SET007 Axioms: SET007+19.ax
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% File : SET007+19 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Zermelo Theorem and Axiom of Choice
% 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 : wellord2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 39 ( 14 unt; 0 def)
% Number of atoms : 128 ( 19 equ)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 107 ( 18 ~; 1 |; 38 &)
% ( 6 <=>; 44 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 24 ( 22 usr; 1 prp; 0-3 aty)
% Number of functors : 17 ( 17 usr; 6 con; 0-2 aty)
% Number of variables : 72 ( 66 !; 6 ?)
% 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.
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fof(d1_wellord2,axiom,
! [A,B] :
( v1_relat_1(B)
=> ( B = k1_wellord2(A)
<=> ( k3_relat_1(B) = A
& ! [C,D] :
( ( r2_hidden(C,A)
& r2_hidden(D,A) )
=> ( r2_hidden(k4_tarski(C,D),B)
<=> r1_tarski(C,D) ) ) ) ) ) ).
fof(t1_wellord2,axiom,
$true ).
fof(t2_wellord2,axiom,
! [A] : v1_relat_2(k1_wellord2(A)) ).
fof(t3_wellord2,axiom,
! [A] : v8_relat_2(k1_wellord2(A)) ).
fof(t4_wellord2,axiom,
! [A] :
( v3_ordinal1(A)
=> v6_relat_2(k1_wellord2(A)) ) ).
fof(t5_wellord2,axiom,
! [A] : v4_relat_2(k1_wellord2(A)) ).
fof(t6_wellord2,axiom,
! [A] :
( v3_ordinal1(A)
=> v1_wellord1(k1_wellord2(A)) ) ).
fof(t7_wellord2,axiom,
! [A] :
( v3_ordinal1(A)
=> v2_wellord1(k1_wellord2(A)) ) ).
fof(t8_wellord2,axiom,
! [A,B] :
( r1_tarski(A,B)
=> k2_wellord1(k1_wellord2(B),A) = k1_wellord2(A) ) ).
fof(t9_wellord2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( r1_tarski(B,A)
=> v2_wellord1(k1_wellord2(B)) ) ) ).
fof(t10_wellord2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( r2_hidden(A,B)
=> A = k1_wellord1(k1_wellord2(B),A) ) ) ) ).
fof(t11_wellord2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( r4_wellord1(k1_wellord2(A),k1_wellord2(B))
=> A = B ) ) ) ).
fof(t12_wellord2,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ! [C] :
( v3_ordinal1(C)
=> ( ( r4_wellord1(A,k1_wellord2(B))
& r4_wellord1(A,k1_wellord2(C)) )
=> B = C ) ) ) ) ).
fof(t13_wellord2,axiom,
! [A] :
( v1_relat_1(A)
=> ~ ( v2_wellord1(A)
& ! [B] :
~ ( r2_hidden(B,k3_relat_1(A))
& ! [C] :
( v3_ordinal1(C)
=> ~ r4_wellord1(k2_wellord1(A,k1_wellord1(A,B)),k1_wellord2(C)) ) )
& ! [B] :
( v3_ordinal1(B)
=> ~ r4_wellord1(A,k1_wellord2(B)) ) ) ) ).
fof(t14_wellord2,axiom,
! [A] :
( v1_relat_1(A)
=> ~ ( v2_wellord1(A)
& ! [B] :
( v3_ordinal1(B)
=> ~ r4_wellord1(A,k1_wellord2(B)) ) ) ) ).
fof(d2_wellord2,axiom,
! [A] :
( v1_relat_1(A)
=> ( v2_wellord1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( B = k2_wellord2(A)
<=> r4_wellord1(A,k1_wellord2(B)) ) ) ) ) ).
fof(d3_wellord2,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v1_relat_1(B)
=> ( r1_wellord2(A,B)
<=> A = k2_wellord2(B) ) ) ) ).
fof(t15_wellord2,axiom,
$true ).
fof(t16_wellord2,axiom,
$true ).
fof(t17_wellord2,axiom,
! [A,B] :
( v3_ordinal1(B)
=> ( r1_tarski(A,B)
=> r1_ordinal1(k2_wellord2(k1_wellord2(A)),B) ) ) ).
fof(d4_wellord2,axiom,
! [A,B] :
( r2_wellord2(A,B)
<=> ? [C] :
( v1_relat_1(C)
& v1_funct_1(C)
& v2_funct_1(C)
& k1_relat_1(C) = A
& k2_relat_1(C) = B ) ) ).
fof(t18_wellord2,axiom,
$true ).
fof(t19_wellord2,axiom,
$true ).
fof(t20_wellord2,axiom,
$true ).
fof(t21_wellord2,axiom,
$true ).
fof(t22_wellord2,axiom,
! [A,B,C] :
( ( r2_wellord2(A,B)
& r2_wellord2(B,C) )
=> r2_wellord2(A,C) ) ).
fof(t23_wellord2,axiom,
$true ).
fof(t24_wellord2,axiom,
$true ).
fof(t25_wellord2,axiom,
! [A,B] :
( v1_relat_1(B)
=> ( r2_wellord1(B,A)
=> ( k3_relat_1(k2_wellord1(B,A)) = A
& v2_wellord1(k2_wellord1(B,A)) ) ) ) ).
fof(t26_wellord2,axiom,
! [A] :
? [B] :
( v1_relat_1(B)
& r2_wellord1(B,A) ) ).
fof(t27_wellord2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ~ ( ! [B] :
~ ( r2_hidden(B,A)
& B = k1_xboole_0 )
& ! [B,C] :
( ( r2_hidden(B,A)
& r2_hidden(C,A) )
=> ( B = C
| r1_xboole_0(B,C) ) )
& ! [B] :
? [C] :
( r2_hidden(C,A)
& ! [D] : k3_xboole_0(B,C) != k1_tarski(D) ) ) ) ).
fof(t28_wellord2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ~ ( ! [B] :
~ ( r2_hidden(B,A)
& B = k1_xboole_0 )
& ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ~ ( k1_relat_1(B) = A
& ! [C] :
( r2_hidden(C,A)
=> r2_hidden(k1_funct_1(B,C),C) ) ) ) ) ) ).
fof(s1_wellord2,axiom,
( ! [A] :
~ ( r2_hidden(A,f1_s1_wellord2)
& ! [B] :
~ ( r2_hidden(B,f2_s1_wellord2)
& p1_s1_wellord2(A,B) ) )
=> ? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& k1_relat_1(A) = f1_s1_wellord2
& r1_tarski(k2_relat_1(A),f2_s1_wellord2)
& ! [B] :
( r2_hidden(B,f1_s1_wellord2)
=> p1_s1_wellord2(B,k1_funct_1(A,B)) ) ) ) ).
fof(s2_wellord2,axiom,
( ! [A] :
~ ( r2_hidden(A,f1_s2_wellord2)
& ! [B,C] :
~ ( r2_hidden(B,f2_s2_wellord2)
& r2_hidden(C,f3_s2_wellord2)
& p1_s2_wellord2(A,B,C) ) )
=> ? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& ? [B] :
( v1_relat_1(B)
& v1_funct_1(B)
& k1_relat_1(A) = f1_s2_wellord2
& k1_relat_1(B) = f1_s2_wellord2
& ! [C] :
( r2_hidden(C,f1_s2_wellord2)
=> p1_s2_wellord2(C,k1_funct_1(A,C),k1_funct_1(B,C)) ) ) ) ) ).
fof(symmetry_r2_wellord2,axiom,
! [A,B] :
( r2_wellord2(A,B)
=> r2_wellord2(B,A) ) ).
fof(reflexivity_r2_wellord2,axiom,
! [A,B] : r2_wellord2(A,A) ).
fof(redefinition_r2_wellord2,axiom,
! [A,B] :
( r2_wellord2(A,B)
<=> r2_tarski(A,B) ) ).
fof(dt_k1_wellord2,axiom,
! [A] : v1_relat_1(k1_wellord2(A)) ).
fof(dt_k2_wellord2,axiom,
! [A] :
( v1_relat_1(A)
=> v3_ordinal1(k2_wellord2(A)) ) ).
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