SET007 Axioms: SET007+136.ax
%------------------------------------------------------------------------------
% File : SET007+136 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Mostowski's Fundamental Operations - Part II
% 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_fund2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 11 ( 0 unt; 0 def)
% Number of atoms : 123 ( 10 equ)
% Maximal formula atoms : 22 ( 11 avg)
% Number of connectives : 126 ( 14 ~; 2 |; 57 &)
% ( 3 <=>; 50 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 11 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 25 ( 24 usr; 0 prp; 1-3 aty)
% Number of functors : 34 ( 34 usr; 9 con; 0-5 aty)
% Number of variables : 46 ( 44 !; 2 ?)
% 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(d2_zf_fund2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_zf_fund2(A)
<=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k1_zf_lang,C)
& m2_relset_1(D,k1_zf_lang,C) )
=> ( r2_hidden(C,A)
=> r2_hidden(k1_zf_fund2(B,C,D),A) ) ) ) ) ) ) ).
fof(t1_zf_fund2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k1_zf_lang,A)
& m2_relset_1(C,k1_zf_lang,A) )
=> ( v1_ordinal1(A)
=> k1_zf_fund2(k8_zf_lang(k2_zf_lang(np__2),k11_zf_lang(k5_zf_lang(k2_zf_lang(np__2),k2_zf_lang(np__0)),k5_zf_lang(k2_zf_lang(np__2),k2_zf_lang(np__1)))),A,k1_zf_lang1(A,C,k2_zf_lang(np__1),B)) = k3_xboole_0(A,k1_zfmisc_1(B)) ) ) ) ) ).
fof(t2_zf_fund2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ( v2_relat_1(B)
& v1_zf_refle(B,A)
& m1_ordinal1(B,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)
=> ( r2_hidden(k5_zf_refle(A,B,C),k4_zf_refle(A,B))
& v1_ordinal1(k5_zf_refle(A,B,C)) ) )
& v1_zf_fund2(k4_zf_refle(A,B)) )
=> r2_zf_model(k4_zf_refle(A,B),k10_zf_model) ) ) ) ).
fof(t3_zf_fund2,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] :
( m1_ordinal4(C,A)
=> ( r2_hidden(k5_zf_refle(A,B,C),k4_zf_refle(A,B))
& v1_ordinal1(k5_zf_refle(A,B,C)) ) )
& v1_zf_fund2(k4_zf_refle(A,B)) )
=> ! [C] :
( ( v1_zf_lang(C)
& m2_finseq_1(C,k5_numbers) )
=> ( r1_xboole_0(k1_enumset1(k2_zf_lang(np__0),k2_zf_lang(np__1),k2_zf_lang(np__2)),k6_zf_fund1(C))
=> r2_zf_model(k4_zf_refle(A,B),k11_zf_model(C)) ) ) ) ) ) ).
fof(t5_zf_fund2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ( ( v8_zf_fund1(B,A)
& v1_ordinal1(B) )
=> v1_zf_fund2(B) ) ) ) ).
fof(t6_zf_fund2,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] :
( m1_ordinal4(C,A)
=> ( r2_hidden(k5_zf_refle(A,B,C),k4_zf_refle(A,B))
& v1_ordinal1(k5_zf_refle(A,B,C)) ) )
& v8_zf_fund1(k4_zf_refle(A,B),A) )
=> v1_zf_model(k4_zf_refle(A,B)) ) ) ) ).
fof(dt_k1_zf_fund2,axiom,
! [A,B,C] :
( ( v1_zf_lang(A)
& m1_finseq_1(A,k5_numbers)
& ~ v1_xboole_0(B)
& v1_funct_1(C)
& v1_funct_2(C,k1_zf_lang,B)
& m1_relset_1(C,k1_zf_lang,B) )
=> m1_subset_1(k1_zf_fund2(A,B,C),k1_zfmisc_1(B)) ) ).
fof(d1_zf_fund2,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k1_zf_lang,B)
& m2_relset_1(C,k1_zf_lang,B) )
=> ( ( r2_hidden(k2_zf_lang(np__0),k6_zf_fund1(A))
=> k1_zf_fund2(A,B,C) = a_3_0_zf_fund2(A,B,C) )
& ( ~ r2_hidden(k2_zf_lang(np__0),k6_zf_fund1(A))
=> k1_zf_fund2(A,B,C) = k1_xboole_0 ) ) ) ) ) ).
fof(t4_zf_fund2,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k1_zf_lang,B)
& m2_relset_1(C,k1_zf_lang,B) )
=> k1_zf_fund2(A,B,C) = a_3_1_zf_fund2(A,B,C) ) ) ) ).
fof(fraenkel_a_3_0_zf_fund2,axiom,
! [A,B,C,D] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers)
& ~ v1_xboole_0(C)
& v1_funct_1(D)
& v1_funct_2(D,k1_zf_lang,C)
& m2_relset_1(D,k1_zf_lang,C) )
=> ( r2_hidden(A,a_3_0_zf_fund2(B,C,D))
<=> ? [E] :
( m1_subset_1(E,C)
& A = E
& r1_zf_model(C,k1_zf_lang1(C,D,k2_zf_lang(np__0),E),B) ) ) ) ).
fof(fraenkel_a_3_1_zf_fund2,axiom,
! [A,B,C,D] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers)
& ~ v1_xboole_0(C)
& v1_funct_1(D)
& v1_funct_2(D,k1_zf_lang,C)
& m2_relset_1(D,k1_zf_lang,C) )
=> ( r2_hidden(A,a_3_1_zf_fund2(B,C,D))
<=> ? [E] :
( m1_subset_1(E,C)
& A = E
& r2_hidden(k2_xboole_0(k1_tarski(k4_tarski(k1_xboole_0,E)),k7_relat_1(k7_funct_2(k5_ordinal2,k1_zf_lang,C,k3_zf_fund1,D),k4_xboole_0(k5_zf_fund1(k6_zf_fund1(B)),k1_tarski(k1_xboole_0)))),k10_zf_fund1(B,C)) ) ) ) ).
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