_setfam_1(A,B) = a_2_0_card_fil(A,B) ) ) ).
fof(t9_card_fil,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> k7_setfam_1(A,B) = a_2_1_card_fil(A,B) ) ) ).
fof(d8_card_fil,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_card_fil(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> k3_card_fil(A,B,C) = a_3_0_card_fil(A,B,C) ) ) ) ).
fof(d9_card_fil,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k4_card_fil(A) = a_1_0_card_fil(A) ) ).
fof(d10_card_fil,axiom,
! [A] :
( ~ v1_finset_1(A)
=> k5_card_fil(A) = a_1_1_card_fil(A) ) ).
fof(d18_card_fil,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A)
& ~ v2_card_1(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(k7_card_fil(A),A),k1_zfmisc_1(A))
& m2_relset_1(B,k2_zfmisc_1(k7_card_fil(A),A),k1_zfmisc_1(A)) )
=> ( r4_card_fil(A,B)
<=> ( ! [C] :
( m1_subset_1(C,k7_card_fil(A))
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> ( D != E
=> v1_xboole_0(k5_subset_1(A,k2_binop_1(k7_card_fil(A),A,k1_zfmisc_1(A),B,C,D),k2_binop_1(k7_card_fil(A),A,k1_zfmisc_1(A),B,C,E))) ) ) ) )
& ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,k7_card_fil(A))
=> ! [E] :
( m1_subset_1(E,k7_card_fil(A))
=> ( D != E
=> v1_xboole_0(k5_subset_1(A,k2_binop_1(k7_card_fil(A),A,k1_zfmisc_1(A),B,D,C),k2_binop_1(k7_card_fil(A),A,k1_zfmisc_1(A),B,E,C))) ) ) ) )
& ! [C] :
( m1_subset_1(C,k7_card_fil(A))
=> r1_tarski(k1_card_1(k1_card_fil(A,k3_tarski(a_3_2_card_fil(A,B,C)))),k7_card_fil(A)) )
& ! [C] :
( m1_subset_1(C,A)
=> r1_tarski(k1_card_1(k1_card_fil(A,k3_tarski(a_3_3_card_fil(A,B,C)))),k7_card_fil(A)) ) ) ) ) ) ).
fof(s1_card_fil,axiom,
( r2_hidden(f2_s1_card_fil,a_0_0_card_fil)
=> p1_s1_card_fil(f2_s1_card_fil) ) ).
fof(fraenkel_a_2_0_card_fil,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(B))) )
=> ( r2_hidden(A,a_2_0_card_fil(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(B))
& A = D
& r2_hidden(k3_subset_1(B,D),C) ) ) ) ).
fof(fraenkel_a_2_1_card_fil,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(B))) )
=> ( r2_hidden(A,a_2_1_card_fil(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(B))
& A = k3_subset_1(B,D)
& r2_hidden(D,C) ) ) ) ).
fof(fraenkel_a_3_0_card_fil,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& m1_card_fil(C,B)
& m1_subset_1(D,k1_zfmisc_1(B)) )
=> ( r2_hidden(A,a_3_0_card_fil(B,C,D))
<=> ? [E] :
( m1_subset_1(E,k1_zfmisc_1(B))
& A = E
& ? [F] :
( m1_subset_1(F,k1_zfmisc_1(B))
& r2_hidden(F,a_3_1_card_fil(B,C,D))
& r1_tarski(F,E) ) ) ) ) ).
fof(fraenkel_a_3_1_card_fil,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& m1_card_fil(C,B)
& m1_subset_1(D,k1_zfmisc_1(B)) )
=> ( r2_hidden(A,a_3_1_card_fil(B,C,D))
<=> ? [E] :
( m1_subset_1(E,k1_zfmisc_1(B))
& A = k5_subset_1(B,E,D)
& r2_hidden(E,C) ) ) ) ).
fof(fraenkel_a_1_0_card_fil,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ( r2_hidden(A,a_1_0_card_fil(B))
<=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(B)))
& A = C
& m1_card_fil(C,B) ) ) ) ).
fof(fraenkel_a_1_1_card_fil,axiom,
! [A,B] :
( ~ v1_finset_1(B)
=> ( r2_hidden(A,a_1_1_card_fil(B))
<=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(B))
& A = C
& r2_hidden(k1_card_1(k1_card_fil(B,C)),k1_card_1(B)) ) ) ) ).
fof(fraenkel_a_3_2_card_fil,axiom,
! [A,B,C,D] :
( ( ~ v1_finset_1(B)
& v1_card_1(B)
& ~ v2_card_1(B)
& v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(k7_card_fil(B),B),k1_zfmisc_1(B))
& m2_relset_1(C,k2_zfmisc_1(k7_card_fil(B),B),k1_zfmisc_1(B))
& m1_subset_1(D,k7_card_fil(B)) )
=> ( r2_hidden(A,a_3_2_card_fil(B,C,D))
<=> ? [E] :
( m1_subset_1(E,B)
& A = k2_binop_1(k7_card_fil(B),B,k1_zfmisc_1(B),C,D,E)
& r2_hidden(E,B) ) ) ) ).
fof(fraenkel_a_3_3_card_fil,axiom,
! [A,B,C,D] :
( ( ~ v1_finset_1(B)
& v1_card_1(B)
& ~ v2_card_1(B)
& v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(k7_card_fil(B),B),k1_zfmisc_1(B))
& m2_relset_1(C,k2_zfmisc_1(k7_card_fil(B),B),k1_zfmisc_1(B))
& m1_subset_1(D,B) )
=> ( r2_hidden(A,a_3_3_card_fil(B,C,D))
<=> ? [E] :
( m1_subset_1(E,k7_card_fil(B))
& A = k2_binop_1(k7_card_fil(B),B,k1_zfmisc_1(B),C,E,D)
& r2_hidden(E,k7_card_fil(B)) ) ) ) ).
fof(fraenkel_a_0_0_card_fil,axiom,
! [A] :
( r2_hidden(A,a_0_0_card_fil)
<=> ? [B] :
( m1_subset_1(B,f1_s1_card_fil)
& A = B
& p1_s1_card_fil(B) ) ) ).
%------------------------------------------------------------------------------