SET007 Axioms: SET007+153.ax
%------------------------------------------------------------------------------
% File : SET007+153 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On Powers of 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_5 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 66 ( 13 unt; 0 def)
% Number of atoms : 326 ( 43 equ)
% Maximal formula atoms : 15 ( 4 avg)
% Number of connectives : 328 ( 68 ~; 2 |; 160 &)
% ( 6 <=>; 92 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 23 ( 21 usr; 1 prp; 0-3 aty)
% Number of functors : 34 ( 34 usr; 6 con; 0-2 aty)
% Number of variables : 98 ( 91 !; 7 ?)
% 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_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) )
=> ( v1_ordinal1(k3_card_3(A))
& v2_ordinal1(k3_card_3(A))
& v3_ordinal1(k3_card_3(A)) ) ) ).
fof(rc1_card_5,axiom,
? [A] : ~ v1_finset_1(A) ).
fof(rc2_card_5,axiom,
? [A] :
( v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A)
& ~ v1_finset_1(A)
& v1_card_1(A) ) ).
fof(cc1_card_5,axiom,
! [A] :
( ~ v1_finset_1(A)
=> ~ v1_xboole_0(A) ) ).
fof(fc2_card_5,axiom,
! [A] :
( v3_ordinal1(A)
=> ( ~ v1_xboole_0(k3_card_1(A))
& v1_ordinal1(k3_card_1(A))
& v2_ordinal1(k3_card_1(A))
& v3_ordinal1(k3_card_1(A))
& ~ v1_finset_1(k3_card_1(A))
& v1_card_1(k3_card_1(A)) ) ) ).
fof(fc3_card_5,axiom,
! [A,B] :
( ( ~ v1_finset_1(A)
& v1_card_1(A)
& v1_card_1(B) )
=> ( ~ v1_xboole_0(k1_card_2(A,B))
& v1_ordinal1(k1_card_2(A,B))
& v2_ordinal1(k1_card_2(A,B))
& v3_ordinal1(k1_card_2(A,B))
& ~ v1_finset_1(k1_card_2(A,B))
& v1_card_1(k1_card_2(A,B)) ) ) ).
fof(fc4_card_5,axiom,
! [A,B] :
( ( v1_card_1(A)
& ~ v1_finset_1(B)
& v1_card_1(B) )
=> ( ~ v1_xboole_0(k1_card_2(A,B))
& v1_ordinal1(k1_card_2(A,B))
& v2_ordinal1(k1_card_2(A,B))
& v3_ordinal1(k1_card_2(A,B))
& ~ v1_finset_1(k1_card_2(A,B))
& v1_card_1(k1_card_2(A,B)) ) ) ).
fof(fc5_card_5,axiom,
! [A,B] :
( ( ~ v1_finset_1(A)
& v1_card_1(A)
& ~ v1_finset_1(B)
& v1_card_1(B) )
=> ( ~ v1_xboole_0(k2_card_2(A,B))
& v1_ordinal1(k2_card_2(A,B))
& v2_ordinal1(k2_card_2(A,B))
& v3_ordinal1(k2_card_2(A,B))
& ~ v1_finset_1(k2_card_2(A,B))
& v1_card_1(k2_card_2(A,B)) ) ) ).
fof(fc6_card_5,axiom,
! [A,B] :
( ( ~ v1_finset_1(A)
& v1_card_1(A)
& ~ v1_finset_1(B)
& v1_card_1(B) )
=> ( ~ v1_xboole_0(k3_card_2(A,B))
& v1_ordinal1(k3_card_2(A,B))
& v2_ordinal1(k3_card_2(A,B))
& v3_ordinal1(k3_card_2(A,B))
& ~ v1_finset_1(k3_card_2(A,B))
& v1_card_1(k3_card_2(A,B)) ) ) ).
fof(fc7_card_5,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A) )
=> ( ~ v1_xboole_0(k2_card_1(A))
& v1_ordinal1(k2_card_1(A))
& v2_ordinal1(k2_card_1(A))
& v3_ordinal1(k2_card_1(A))
& ~ v1_finset_1(k2_card_1(A))
& v1_card_1(k2_card_1(A)) ) ) ).
fof(cc2_card_5,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A) )
=> ! [B] :
( m1_subset_1(B,A)
=> ( v1_ordinal1(B)
& v2_ordinal1(B)
& v3_ordinal1(B) ) ) ) ).
fof(fc8_card_5,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A) )
=> ( ~ v1_xboole_0(k1_card_5(A))
& v1_ordinal1(k1_card_5(A))
& v2_ordinal1(k1_card_5(A))
& v3_ordinal1(k1_card_5(A))
& ~ v1_finset_1(k1_card_5(A))
& v1_card_1(k1_card_5(A)) ) ) ).
fof(t1_card_5,axiom,
( np__1 = k1_tarski(np__0)
& np__2 = k2_tarski(np__0,np__1) ) ).
fof(t2_card_5,axiom,
$true ).
fof(t3_card_5,axiom,
$true ).
fof(t4_card_5,axiom,
$true ).
fof(t5_card_5,axiom,
$true ).
fof(t6_card_5,axiom,
$true ).
fof(t7_card_5,axiom,
$true ).
fof(t8_card_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k2_finseq_1(A) = k4_xboole_0(k1_nat_1(A,np__1),k1_tarski(np__0)) ) ).
fof(t9_card_5,axiom,
! [A] : k2_card_1(k1_card_1(A)) = k2_card_1(A) ).
fof(t10_card_5,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r2_hidden(A,k3_card_3(B))
<=> ? [C] :
( r2_hidden(C,k1_relat_1(B))
& r2_hidden(A,k1_funct_1(B,C)) ) ) ) ).
fof(t11_card_5,axiom,
! [A] :
( v3_ordinal1(A)
=> ~ v1_finset_1(k3_card_1(A)) ) ).
fof(t12_card_5,axiom,
! [A] :
( v1_card_1(A)
=> ~ ( ~ v1_finset_1(A)
& ! [B] :
( v3_ordinal1(B)
=> A != k3_card_1(B) ) ) ) ).
fof(t13_card_5,axiom,
! [A] :
( v1_card_1(A)
=> ~ ( ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> A != k1_card_1(B) )
& ! [B] :
( v3_ordinal1(B)
=> A != k3_card_1(B) ) ) ) ).
fof(t14_card_5,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
~ ( r1_tarski(B,A)
& ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& v1_ordinal2(C) )
=> ~ ( C = k3_wellord1(k1_wellord2(k2_wellord2(k1_wellord2(B))),k1_wellord2(B))
& v2_ordinal2(C)
& k1_relat_1(C) = k2_wellord2(k1_wellord2(B))
& k2_relat_1(C) = B ) ) ) ) ).
fof(t15_card_5,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( r1_tarski(B,A)
=> r4_zfrefle1(k7_ordinal2(B),k2_wellord2(k1_wellord2(B))) ) ) ).
fof(t16_card_5,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( r1_tarski(B,A)
=> k1_card_1(B) = k1_card_1(k2_wellord2(k1_wellord2(B))) ) ) ).
fof(t17_card_5,axiom,
! [A] :
( v3_ordinal1(A)
=> ? [B] :
( v3_ordinal1(B)
& r1_ordinal1(B,k1_card_1(A))
& r4_zfrefle1(A,B) ) ) ).
fof(t18_card_5,axiom,
! [A] :
( v3_ordinal1(A)
=> ? [B] :
( v1_card_1(B)
& r1_tarski(B,k1_card_1(A))
& r4_zfrefle1(A,B)
& ! [C] :
( v3_ordinal1(C)
=> ( r4_zfrefle1(A,C)
=> r1_ordinal1(B,C) ) ) ) ) ).
fof(t19_card_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_ordinal2(B) )
=> ( ( k2_relat_1(A) = k2_relat_1(B)
& v2_ordinal2(A)
& v2_ordinal2(B) )
=> A = B ) ) ) ).
fof(t20_card_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) )
=> ( v2_ordinal2(A)
=> v2_funct_1(A) ) ) ).
fof(t21_card_5,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_ordinal2(B) )
=> k2_ordinal1(k1_ordinal4(A,B),k1_relat_1(A)) = A ) ) ).
fof(t23_card_5,axiom,
! [A] :
( v1_card_1(A)
=> r2_hidden(A,k3_card_2(np__2,A)) ) ).
fof(d1_card_5,axiom,
$true ).
fof(d2_card_5,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( v1_card_1(B)
=> ( B = k1_card_5(A)
<=> ( r4_zfrefle1(A,B)
& ! [C] :
( v1_card_1(C)
=> ( r4_zfrefle1(A,C)
=> r1_tarski(B,C) ) ) ) ) ) ) ).
fof(d3_card_5,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( v1_card_1(B)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_card_3(C) )
=> ( C = k2_card_5(A,B)
<=> ( ! [D] :
( r2_hidden(D,k1_relat_1(C))
<=> ( r2_hidden(D,A)
& v1_card_1(D) ) )
& ! [D] :
( v1_card_1(D)
=> ( r2_hidden(D,A)
=> k1_funct_1(C,D) = k3_card_2(D,B) ) ) ) ) ) ) ) ).
fof(t24_card_5,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A) )
=> ? [B] :
( v3_ordinal1(B)
& A = k3_card_1(B) ) ) ).
fof(t25_card_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( ~ v1_finset_1(B)
& v1_card_1(B) )
=> ( B != np__0
& B != np__1
& B != np__2
& B != k1_card_1(A)
& r2_hidden(k1_card_1(A),B)
& r1_tarski(k3_card_1(np__0),B) ) ) ) ).
fof(t26_card_5,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( ( ~ v1_finset_1(B)
& v1_card_1(B) )
=> ( ( r1_tarski(B,A)
| r2_hidden(B,A) )
=> ( ~ v1_finset_1(A)
& v1_card_1(A) ) ) ) ) ).
fof(t27_card_5,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( ( ~ v1_finset_1(B)
& v1_card_1(B) )
=> ( ( r1_tarski(B,A)
| r2_hidden(B,A) )
=> ( k1_card_2(B,A) = A
& k1_card_2(A,B) = A
& k2_card_2(B,A) = A
& k2_card_2(A,B) = A ) ) ) ) ).
fof(t28_card_5,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A) )
=> ( k1_card_2(A,A) = A
& k2_card_2(A,A) = A ) ) ).
fof(t29_card_5,axiom,
$true ).
fof(t30_card_5,axiom,
$true ).
fof(t31_card_5,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( ( ~ v1_finset_1(B)
& v1_card_1(B) )
=> r1_tarski(A,k3_card_2(A,B)) ) ) ).
fof(t32_card_5,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A) )
=> k3_tarski(A) = A ) ).
fof(d4_card_5,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A) )
=> ( v1_card_5(A)
<=> k1_card_5(A) = A ) ) ).
fof(t33_card_5,axiom,
$true ).
fof(t34_card_5,axiom,
k1_card_5(k3_card_1(np__0)) = k3_card_1(np__0) ).
fof(t35_card_5,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A) )
=> k1_card_5(k2_card_1(A)) = k2_card_1(A) ) ).
fof(t36_card_5,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A) )
=> r1_tarski(k3_card_1(np__0),k1_card_5(A)) ) ).
fof(t37_card_5,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( k1_card_5(np__0) = np__0
& k1_card_5(k1_card_1(k1_nat_1(A,np__1))) = np__1 ) ) ).
fof(t38_card_5,axiom,
! [A,B] :
( v1_card_1(B)
=> ( ( r1_tarski(A,B)
& r2_hidden(k1_card_1(A),k1_card_5(B)) )
=> ( r2_hidden(k7_ordinal2(A),B)
& r2_hidden(k3_tarski(A),B) ) ) ) ).
fof(t39_card_5,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( v1_card_1(B)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& v1_ordinal2(C) )
=> ( ( k1_relat_1(C) = A
& r1_tarski(k2_relat_1(C),B)
& r2_hidden(A,k1_card_5(B)) )
=> ( r2_hidden(k8_ordinal2(C),B)
& r2_hidden(k3_card_3(C),B) ) ) ) ) ) ).
fof(t40_card_5,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A) )
=> ( r2_hidden(k1_card_5(A),A)
=> v2_card_1(A) ) ) ).
fof(t41_card_5,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A) )
=> ~ ( r2_hidden(k1_card_5(A),A)
& ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_ordinal2(B) )
=> ~ ( k1_relat_1(B) = k1_card_5(A)
& r1_tarski(k2_relat_1(B),A)
& v2_ordinal2(B)
& A = k8_ordinal2(B)
& v1_relat_1(B)
& v1_funct_1(B)
& v1_card_3(B)
& ~ r2_hidden(np__0,k2_relat_1(B)) ) ) ) ) ).
fof(t42_card_5,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A) )
=> ( v1_card_5(k3_card_1(np__0))
& v1_card_5(k2_card_1(A)) ) ) ).
fof(t43_card_5,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A) )
=> ! [B] :
( ( ~ v1_finset_1(B)
& v1_card_1(B) )
=> ( r1_tarski(A,B)
=> k3_card_2(A,B) = k3_card_2(np__2,B) ) ) ) ).
fof(t44_card_5,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A) )
=> ! [B] :
( ( ~ v1_finset_1(B)
& v1_card_1(B) )
=> k3_card_2(k2_card_1(A),B) = k2_card_2(k3_card_2(A,B),k2_card_1(A)) ) ) ).
fof(t45_card_5,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A) )
=> ! [B] :
( ( ~ v1_finset_1(B)
& v1_card_1(B) )
=> r1_tarski(k6_card_3(k2_card_5(B,A)),k3_card_2(B,A)) ) ) ).
fof(t46_card_5,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A) )
=> ! [B] :
( ( ~ v1_finset_1(B)
& v1_card_1(B) )
=> ( ( v2_card_1(A)
& r2_hidden(B,k1_card_5(A)) )
=> k3_card_2(A,B) = k6_card_3(k2_card_5(A,B)) ) ) ) ).
fof(t47_card_5,axiom,
! [A] :
( ( ~ v1_finset_1(A)
& v1_card_1(A) )
=> ! [B] :
( ( ~ v1_finset_1(B)
& v1_card_1(B) )
=> ( ( r1_tarski(k1_card_5(A),B)
& r2_hidden(B,A) )
=> k3_card_2(A,B) = k3_card_2(k6_card_3(k2_card_5(A,B)),k1_card_5(A)) ) ) ) ).
fof(dt_k1_card_5,axiom,
! [A] :
( v1_card_1(A)
=> v1_card_1(k1_card_5(A)) ) ).
fof(dt_k2_card_5,axiom,
! [A,B] :
( ( v1_card_1(A)
& v1_card_1(B) )
=> ( v1_relat_1(k2_card_5(A,B))
& v1_funct_1(k2_card_5(A,B))
& v1_card_3(k2_card_5(A,B)) ) ) ).
fof(t22_card_5,axiom,
! [A,B] :
( v1_card_1(B)
=> ( A != k1_xboole_0
=> r1_tarski(k1_card_1(a_2_0_card_5(A,B)),k2_card_2(B,k3_card_2(k1_card_1(A),B))) ) ) ).
fof(fraenkel_a_2_0_card_5,axiom,
! [A,B,C] :
( v1_card_1(C)
=> ( r2_hidden(A,a_2_0_card_5(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(B))
& A = D
& r2_hidden(k1_card_1(D),C) ) ) ) ).
%------------------------------------------------------------------------------