SET007 Axioms: SET007+568.ax
%------------------------------------------------------------------------------
% File : SET007+568 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : A Theory of Partitions. Part I
% 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 : partit1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 60 ( 7 unt; 0 def)
% Number of atoms : 314 ( 42 equ)
% Maximal formula atoms : 22 ( 5 avg)
% Number of connectives : 316 ( 62 ~; 3 |; 89 &)
% ( 12 <=>; 150 =>; 0 <=; 0 <~>)
% Maximal formula depth : 33 ( 8 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 21 ( 19 usr; 1 prp; 0-4 aty)
% Number of functors : 30 ( 30 usr; 4 con; 0-3 aty)
% Number of variables : 183 ( 175 !; 8 ?)
% 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(fc1_partit1,axiom,
! [A] : ~ v1_xboole_0(k1_partit1(A)) ).
fof(t1_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C,D] :
( ( r2_hidden(C,B)
& r2_hidden(D,B)
& r1_tarski(C,D) )
=> C = D ) ) ) ).
fof(t2_partit1,axiom,
$true ).
fof(t3_partit1,axiom,
! [A] : k3_tarski(k4_xboole_0(A,k1_tarski(k1_xboole_0))) = k3_tarski(A) ).
fof(t4_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( m1_eqrel_1(C,A)
=> ( ( r1_setfam_1(C,B)
& r1_setfam_1(B,C) )
=> r1_tarski(C,B) ) ) ) ) ).
fof(t5_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( m1_eqrel_1(C,A)
=> ( ( r1_setfam_1(C,B)
& r1_setfam_1(B,C) )
=> B = C ) ) ) ) ).
fof(t6_partit1,axiom,
$true ).
fof(t7_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( m1_eqrel_1(C,A)
=> ( r1_setfam_1(C,B)
=> r2_setfam_1(C,B) ) ) ) ) ).
fof(d1_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( r1_partit1(A,B,C)
<=> ? [D] :
( r1_tarski(D,B)
& D != k1_xboole_0
& C = k3_tarski(D) ) ) ) ) ).
fof(d2_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( m1_eqrel_1(C,A)
=> ! [D] :
( r2_partit1(A,B,C,D)
<=> ( r1_partit1(A,B,D)
& r1_partit1(A,C,D)
& ! [E] :
( ( r1_tarski(E,D)
& r1_partit1(A,B,E)
& r1_partit1(A,C,E) )
=> E = D ) ) ) ) ) ) ).
fof(t8_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( m1_eqrel_1(C,A)
=> ( r1_setfam_1(C,B)
=> ! [D] :
( r2_hidden(D,B)
=> r1_partit1(A,C,D) ) ) ) ) ) ).
fof(t9_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> r1_partit1(A,B,A) ) ) ).
fof(t10_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( ! [D] :
( r2_hidden(D,C)
=> r1_partit1(A,B,D) )
=> ( k8_setfam_1(A,C) = k1_xboole_0
| r1_partit1(A,B,k8_setfam_1(A,C)) ) ) ) ) ) ).
fof(t11_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( ( r1_partit1(A,B,C)
& r1_partit1(A,B,D) )
=> ( r1_xboole_0(C,D)
| r1_partit1(A,B,k5_subset_1(A,C,D)) ) ) ) ) ) ) ).
fof(t12_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r1_partit1(A,B,C)
=> ( C = A
| r1_partit1(A,B,k3_subset_1(A,C)) ) ) ) ) ) ).
fof(t13_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( m1_eqrel_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ? [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
& r2_hidden(D,E)
& r2_partit1(A,B,C,E) ) ) ) ) ) ).
fof(t14_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
& r2_hidden(C,D)
& r2_hidden(D,B) ) ) ) ) ).
fof(d3_partit1,axiom,
! [A,B] :
( B = k1_partit1(A)
<=> ! [C] :
( r2_hidden(C,B)
<=> m1_eqrel_1(C,A) ) ) ).
fof(d4_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( m1_eqrel_1(C,A)
=> k2_partit1(A,B,C) = k4_xboole_0(k3_setfam_1(B,C),k1_tarski(k1_xboole_0)) ) ) ) ).
fof(t15_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> k2_partit1(A,B,B) = B ) ) ).
fof(t16_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( m1_eqrel_1(C,A)
=> ! [D] :
( m1_eqrel_1(D,A)
=> k2_partit1(A,k2_partit1(A,B,C),D) = k2_partit1(A,B,k2_partit1(A,C,D)) ) ) ) ) ).
fof(t17_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( m1_eqrel_1(C,A)
=> r1_setfam_1(k2_partit1(A,B,C),B) ) ) ) ).
fof(d5_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( m1_eqrel_1(C,A)
=> ! [D] :
( m1_eqrel_1(D,A)
=> ( D = k3_partit1(A,B,C)
<=> ! [E] :
( r2_hidden(E,D)
<=> r2_partit1(A,B,C,E) ) ) ) ) ) ) ).
fof(t18_partit1,axiom,
$true ).
fof(t19_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( m1_eqrel_1(C,A)
=> r1_setfam_1(B,k3_partit1(A,B,C)) ) ) ) ).
fof(t20_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> k3_partit1(A,B,B) = B ) ) ).
fof(t21_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C,D,E] :
( m1_eqrel_1(E,A)
=> ! [F] :
( m1_eqrel_1(F,A)
=> ( ( r1_setfam_1(E,F)
& r2_hidden(B,F)
& r2_hidden(C,E)
& r2_hidden(D,B)
& r2_hidden(D,C) )
=> r1_tarski(C,B) ) ) ) ) ).
fof(t22_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C,D,E] :
( m1_eqrel_1(E,A)
=> ! [F] :
( m1_eqrel_1(F,A)
=> ( ( r2_hidden(B,k3_partit1(A,E,F))
& r2_hidden(C,E)
& r2_hidden(D,B)
& r2_hidden(D,C) )
=> r1_tarski(C,B) ) ) ) ) ).
fof(t23_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ? [C] :
( v3_relat_2(C)
& v8_relat_2(C)
& v1_partfun1(C,A,A)
& m2_relset_1(C,A,A)
& ! [D,E] :
( r2_hidden(k4_tarski(D,E),C)
<=> ? [F] :
( m1_subset_1(F,k1_zfmisc_1(A))
& r2_hidden(F,B)
& r2_hidden(D,F)
& r2_hidden(E,F) ) ) ) ) ) ).
fof(d6_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( ( v3_relat_2(C)
& v8_relat_2(C)
& v1_partfun1(C,A,A)
& m2_relset_1(C,A,A) )
=> ( C = k4_partit1(A,B)
<=> ! [D,E] :
( r2_hidden(k4_tarski(D,E),C)
<=> ? [F] :
( m1_subset_1(F,k1_zfmisc_1(A))
& r2_hidden(F,B)
& r2_hidden(D,F)
& r2_hidden(E,F) ) ) ) ) ) ) ).
fof(d7_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( B = k5_partit1(A)
<=> ( k1_relat_1(B) = k1_partit1(A)
& ! [C] :
~ ( r2_hidden(C,k1_partit1(A))
& ! [D] :
( m1_eqrel_1(D,A)
=> ~ ( D = C
& k1_funct_1(B,C) = k4_partit1(A,D) ) ) ) ) ) ) ) ).
fof(t24_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( m1_eqrel_1(C,A)
=> ( r1_setfam_1(B,C)
<=> r1_tarski(k4_partit1(A,B),k4_partit1(A,C)) ) ) ) ) ).
fof(t25_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( m1_eqrel_1(C,A)
=> ! [D,E,F,G] :
( m2_finseq_1(G,A)
=> ( ( r1_tarski(D,A)
& r2_hidden(E,D)
& k1_funct_1(G,np__1) = E
& k1_funct_1(G,k3_finseq_1(G)) = F
& r1_xreal_0(np__1,k3_finseq_1(G))
& ! [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,H)
& ~ r1_xreal_0(k3_finseq_1(G),H)
& ! [I,J,K] :
~ ( r2_hidden(I,B)
& r2_hidden(J,C)
& r2_hidden(k1_funct_1(G,H),I)
& r2_hidden(K,I)
& r2_hidden(K,J)
& r2_hidden(k1_funct_1(G,k1_nat_1(H,np__1)),J) ) ) )
& r1_partit1(A,B,D)
& r1_partit1(A,C,D) )
=> r2_hidden(F,D) ) ) ) ) ) ).
fof(t26_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v3_relat_2(B)
& v8_relat_2(B)
& v1_partfun1(B,A,A)
& m2_relset_1(B,A,A) )
=> ! [C] :
( ( v3_relat_2(C)
& v8_relat_2(C)
& v1_partfun1(C,A,A)
& m2_relset_1(C,A,A) )
=> ! [D] :
( m2_finseq_1(D,A)
=> ! [E,F] :
( ( r2_hidden(E,A)
& r2_hidden(F,A)
& k1_funct_1(D,np__1) = E
& k1_funct_1(D,k3_finseq_1(D)) = F
& r1_xreal_0(np__1,k3_finseq_1(D))
& ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__1,G)
& ~ r1_xreal_0(k3_finseq_1(D),G)
& ! [H] :
~ ( r2_hidden(H,A)
& r2_hidden(k4_tarski(k1_funct_1(D,G),H),k3_eqrel_1(A,B,C))
& r2_hidden(k4_tarski(H,k1_funct_1(D,k1_nat_1(G,np__1))),k3_eqrel_1(A,B,C)) ) ) ) )
=> r2_hidden(k4_tarski(E,F),k5_eqrel_1(A,B,C)) ) ) ) ) ) ).
fof(t27_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( m1_eqrel_1(C,A)
=> k4_partit1(A,k3_partit1(A,B,C)) = k5_eqrel_1(A,k4_partit1(A,B),k4_partit1(A,C)) ) ) ) ).
fof(t28_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( m1_eqrel_1(C,A)
=> k4_partit1(A,k2_partit1(A,B,C)) = k4_eqrel_1(A,k4_partit1(A,B),k4_partit1(A,C)) ) ) ) ).
fof(t29_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( m1_eqrel_1(C,A)
=> ( k4_partit1(A,B) = k4_partit1(A,C)
=> B = C ) ) ) ) ).
fof(t30_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( m1_eqrel_1(C,A)
=> ! [D] :
( m1_eqrel_1(D,A)
=> k3_partit1(A,k3_partit1(A,B,C),D) = k3_partit1(A,B,k3_partit1(A,C,D)) ) ) ) ) ).
fof(t31_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( m1_eqrel_1(C,A)
=> k2_partit1(A,B,k3_partit1(A,B,C)) = B ) ) ) ).
fof(t32_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( m1_eqrel_1(C,A)
=> k3_partit1(A,B,k2_partit1(A,B,C)) = B ) ) ) ).
fof(t33_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( m1_eqrel_1(C,A)
=> ! [D] :
( m1_eqrel_1(D,A)
=> ( ( r1_setfam_1(B,D)
& r1_setfam_1(C,D) )
=> r1_setfam_1(k3_partit1(A,B,C),D) ) ) ) ) ) ).
fof(t34_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ! [C] :
( m1_eqrel_1(C,A)
=> ! [D] :
( m1_eqrel_1(D,A)
=> ( ( r1_setfam_1(D,B)
& r1_setfam_1(D,C) )
=> r1_setfam_1(D,k2_partit1(A,B,C)) ) ) ) ) ) ).
fof(d8_partit1,axiom,
$true ).
fof(d9_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k6_partit1(A) = k1_tarski(A) ) ).
fof(t36_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ( r1_setfam_1(B,k6_partit1(A))
& r1_setfam_1(k3_pua2mss1(A),B) ) ) ) ).
fof(t37_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k4_partit1(A,k6_partit1(A)) = k1_eqrel_1(A) ) ).
fof(t38_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k4_partit1(A,k3_pua2mss1(A)) = k6_relat_1(A) ) ).
fof(t39_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> r1_setfam_1(k3_pua2mss1(A),k6_partit1(A)) ) ).
fof(t40_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ( k3_partit1(A,k6_partit1(A),B) = k6_partit1(A)
& k2_partit1(A,k6_partit1(A),B) = B ) ) ) ).
fof(t41_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_eqrel_1(B,A)
=> ( k3_partit1(A,k3_pua2mss1(A),B) = B
& k2_partit1(A,k3_pua2mss1(A),B) = k3_pua2mss1(A) ) ) ) ).
fof(dt_k1_partit1,axiom,
$true ).
fof(dt_k2_partit1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_eqrel_1(B,A)
& m1_eqrel_1(C,A) )
=> m1_eqrel_1(k2_partit1(A,B,C),A) ) ).
fof(commutativity_k2_partit1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_eqrel_1(B,A)
& m1_eqrel_1(C,A) )
=> k2_partit1(A,B,C) = k2_partit1(A,C,B) ) ).
fof(dt_k3_partit1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_eqrel_1(B,A)
& m1_eqrel_1(C,A) )
=> m1_eqrel_1(k3_partit1(A,B,C),A) ) ).
fof(commutativity_k3_partit1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_eqrel_1(B,A)
& m1_eqrel_1(C,A) )
=> k3_partit1(A,B,C) = k3_partit1(A,C,B) ) ).
fof(dt_k4_partit1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_eqrel_1(B,A) )
=> ( v3_relat_2(k4_partit1(A,B))
& v8_relat_2(k4_partit1(A,B))
& v1_partfun1(k4_partit1(A,B),A,A)
& m2_relset_1(k4_partit1(A,B),A,A) ) ) ).
fof(dt_k5_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_relat_1(k5_partit1(A))
& v1_funct_1(k5_partit1(A)) ) ) ).
fof(dt_k6_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> m1_eqrel_1(k6_partit1(A),A) ) ).
fof(t35_partit1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> k3_pua2mss1(A) = a_1_0_partit1(A) ) ).
fof(fraenkel_a_1_0_partit1,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ( r2_hidden(A,a_1_0_partit1(B))
<=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(B))
& A = C
& ? [D] :
( C = k1_tarski(D)
& r2_hidden(D,B) ) ) ) ) ).
%------------------------------------------------------------------------------