SET007 Axioms: SET007+67.ax
%------------------------------------------------------------------------------
% File : SET007+67 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Koenig'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_3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 87 ( 29 unt; 0 def)
% Number of atoms : 334 ( 59 equ)
% Maximal formula atoms : 11 ( 3 avg)
% Number of connectives : 261 ( 14 ~; 4 |; 131 &)
% ( 12 <=>; 100 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 5 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 23 ( 21 usr; 1 prp; 0-3 aty)
% Number of functors : 40 ( 40 usr; 10 con; 0-2 aty)
% Number of variables : 155 ( 148 !; 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(rc1_card_3,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v1_card_3(A) ) ).
fof(fc1_card_3,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_card_3(A) )
=> ( v1_relat_1(k7_relat_1(A,B))
& v1_funct_1(k7_relat_1(A,B))
& v1_card_3(k7_relat_1(A,B)) ) ) ).
fof(fc2_card_3,axiom,
! [A,B] :
( v1_card_1(B)
=> ( v1_relat_1(k2_funcop_1(A,B))
& v1_funct_1(k2_funcop_1(A,B))
& v1_ordinal2(k2_funcop_1(A,B))
& v1_card_3(k2_funcop_1(A,B)) ) ) ).
fof(fc3_card_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> v1_fraenkel(k4_card_3(A)) ) ).
fof(fc4_card_3,axiom,
! [A,B,C] :
( ( v1_setfam_1(B)
& v1_funct_1(C)
& v1_funct_2(C,A,B)
& m1_relset_1(C,A,B) )
=> ( ~ v1_xboole_0(k4_card_3(C))
& v1_fraenkel(k4_card_3(C)) ) ) ).
fof(fc5_card_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v2_relat_1(A)
& v1_funct_1(A) )
=> ( ~ v1_xboole_0(k4_card_3(A))
& v1_fraenkel(k4_card_3(A)) ) ) ).
fof(d1_card_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v1_card_3(A)
<=> ! [B] :
( r2_hidden(B,k1_relat_1(A))
=> v1_card_1(k1_funct_1(A,B)) ) ) ) ).
fof(t1_card_3,axiom,
$true ).
fof(t2_card_3,axiom,
$true ).
fof(t3_card_3,axiom,
( v1_relat_1(k1_xboole_0)
& v1_funct_1(k1_xboole_0)
& v1_card_3(k1_xboole_0) ) ).
fof(d2_card_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_card_3(B) )
=> ( B = k1_card_3(A)
<=> ( k1_relat_1(B) = k1_relat_1(A)
& ! [C] :
( r2_hidden(C,k1_relat_1(A))
=> k1_funct_1(B,C) = k1_card_1(k1_funct_1(A,C)) ) ) ) ) ) ).
fof(d3_card_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( B = k2_card_3(A)
<=> ( k1_relat_1(B) = k1_relat_1(A)
& ! [C] :
( r2_hidden(C,k1_relat_1(A))
=> k1_funct_1(B,C) = k2_zfmisc_1(k1_funct_1(A,C),k1_tarski(C)) ) ) ) ) ) ).
fof(d4_card_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> k3_card_3(A) = k3_tarski(k2_relat_1(A)) ) ).
fof(d5_card_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( B = k4_card_3(A)
<=> ! [C] :
( r2_hidden(C,B)
<=> ? [D] :
( v1_relat_1(D)
& v1_funct_1(D)
& C = D
& k1_relat_1(D) = k1_relat_1(A)
& ! [E] :
( r2_hidden(E,k1_relat_1(A))
=> r2_hidden(k1_funct_1(D,E),k1_funct_1(A,E)) ) ) ) ) ) ).
fof(t4_card_3,axiom,
$true ).
fof(t5_card_3,axiom,
$true ).
fof(t6_card_3,axiom,
$true ).
fof(t7_card_3,axiom,
$true ).
fof(t8_card_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_card_3(A) )
=> k1_card_3(A) = A ) ).
fof(t9_card_3,axiom,
k1_card_3(k1_xboole_0) = k1_xboole_0 ).
fof(t10_card_3,axiom,
! [A,B] : k1_card_3(k2_funcop_1(A,B)) = k2_funcop_1(A,k1_card_1(B)) ).
fof(t11_card_3,axiom,
k2_card_3(k1_xboole_0) = k1_xboole_0 ).
fof(t12_card_3,axiom,
! [A,B] : k2_card_3(k2_funcop_1(k1_tarski(A),B)) = k2_funcop_1(k1_tarski(A),k2_zfmisc_1(B,k1_tarski(A))) ).
fof(t13_card_3,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( r2_hidden(A,k1_relat_1(C))
& r2_hidden(B,k1_relat_1(C)) )
=> ( A = B
| r1_xboole_0(k1_funct_1(k2_card_3(C),A),k1_funct_1(k2_card_3(C),B)) ) ) ) ).
fof(t14_card_3,axiom,
k3_card_3(k1_xboole_0) = k1_xboole_0 ).
fof(t15_card_3,axiom,
! [A,B] : r1_tarski(k3_card_3(k2_funcop_1(A,B)),B) ).
fof(t16_card_3,axiom,
! [A,B] :
( A != k1_xboole_0
=> k3_card_3(k2_funcop_1(A,B)) = B ) ).
fof(t17_card_3,axiom,
! [A,B] : k3_card_3(k2_funcop_1(k1_tarski(A),B)) = B ).
fof(t18_card_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r2_hidden(A,k4_card_3(B))
<=> ( k1_relat_1(A) = k1_relat_1(B)
& ! [C] :
( r2_hidden(C,k1_relat_1(B))
=> r2_hidden(k1_funct_1(A,C),k1_funct_1(B,C)) ) ) ) ) ) ).
fof(t19_card_3,axiom,
k4_card_3(k1_xboole_0) = k1_tarski(k1_xboole_0) ).
fof(t20_card_3,axiom,
! [A,B] : k1_funct_2(A,B) = k4_card_3(k2_funcop_1(A,B)) ).
fof(d6_card_3,axiom,
! [A,B,C] :
( C = k5_card_3(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ? [E] :
( v1_relat_1(E)
& v1_funct_1(E)
& r2_hidden(E,B)
& D = k1_funct_1(E,A) ) ) ) ).
fof(t21_card_3,axiom,
$true ).
fof(t22_card_3,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r2_hidden(A,k1_relat_1(B))
=> ( k4_card_3(B) = k1_xboole_0
| k5_card_3(A,k4_card_3(B)) = k1_funct_1(B,A) ) ) ) ).
fof(t23_card_3,axiom,
$true ).
fof(t24_card_3,axiom,
! [A] : k5_card_3(A,k1_xboole_0) = k1_xboole_0 ).
fof(t25_card_3,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> k5_card_3(A,k1_tarski(B)) = k1_tarski(k1_funct_1(B,A)) ) ).
fof(t26_card_3,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> k5_card_3(A,k2_tarski(B,C)) = k2_tarski(k1_funct_1(B,A),k1_funct_1(C,A)) ) ) ).
fof(t27_card_3,axiom,
! [A,B,C] : k5_card_3(C,k2_xboole_0(A,B)) = k2_xboole_0(k5_card_3(C,A),k5_card_3(C,B)) ).
fof(t28_card_3,axiom,
! [A,B,C] : r1_tarski(k5_card_3(C,k3_xboole_0(A,B)),k3_xboole_0(k5_card_3(C,A),k5_card_3(C,B))) ).
fof(t29_card_3,axiom,
! [A,B,C] : r1_tarski(k4_xboole_0(k5_card_3(B,A),k5_card_3(B,C)),k5_card_3(B,k4_xboole_0(A,C))) ).
fof(t30_card_3,axiom,
! [A,B,C] : r1_tarski(k5_xboole_0(k5_card_3(B,A),k5_card_3(B,C)),k5_card_3(B,k5_xboole_0(A,C))) ).
fof(t31_card_3,axiom,
! [A,B] : r1_tarski(k1_card_1(k5_card_3(B,A)),k1_card_1(A)) ).
fof(t32_card_3,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ~ ( r2_hidden(A,k3_card_3(k2_card_3(B)))
& ! [C,D] : A != k4_tarski(C,D) ) ) ).
fof(t33_card_3,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r2_hidden(A,k3_card_3(k2_card_3(B)))
<=> ( r2_hidden(k2_mcart_1(A),k1_relat_1(B))
& r2_hidden(k1_mcart_1(A),k1_funct_1(B,k2_mcart_1(A)))
& A = k4_tarski(k1_mcart_1(A),k2_mcart_1(A)) ) ) ) ).
fof(t34_card_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_tarski(A,B)
=> r1_tarski(k2_card_3(A),k2_card_3(B)) ) ) ) ).
fof(t35_card_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_tarski(A,B)
=> r1_tarski(k3_card_3(A),k3_card_3(B)) ) ) ) ).
fof(t36_card_3,axiom,
! [A,B] : k3_card_3(k2_card_3(k2_funcop_1(A,B))) = k2_zfmisc_1(B,A) ).
fof(t37_card_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( k4_card_3(A) = k1_xboole_0
<=> r2_hidden(k1_xboole_0,k2_relat_1(A)) ) ) ).
fof(t38_card_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( ( k1_relat_1(A) = k1_relat_1(B)
& ! [C] :
( r2_hidden(C,k1_relat_1(A))
=> r1_tarski(k1_funct_1(A,C),k1_funct_1(B,C)) ) )
=> r1_tarski(k4_card_3(A),k4_card_3(B)) ) ) ) ).
fof(t39_card_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_card_3(A) )
=> ! [B] :
( r2_hidden(B,k1_relat_1(A))
=> k1_card_1(k1_funct_1(A,B)) = k1_funct_1(A,B) ) ) ).
fof(t40_card_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_card_3(A) )
=> ! [B] :
( r2_hidden(B,k1_relat_1(A))
=> k1_card_1(k1_funct_1(k2_card_3(A),B)) = k1_funct_1(A,B) ) ) ).
fof(d7_card_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_card_3(A) )
=> k6_card_3(A) = k1_card_1(k3_card_3(k2_card_3(A))) ) ).
fof(d8_card_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_card_3(A) )
=> k7_card_3(A) = k1_card_1(k4_card_3(A)) ) ).
fof(t41_card_3,axiom,
$true ).
fof(t42_card_3,axiom,
$true ).
fof(t43_card_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_card_3(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_card_3(B) )
=> ( ( k1_relat_1(A) = k1_relat_1(B)
& ! [C] :
( r2_hidden(C,k1_relat_1(A))
=> r1_tarski(k1_funct_1(A,C),k1_funct_1(B,C)) ) )
=> r1_tarski(k6_card_3(A),k6_card_3(B)) ) ) ) ).
fof(t44_card_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_card_3(A) )
=> ( r2_hidden(k1_xboole_0,k2_relat_1(A))
<=> k7_card_3(A) = np__0 ) ) ).
fof(t45_card_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_card_3(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_card_3(B) )
=> ( ( k1_relat_1(A) = k1_relat_1(B)
& ! [C] :
( r2_hidden(C,k1_relat_1(A))
=> r1_tarski(k1_funct_1(A,C),k1_funct_1(B,C)) ) )
=> r1_tarski(k7_card_3(A),k7_card_3(B)) ) ) ) ).
fof(t46_card_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_card_3(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_card_3(B) )
=> ( r1_tarski(A,B)
=> r1_tarski(k6_card_3(A),k6_card_3(B)) ) ) ) ).
fof(t47_card_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_card_3(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_card_3(B) )
=> ( r1_tarski(A,B)
=> ( r2_hidden(np__0,k2_relat_1(B))
| r1_tarski(k7_card_3(A),k7_card_3(B)) ) ) ) ) ).
fof(t48_card_3,axiom,
! [A] :
( v1_card_1(A)
=> k6_card_3(k2_funcop_1(k1_xboole_0,A)) = np__0 ) ).
fof(t49_card_3,axiom,
! [A] :
( v1_card_1(A)
=> k7_card_3(k2_funcop_1(k1_xboole_0,A)) = np__1 ) ).
fof(t50_card_3,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] : k6_card_3(k2_funcop_1(k1_tarski(B),A)) = A ) ).
fof(t51_card_3,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] : k7_card_3(k2_funcop_1(k1_tarski(B),A)) = A ) ).
fof(t52_card_3,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( v1_card_1(B)
=> k6_card_3(k2_funcop_1(A,B)) = k2_card_2(A,B) ) ) ).
fof(t53_card_3,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( v1_card_1(B)
=> k7_card_3(k2_funcop_1(A,B)) = k3_card_2(B,A) ) ) ).
fof(t54_card_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> r1_tarski(k1_card_1(k3_card_3(A)),k6_card_3(k1_card_3(A))) ) ).
fof(t55_card_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_card_3(A) )
=> r1_tarski(k1_card_1(k3_card_3(A)),k6_card_3(A)) ) ).
fof(t56_card_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_card_3(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_card_3(B) )
=> ( ( k1_relat_1(A) = k1_relat_1(B)
& ! [C] :
( r2_hidden(C,k1_relat_1(A))
=> r2_hidden(k1_funct_1(A,C),k1_funct_1(B,C)) ) )
=> r2_hidden(k6_card_3(A),k7_card_3(B)) ) ) ) ).
fof(t57_card_3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> v1_finset_1(k4_classes1(A)) ) ).
fof(t58_card_3,axiom,
! [A] :
( v1_finset_1(A)
=> r2_hidden(k1_card_1(A),k1_card_1(k5_ordinal2)) ) ).
fof(t59_card_3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( r2_hidden(k1_card_1(A),k1_card_1(B))
=> r2_hidden(A,B) ) ) ) ).
fof(t60_card_3,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v1_card_1(B)
=> ( r2_hidden(k1_card_1(A),B)
=> r2_hidden(A,B) ) ) ) ).
fof(t61_card_3,axiom,
! [A] :
~ ( v6_ordinal1(A)
& ! [B] :
~ ( r1_tarski(B,A)
& k3_tarski(B) = k3_tarski(A)
& ! [C] :
~ ( r1_tarski(C,B)
& C != k1_xboole_0
& ! [D] :
~ ( r2_hidden(D,C)
& ! [E] :
( r2_hidden(E,C)
=> r1_tarski(D,E) ) ) ) ) ) ).
fof(t62_card_3,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( ( ! [C] :
( r2_hidden(C,B)
=> r2_hidden(k1_card_1(C),A) )
& v6_ordinal1(B) )
=> r1_tarski(k1_card_1(k3_tarski(B)),A) ) ) ).
fof(s1_card_3,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v1_card_3(A)
& k1_relat_1(A) = f1_s1_card_3
& ! [B] :
( r2_hidden(B,f1_s1_card_3)
=> k1_funct_1(A,B) = f2_s1_card_3(B) ) ) ).
fof(s2_card_3,axiom,
( ( f1_s2_card_3 != k1_xboole_0
& ! [A,B] :
( ( p1_s2_card_3(A,B)
& p1_s2_card_3(B,A) )
=> A = B )
& ! [A,B,C] :
( ( p1_s2_card_3(A,B)
& p1_s2_card_3(B,C) )
=> p1_s2_card_3(A,C) ) )
=> ? [A] :
( r2_hidden(A,f1_s2_card_3)
& ! [B] :
~ ( r2_hidden(B,f1_s2_card_3)
& B != A
& p1_s2_card_3(B,A) ) ) ) ).
fof(s3_card_3,axiom,
( ( f1_s3_card_3 != k1_xboole_0
& ! [A,B] :
( p1_s3_card_3(A,B)
| p1_s3_card_3(B,A) )
& ! [A,B,C] :
( ( p1_s3_card_3(A,B)
& p1_s3_card_3(B,C) )
=> p1_s3_card_3(A,C) ) )
=> ? [A] :
( r2_hidden(A,f1_s3_card_3)
& ! [B] :
( r2_hidden(B,f1_s3_card_3)
=> p1_s3_card_3(A,B) ) ) ) ).
fof(s4_card_3,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& k1_relat_1(A) = f1_s4_card_3
& ! [B] :
( r2_hidden(B,f1_s4_card_3)
=> ! [C] :
( r2_hidden(C,k1_funct_1(A,B))
<=> ( r2_hidden(C,f2_s4_card_3(B))
& p1_s4_card_3(B,C) ) ) ) ) ).
fof(dt_k1_card_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v1_relat_1(k1_card_3(A))
& v1_funct_1(k1_card_3(A))
& v1_card_3(k1_card_3(A)) ) ) ).
fof(dt_k2_card_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v1_relat_1(k2_card_3(A))
& v1_funct_1(k2_card_3(A)) ) ) ).
fof(dt_k3_card_3,axiom,
$true ).
fof(dt_k4_card_3,axiom,
$true ).
fof(dt_k5_card_3,axiom,
$true ).
fof(dt_k6_card_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_card_3(A) )
=> v1_card_1(k6_card_3(A)) ) ).
fof(dt_k7_card_3,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_card_3(A) )
=> v1_card_1(k7_card_3(A)) ) ).
%------------------------------------------------------------------------------