SET007 Axioms: SET007+54.ax
%------------------------------------------------------------------------------
% File : SET007+54 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Cardinal Numbers
% 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_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 111 ( 49 unt; 0 def)
% Number of atoms : 307 ( 50 equ)
% Maximal formula atoms : 16 ( 2 avg)
% Number of connectives : 206 ( 10 ~; 3 |; 71 &)
% ( 20 <=>; 102 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 4 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 28 ( 26 usr; 1 prp; 0-3 aty)
% Number of functors : 27 ( 27 usr; 7 con; 0-2 aty)
% Number of variables : 131 ( 121 !; 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(rc1_card_1,axiom,
? [A] : v1_card_1(A) ).
fof(cc1_card_1,axiom,
! [A] :
( v1_card_1(A)
=> ( v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A) ) ) ).
fof(fc1_card_1,axiom,
! [A] :
( v3_ordinal1(A)
=> ( v1_ordinal1(k3_card_1(A))
& v2_ordinal1(k3_card_1(A))
& v3_ordinal1(k3_card_1(A))
& v1_card_1(k3_card_1(A)) ) ) ).
fof(cc2_card_1,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A)
& v4_ordinal2(A)
& v1_xcmplx_0(A)
& v1_xreal_0(A)
& ~ v3_xreal_0(A)
& v1_card_1(A) ) ) ).
fof(cc3_card_1,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> ( v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A)
& v4_ordinal2(A)
& v1_xcmplx_0(A)
& v1_finset_1(A)
& v1_xreal_0(A)
& ~ v3_xreal_0(A)
& v1_card_1(A) ) ) ).
fof(rc2_card_1,axiom,
? [A] :
( v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A)
& v1_finset_1(A)
& v1_card_1(A) ) ).
fof(fc2_card_1,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_finset_1(k1_card_1(A))
& v1_card_1(k1_card_1(A)) ) ) ).
fof(d1_card_1,axiom,
! [A] :
( v1_card_1(A)
<=> ? [B] :
( v3_ordinal1(B)
& A = B
& ! [C] :
( v3_ordinal1(C)
=> ( r2_wellord2(C,B)
=> r1_ordinal1(B,C) ) ) ) ) ).
fof(t1_card_1,axiom,
$true ).
fof(t2_card_1,axiom,
$true ).
fof(t3_card_1,axiom,
$true ).
fof(t4_card_1,axiom,
! [A] :
? [B] :
( v3_ordinal1(B)
& r2_wellord2(A,B) ) ).
fof(t5_card_1,axiom,
$true ).
fof(t6_card_1,axiom,
$true ).
fof(t7_card_1,axiom,
$true ).
fof(t8_card_1,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( v1_card_1(B)
=> ( A = B
<=> r2_wellord2(A,B) ) ) ) ).
fof(t9_card_1,axiom,
$true ).
fof(t10_card_1,axiom,
$true ).
fof(t11_card_1,axiom,
$true ).
fof(t12_card_1,axiom,
$true ).
fof(t13_card_1,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( v1_card_1(B)
=> ( r2_hidden(A,B)
<=> ( r1_tarski(A,B)
& A != B ) ) ) ) ).
fof(t14_card_1,axiom,
! [A] :
( v1_card_1(A)
=> ! [B] :
( v1_card_1(B)
=> ( r2_hidden(A,B)
<=> ~ r1_tarski(B,A) ) ) ) ).
fof(d2_card_1,axiom,
$true ).
fof(d3_card_1,axiom,
$true ).
fof(d4_card_1,axiom,
$true ).
fof(d5_card_1,axiom,
! [A,B] :
( v1_card_1(B)
=> ( B = k1_card_1(A)
<=> r2_wellord2(A,B) ) ) ).
fof(t15_card_1,axiom,
$true ).
fof(t16_card_1,axiom,
$true ).
fof(t17_card_1,axiom,
$true ).
fof(t18_card_1,axiom,
$true ).
fof(t19_card_1,axiom,
$true ).
fof(t20_card_1,axiom,
$true ).
fof(t21_card_1,axiom,
! [A,B] :
( r2_wellord2(A,B)
<=> k1_card_1(A) = k1_card_1(B) ) ).
fof(t22_card_1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v2_wellord1(A)
=> r2_wellord2(k3_relat_1(A),k2_wellord2(A)) ) ) ).
fof(t23_card_1,axiom,
! [A,B] :
( v1_card_1(B)
=> ( r1_tarski(A,B)
=> r1_tarski(k1_card_1(A),B) ) ) ).
fof(t24_card_1,axiom,
! [A] :
( v3_ordinal1(A)
=> r1_ordinal1(k1_card_1(A),A) ) ).
fof(t25_card_1,axiom,
! [A,B] :
( v1_card_1(B)
=> ( r2_hidden(A,B)
=> r2_hidden(k1_card_1(A),B) ) ) ).
fof(t26_card_1,axiom,
! [A,B] :
( r1_tarski(k1_card_1(A),k1_card_1(B))
<=> ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& v2_funct_1(C)
& k1_relat_1(C) = A
& r1_tarski(k2_relat_1(C),B) ) ) ).
fof(t27_card_1,axiom,
! [A,B] :
( r1_tarski(A,B)
=> r1_tarski(k1_card_1(A),k1_card_1(B)) ) ).
fof(t28_card_1,axiom,
! [A,B] :
( r1_tarski(k1_card_1(A),k1_card_1(B))
<=> ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& k1_relat_1(C) = B
& r1_tarski(A,k2_relat_1(C)) ) ) ).
fof(t29_card_1,axiom,
! [A] : ~ r2_wellord2(A,k1_zfmisc_1(A)) ).
fof(t30_card_1,axiom,
! [A] : r2_hidden(k1_card_1(A),k1_card_1(k1_zfmisc_1(A))) ).
fof(d6_card_1,axiom,
! [A,B] :
( v1_card_1(B)
=> ( B = k2_card_1(A)
<=> ( r2_hidden(k1_card_1(A),B)
& ! [C] :
( v1_card_1(C)
=> ( r2_hidden(k1_card_1(A),C)
=> r1_tarski(B,C) ) ) ) ) ) ).
fof(t31_card_1,axiom,
$true ).
fof(t32_card_1,axiom,
! [A] :
( v1_card_1(A)
=> r2_hidden(A,k2_card_1(A)) ) ).
fof(t33_card_1,axiom,
! [A] : r2_hidden(k1_card_1(k1_xboole_0),k2_card_1(A)) ).
fof(t34_card_1,axiom,
! [A,B] :
( k1_card_1(A) = k1_card_1(B)
=> k2_card_1(A) = k2_card_1(B) ) ).
fof(t35_card_1,axiom,
! [A,B] :
( r2_wellord2(A,B)
=> k2_card_1(A) = k2_card_1(B) ) ).
fof(t36_card_1,axiom,
! [A] :
( v3_ordinal1(A)
=> r2_hidden(A,k2_card_1(A)) ) ).
fof(d7_card_1,axiom,
! [A] :
( v1_card_1(A)
=> ( v2_card_1(A)
<=> ! [B] :
( v1_card_1(B)
=> A != k2_card_1(B) ) ) ) ).
fof(d8_card_1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( B = k3_card_1(A)
<=> ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& v5_ordinal1(C)
& B = k1_ordinal2(C)
& k1_relat_1(C) = k1_ordinal1(A)
& k1_funct_1(C,k1_xboole_0) = k1_card_1(k5_numbers)
& ! [D] :
( v3_ordinal1(D)
=> ( r2_hidden(k1_ordinal1(D),k1_ordinal1(A))
=> k1_funct_1(C,k1_ordinal1(D)) = k2_card_1(k3_tarski(k1_tarski(k1_funct_1(C,D)))) ) )
& ! [D] :
( v3_ordinal1(D)
=> ( ( r2_hidden(D,k1_ordinal1(A))
& v4_ordinal1(D) )
=> ( D = k1_xboole_0
| k1_funct_1(C,D) = k1_card_1(k8_ordinal2(k2_ordinal1(C,D))) ) ) ) ) ) ) ).
fof(t37_card_1,axiom,
$true ).
fof(t38_card_1,axiom,
k3_card_1(np__0) = k1_card_1(k5_numbers) ).
fof(t39_card_1,axiom,
! [A] :
( v3_ordinal1(A)
=> k3_card_1(k1_ordinal1(A)) = k2_card_1(k3_card_1(A)) ) ).
fof(t40_card_1,axiom,
! [A] :
( v3_ordinal1(A)
=> ( v4_ordinal1(A)
=> ( A = k1_xboole_0
| ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B) )
=> ( ( k1_relat_1(B) = A
& ! [C] :
( v3_ordinal1(C)
=> ( r2_hidden(C,A)
=> k1_funct_1(B,C) = k3_card_1(C) ) ) )
=> k3_card_1(A) = k1_card_1(k8_ordinal2(B)) ) ) ) ) ) ).
fof(t41_card_1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( r2_hidden(A,B)
<=> r2_hidden(k3_card_1(A),k3_card_1(B)) ) ) ) ).
fof(t42_card_1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( k3_card_1(A) = k3_card_1(B)
=> A = B ) ) ) ).
fof(t43_card_1,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( r1_ordinal1(A,B)
<=> r1_tarski(k3_card_1(A),k3_card_1(B)) ) ) ) ).
fof(t44_card_1,axiom,
! [A,B,C] :
( ( r1_tarski(A,B)
& r1_tarski(B,C)
& r2_wellord2(A,C) )
=> ( r2_wellord2(A,B)
& r2_wellord2(B,C) ) ) ).
fof(t45_card_1,axiom,
! [A,B] :
( r1_tarski(k1_zfmisc_1(A),B)
=> ( r2_hidden(k1_card_1(A),k1_card_1(B))
& ~ r2_wellord2(A,B) ) ) ).
fof(t46_card_1,axiom,
! [A] :
( r2_wellord2(A,k1_xboole_0)
<=> A = k1_xboole_0 ) ).
fof(t47_card_1,axiom,
k1_card_1(k1_xboole_0) = k1_xboole_0 ).
fof(t48_card_1,axiom,
! [A,B] :
( r2_wellord2(A,k1_tarski(B))
<=> ? [C] : A = k1_tarski(C) ) ).
fof(t49_card_1,axiom,
! [A,B] :
( k1_card_1(A) = k1_card_1(k1_tarski(B))
<=> ? [C] : A = k1_tarski(C) ) ).
fof(t50_card_1,axiom,
! [A] : k1_card_1(k1_tarski(A)) = k4_ordinal2 ).
fof(t51_card_1,axiom,
np__0 = k1_xboole_0 ).
fof(t52_card_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k1_ordinal1(A) = k1_nat_1(A,np__1) ) ).
fof(t53_card_1,axiom,
$true ).
fof(t54_card_1,axiom,
$true ).
fof(t55_card_1,axiom,
$true ).
fof(t56_card_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(A,B)
<=> r1_ordinal1(A,B) ) ) ) ).
fof(t57_card_1,axiom,
$true ).
fof(t58_card_1,axiom,
! [A,B,C,D] :
( ( r1_xboole_0(A,B)
& r1_xboole_0(C,D)
& r2_wellord2(A,C)
& r2_wellord2(B,D) )
=> r2_wellord2(k2_xboole_0(A,B),k2_xboole_0(C,D)) ) ).
fof(t59_card_1,axiom,
! [A,B,C] :
( ( r2_hidden(A,B)
& r2_hidden(C,B) )
=> r2_wellord2(k4_xboole_0(B,k1_tarski(A)),k4_xboole_0(B,k1_tarski(C))) ) ).
fof(t60_card_1,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( ( r1_tarski(A,k1_relat_1(B))
& v2_funct_1(B) )
=> r2_wellord2(A,k9_relat_1(B,A)) ) ) ).
fof(t61_card_1,axiom,
! [A,B,C,D] :
( ( r2_wellord2(A,B)
& r2_hidden(C,A)
& r2_hidden(D,B) )
=> r2_wellord2(k4_xboole_0(A,k1_tarski(C)),k4_xboole_0(B,k1_tarski(D))) ) ).
fof(t62_card_1,axiom,
$true ).
fof(t63_card_1,axiom,
$true ).
fof(t64_card_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_wellord2(A,B)
=> A = B ) ) ) ).
fof(t65_card_1,axiom,
! [A] :
( r2_hidden(A,k5_ordinal2)
=> v1_card_1(A) ) ).
fof(t66_card_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> A = k1_card_1(A) ) ).
fof(t67_card_1,axiom,
$true ).
fof(t68_card_1,axiom,
! [A,B] :
( ( r2_wellord2(A,B)
& v1_finset_1(A) )
=> v1_finset_1(B) ) ).
fof(t69_card_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( v1_finset_1(A)
& v1_finset_1(k1_card_1(A)) ) ) ).
fof(t70_card_1,axiom,
$true ).
fof(t71_card_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( k1_card_1(A) = k1_card_1(B)
=> A = B ) ) ) ).
fof(t72_card_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_tarski(k1_card_1(A),k1_card_1(B))
<=> r1_xreal_0(A,B) ) ) ) ).
fof(t73_card_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(k1_card_1(A),k1_card_1(B))
<=> ~ r1_xreal_0(B,A) ) ) ) ).
fof(t74_card_1,axiom,
! [A] :
~ ( v1_finset_1(A)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ r2_wellord2(A,B) ) ) ).
fof(t75_card_1,axiom,
$true ).
fof(t76_card_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k2_card_1(k1_card_1(A)) = k1_card_1(k1_nat_1(A,np__1)) ) ).
fof(d9_card_1,axiom,
$true ).
fof(d10_card_1,axiom,
$true ).
fof(d11_card_1,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( B = k4_card_1(A)
<=> k1_card_1(B) = k1_card_1(A) ) ) ) ).
fof(t77_card_1,axiom,
$true ).
fof(t78_card_1,axiom,
k4_card_1(k1_xboole_0) = np__0 ).
fof(t79_card_1,axiom,
! [A] : k4_card_1(k1_tarski(A)) = np__1 ).
fof(t80_card_1,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( v1_finset_1(B)
=> ( r1_tarski(A,B)
=> r1_xreal_0(k4_card_1(A),k4_card_1(B)) ) ) ) ).
fof(t81_card_1,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( v1_finset_1(B)
=> ( r2_wellord2(A,B)
=> k4_card_1(A) = k4_card_1(B) ) ) ) ).
fof(t82_card_1,axiom,
! [A] :
( v1_finset_1(A)
=> v1_finset_1(k2_card_1(A)) ) ).
fof(t83_card_1,axiom,
k3_card_1(np__0) = k5_ordinal2 ).
fof(t84_card_1,axiom,
k1_card_1(k5_ordinal2) = k5_ordinal2 ).
fof(t85_card_1,axiom,
v2_card_1(k1_card_1(k5_ordinal2)) ).
fof(t86_card_1,axiom,
! [A] :
( ( v1_finset_1(A)
& v1_card_1(A) )
=> ? [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
& A = k1_card_1(B) ) ) ).
fof(s1_card_1,axiom,
( ( p1_s1_card_1(k1_xboole_0)
& ! [A] :
( v1_card_1(A)
=> ( p1_s1_card_1(A)
=> p1_s1_card_1(k2_card_1(A)) ) )
& ! [A] :
( v1_card_1(A)
=> ( ( v2_card_1(A)
& ! [B] :
( v1_card_1(B)
=> ( r2_hidden(B,A)
=> p1_s1_card_1(B) ) ) )
=> ( A = k1_xboole_0
| p1_s1_card_1(A) ) ) ) )
=> ! [A] :
( v1_card_1(A)
=> p1_s1_card_1(A) ) ) ).
fof(s2_card_1,axiom,
( ! [A] :
( v1_card_1(A)
=> ( ! [B] :
( v1_card_1(B)
=> ( r2_hidden(B,A)
=> p1_s2_card_1(B) ) )
=> p1_s2_card_1(A) ) )
=> ! [A] :
( v1_card_1(A)
=> p1_s2_card_1(A) ) ) ).
fof(dt_k1_card_1,axiom,
! [A] : v1_card_1(k1_card_1(A)) ).
fof(dt_k2_card_1,axiom,
! [A] : v1_card_1(k2_card_1(A)) ).
fof(dt_k3_card_1,axiom,
$true ).
fof(dt_k4_card_1,axiom,
! [A] :
( v1_finset_1(A)
=> m2_subset_1(k4_card_1(A),k1_numbers,k5_numbers) ) ).
fof(redefinition_k4_card_1,axiom,
! [A] :
( v1_finset_1(A)
=> k4_card_1(A) = k1_card_1(A) ) ).
%------------------------------------------------------------------------------