SET007 Axioms: SET007+74.ax
%------------------------------------------------------------------------------
% File : SET007+74 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Reflection 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 : zf_refle [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 52 ( 14 unt; 0 def)
% Number of atoms : 278 ( 26 equ)
% Maximal formula atoms : 27 ( 5 avg)
% Number of connectives : 255 ( 29 ~; 4 |; 126 &)
% ( 4 <=>; 92 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 6 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 33 ( 31 usr; 1 prp; 0-3 aty)
% Number of functors : 46 ( 46 usr; 18 con; 0-4 aty)
% Number of variables : 106 ( 94 !; 12 ?)
% 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_zf_refle,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ? [B] :
( m1_subset_1(B,A)
& ~ v1_xboole_0(B) ) ) ).
fof(rc2_zf_refle,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ? [B] :
( m1_ordinal1(B,A)
& v1_relat_1(B)
& v2_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_zf_refle(B,A) ) ) ).
fof(t1_zf_refle,axiom,
$true ).
fof(t2_zf_refle,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> r2_zf_model(A,k7_zf_model) ) ).
fof(t3_zf_refle,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> r2_zf_model(A,k8_zf_model) ) ).
fof(t4_zf_refle,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ( r2_hidden(k5_ordinal2,A)
=> r2_zf_model(A,k9_zf_model) ) ) ).
fof(t5_zf_refle,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> r2_zf_model(A,k10_zf_model) ) ).
fof(t6_zf_refle,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ( r1_xboole_0(k1_enumset1(k2_zf_lang(np__0),k2_zf_lang(np__1),k2_zf_lang(np__2)),k2_zf_model(B))
=> r2_zf_model(A,k11_zf_model(B)) ) ) ) ).
fof(t7_zf_refle,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ( r2_hidden(k5_ordinal2,A)
=> v1_zf_model(A) ) ) ).
fof(d1_zf_refle,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( v3_ordinal1(B)
=> ( r1_tarski(A,B)
<=> ! [C] :
( v3_ordinal1(C)
=> ( r2_hidden(C,A)
=> r2_hidden(C,B) ) ) ) ) ) ).
fof(t8_zf_refle,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( m1_ordinal4(B,A)
<=> r2_hidden(B,k2_ordinal2(A)) ) ) ).
fof(d2_zf_refle,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_classes2(B) )
=> ! [C] :
( m1_ordinal4(C,B)
=> k1_zf_refle(A,B,C) = k3_card_3(k8_relat_1(B,k7_relat_1(A,k4_classes1(C)))) ) ) ) ).
fof(t9_zf_refle,axiom,
$true ).
fof(t10_zf_refle,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A) )
=> ! [B] :
( v3_ordinal1(B)
=> ( v1_relat_1(k7_relat_1(A,k4_classes1(B)))
& v1_funct_1(k7_relat_1(A,k4_classes1(B)))
& v5_ordinal1(k7_relat_1(A,k4_classes1(B))) ) ) ) ).
fof(t11_zf_refle,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) )
=> ! [B] :
( v3_ordinal1(B)
=> ( v1_relat_1(k7_relat_1(A,k4_classes1(B)))
& v1_funct_1(k7_relat_1(A,k4_classes1(B)))
& v5_ordinal1(k7_relat_1(A,k4_classes1(B)))
& v1_ordinal2(k7_relat_1(A,k4_classes1(B))) ) ) ) ).
fof(t12_zf_refle,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A) )
=> v3_ordinal1(k3_card_3(A)) ) ).
fof(t13_zf_refle,axiom,
! [A,B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_ordinal2(B) )
=> v3_ordinal1(k3_card_3(k8_relat_1(A,B))) ) ).
fof(t14_zf_refle,axiom,
! [A] :
( v3_ordinal1(A)
=> k2_ordinal2(k4_classes1(A)) = A ) ).
fof(t15_zf_refle,axiom,
! [A] :
( v3_ordinal1(A)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v5_ordinal1(B)
& v1_ordinal2(B) )
=> k7_relat_1(B,k4_classes1(A)) = k2_ordinal1(B,A) ) ) ).
fof(t16_zf_refle,axiom,
$true ).
fof(t17_zf_refle,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( m1_ordinal4(B,A)
=> ! [C] :
( m2_ordinal4(C,A)
=> ( k2_zf_refle(C,A,B) = k3_card_3(k2_ordinal1(C,B))
& k2_zf_refle(k2_ordinal1(C,B),A,B) = k3_card_3(k2_ordinal1(C,B)) ) ) ) ) ).
fof(d3_zf_refle,axiom,
$true ).
fof(d4_zf_refle,axiom,
$true ).
fof(d5_zf_refle,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( m1_ordinal1(B,A)
=> ( v1_zf_refle(B,A)
<=> k1_relat_1(B) = k2_ordinal2(A) ) ) ) ).
fof(t18_zf_refle,axiom,
$true ).
fof(t19_zf_refle,axiom,
$true ).
fof(t20_zf_refle,axiom,
$true ).
fof(t21_zf_refle,axiom,
$true ).
fof(t22_zf_refle,axiom,
$true ).
fof(t23_zf_refle,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( m1_ordinal4(B,A)
=> ! [C] :
( ( v2_relat_1(C)
& v1_zf_refle(C,A)
& m1_ordinal1(C,A) )
=> r2_hidden(B,k1_relat_1(C)) ) ) ) ).
fof(t24_zf_refle,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( m1_ordinal4(B,A)
=> ! [C] :
( ( v2_relat_1(C)
& v1_zf_refle(C,A)
& m1_ordinal1(C,A) )
=> r1_tarski(k5_zf_refle(A,C,B),k4_zf_refle(A,C)) ) ) ) ).
fof(t25_zf_refle,axiom,
r2_tarski(k5_numbers,k1_zf_lang) ).
fof(t26_zf_refle,axiom,
$true ).
fof(t27_zf_refle,axiom,
! [A] : r1_tarski(k7_ordinal2(A),k1_ordinal1(k3_tarski(k2_ordinal2(A)))) ).
fof(t28_zf_refle,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( r2_hidden(B,A)
=> r2_hidden(k7_ordinal2(B),A) ) ) ).
fof(t29_zf_refle,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ( v2_relat_1(B)
& v1_zf_refle(B,A)
& m1_ordinal1(B,A) )
=> ( ( r2_hidden(k5_ordinal2,A)
& ! [C] :
( m1_ordinal4(C,A)
=> ! [D] :
( m1_ordinal4(D,A)
=> ( r2_hidden(C,D)
=> r1_tarski(k5_zf_refle(A,B,C),k5_zf_refle(A,B,D)) ) ) )
& ! [C] :
( m1_ordinal4(C,A)
=> ( v4_ordinal1(C)
=> ( C = k1_xboole_0
| k5_zf_refle(A,B,C) = k3_card_3(k2_ordinal1(B,C)) ) ) ) )
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ? [D] :
( m2_ordinal4(D,A)
& v2_ordinal2(D)
& v3_ordinal2(D)
& ! [E] :
( m1_ordinal4(E,A)
=> ( k4_ordinal4(A,D,E) = E
=> ( k1_xboole_0 = E
| ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k1_zf_lang,k5_zf_refle(A,B,E))
& m2_relset_1(F,k1_zf_lang,k5_zf_refle(A,B,E)) )
=> ( r1_zf_model(k4_zf_refle(A,B),k2_zf_lang1(k1_zf_lang,k5_zf_refle(A,B,E),k4_zf_refle(A,B),F),C)
<=> r1_zf_model(k5_zf_refle(A,B,E),F,C) ) ) ) ) ) ) ) ) ) ) ).
fof(s1_zf_refle,axiom,
( ! [A] :
( m1_subset_1(A,f1_s1_zf_refle)
=> ? [B] :
( v3_ordinal1(B)
& p1_s1_zf_refle(A,B) ) )
=> ? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& k1_relat_1(A) = f1_s1_zf_refle
& ! [B] :
( m1_subset_1(B,f1_s1_zf_refle)
=> ? [C] :
( v3_ordinal1(C)
& C = k1_funct_1(A,B)
& p1_s1_zf_refle(B,C)
& ! [D] :
( v3_ordinal1(D)
=> ( p1_s1_zf_refle(B,D)
=> r1_tarski(C,D) ) ) ) ) ) ) ).
fof(s2_zf_refle,axiom,
( ! [A] :
( m1_subset_1(A,f2_s2_zf_refle)
=> ? [B] :
( m1_ordinal4(B,f1_s2_zf_refle)
& p1_s2_zf_refle(A,B) ) )
=> ? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& k1_relat_1(A) = f2_s2_zf_refle
& ! [B] :
( m1_subset_1(B,f2_s2_zf_refle)
=> ? [C] :
( m1_ordinal4(C,f1_s2_zf_refle)
& C = k1_funct_1(A,B)
& p1_s2_zf_refle(B,C)
& ! [D] :
( m1_ordinal4(D,f1_s2_zf_refle)
=> ( p1_s2_zf_refle(B,D)
=> r1_tarski(C,D) ) ) ) ) ) ) ).
fof(s3_zf_refle,axiom,
( ( ! [A] :
( m1_ordinal4(A,f1_s3_zf_refle)
=> ! [B] :
( m1_ordinal4(B,f1_s3_zf_refle)
=> ! [C] :
( m1_ordinal4(C,f1_s3_zf_refle)
=> ( ( p1_s3_zf_refle(A,B)
& p1_s3_zf_refle(A,C) )
=> B = C ) ) ) )
& ! [A] :
( m1_ordinal4(A,f1_s3_zf_refle)
=> ? [B] :
( m1_ordinal4(B,f1_s3_zf_refle)
& p1_s3_zf_refle(A,B) ) ) )
=> ? [A] :
( m2_ordinal4(A,f1_s3_zf_refle)
& ! [B] :
( m1_ordinal4(B,f1_s3_zf_refle)
=> p1_s3_zf_refle(B,k4_ordinal4(f1_s3_zf_refle,A,B)) ) ) ) ).
fof(s4_zf_refle,axiom,
? [A] :
( m2_ordinal4(A,f1_s4_zf_refle)
& k4_ordinal4(f1_s4_zf_refle,A,k2_ordinal4(f1_s4_zf_refle)) = f2_s4_zf_refle
& ! [B] :
( m1_ordinal4(B,f1_s4_zf_refle)
=> k4_ordinal4(f1_s4_zf_refle,A,k6_ordinal4(f1_s4_zf_refle,B)) = f3_s4_zf_refle(B,k4_ordinal4(f1_s4_zf_refle,A,B)) )
& ! [B] :
( m1_ordinal4(B,f1_s4_zf_refle)
=> ( v4_ordinal1(B)
=> ( B = k2_ordinal4(f1_s4_zf_refle)
| k4_ordinal4(f1_s4_zf_refle,A,B) = f4_s4_zf_refle(B,k2_ordinal1(A,B)) ) ) ) ) ).
fof(s5_zf_refle,axiom,
( ( p1_s5_zf_refle(k2_ordinal4(f1_s5_zf_refle))
& ! [A] :
( m1_ordinal4(A,f1_s5_zf_refle)
=> ( p1_s5_zf_refle(A)
=> p1_s5_zf_refle(k6_ordinal4(f1_s5_zf_refle,A)) ) )
& ! [A] :
( m1_ordinal4(A,f1_s5_zf_refle)
=> ( ( v4_ordinal1(A)
& ! [B] :
( m1_ordinal4(B,f1_s5_zf_refle)
=> ( r2_hidden(B,A)
=> p1_s5_zf_refle(B) ) ) )
=> ( A = k2_ordinal4(f1_s5_zf_refle)
| p1_s5_zf_refle(A) ) ) ) )
=> ! [A] :
( m1_ordinal4(A,f1_s5_zf_refle)
=> p1_s5_zf_refle(A) ) ) ).
fof(dt_k1_zf_refle,axiom,
$true ).
fof(dt_k2_zf_refle,axiom,
! [A,B,C] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A)
& ~ v1_xboole_0(B)
& v1_classes2(B)
& m1_ordinal4(C,B) )
=> m1_ordinal4(k2_zf_refle(A,B,C),B) ) ).
fof(redefinition_k2_zf_refle,axiom,
! [A,B,C] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v5_ordinal1(A)
& v1_ordinal2(A)
& ~ v1_xboole_0(B)
& v1_classes2(B)
& m1_ordinal4(C,B) )
=> k2_zf_refle(A,B,C) = k1_zf_refle(A,B,C) ) ).
fof(dt_k3_zf_refle,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_ordinal4(B,A)
& m1_ordinal4(C,A) )
=> m1_ordinal4(k3_zf_refle(A,B,C),A) ) ).
fof(commutativity_k3_zf_refle,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_ordinal4(B,A)
& m1_ordinal4(C,A) )
=> k3_zf_refle(A,B,C) = k3_zf_refle(A,C,B) ) ).
fof(idempotence_k3_zf_refle,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_ordinal4(B,A)
& m1_ordinal4(C,A) )
=> k3_zf_refle(A,B,B) = B ) ).
fof(redefinition_k3_zf_refle,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_ordinal4(B,A)
& m1_ordinal4(C,A) )
=> k3_zf_refle(A,B,C) = k2_xboole_0(B,C) ) ).
fof(dt_k4_zf_refle,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& v2_relat_1(B)
& v1_zf_refle(B,A)
& m1_ordinal1(B,A) )
=> ( ~ v1_xboole_0(k4_zf_refle(A,B))
& m1_subset_1(k4_zf_refle(A,B),k1_zfmisc_1(A)) ) ) ).
fof(redefinition_k4_zf_refle,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& v2_relat_1(B)
& v1_zf_refle(B,A)
& m1_ordinal1(B,A) )
=> k4_zf_refle(A,B) = k3_card_3(B) ) ).
fof(dt_k5_zf_refle,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& v2_relat_1(B)
& v1_zf_refle(B,A)
& m1_ordinal1(B,A)
& m1_ordinal4(C,A) )
=> ( ~ v1_xboole_0(k5_zf_refle(A,B,C))
& m1_subset_1(k5_zf_refle(A,B,C),A) ) ) ).
fof(redefinition_k5_zf_refle,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& v2_relat_1(B)
& v1_zf_refle(B,A)
& m1_ordinal1(B,A)
& m1_ordinal4(C,A) )
=> k5_zf_refle(A,B,C) = k1_funct_1(B,C) ) ).
%------------------------------------------------------------------------------