SET007 Axioms: SET007+901.ax
%------------------------------------------------------------------------------
% File : SET007+901 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Properties of First and Second Order Cutting of Binary Relations
% 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 : relset_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 93 ( 4 unt; 0 def)
% Number of atoms : 352 ( 72 equ)
% Maximal formula atoms : 9 ( 3 avg)
% Number of connectives : 280 ( 21 ~; 1 |; 67 &)
% ( 22 <=>; 169 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 8 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 12 ( 10 usr; 1 prp; 0-3 aty)
% Number of functors : 44 ( 44 usr; 1 con; 0-6 aty)
% Number of variables : 378 ( 365 !; 13 ?)
% 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(t1_relset_2,axiom,
! [A,B] :
( r2_hidden(A,k3_pua2mss1(B))
<=> ? [C] :
( A = k1_tarski(C)
& r2_hidden(C,B) ) ) ).
fof(t2_relset_2,axiom,
! [A] :
( A = k1_xboole_0
<=> k3_pua2mss1(A) = k1_xboole_0 ) ).
fof(t3_relset_2,axiom,
! [A,B] : k3_pua2mss1(k2_xboole_0(A,B)) = k2_xboole_0(k3_pua2mss1(A),k3_pua2mss1(B)) ).
fof(t4_relset_2,axiom,
! [A,B] : k3_pua2mss1(k3_xboole_0(A,B)) = k3_xboole_0(k3_pua2mss1(A),k3_pua2mss1(B)) ).
fof(t5_relset_2,axiom,
! [A,B] : k3_pua2mss1(k4_xboole_0(A,B)) = k4_xboole_0(k3_pua2mss1(A),k3_pua2mss1(B)) ).
fof(t6_relset_2,axiom,
! [A,B] :
( r1_tarski(A,B)
<=> r1_tarski(k3_pua2mss1(A),k3_pua2mss1(B)) ) ).
fof(t7_relset_2,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
=> r1_tarski(k5_subset_1(A,k8_setfam_1(A,B),k8_setfam_1(A,C)),k8_setfam_1(A,k1_relset_2(A,B,C))) ) ) ).
fof(t8_relset_2,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( v1_relat_1(B)
=> ! [C] :
( v1_relat_1(C)
=> r1_tarski(k5_relat_1(k3_xboole_0(A,B),C),k3_xboole_0(k5_relat_1(A,C),k5_relat_1(B,C))) ) ) ) ).
fof(d1_relset_2,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
=> ! [D] :
( m1_subset_1(D,A)
=> k2_relset_2(A,B,C,D) = k10_relset_1(A,B,C,k1_tarski(D)) ) ) ).
fof(t9_relset_2,axiom,
! [A,B,C] :
( v1_relat_1(C)
=> ( r2_hidden(A,k9_relat_1(C,k1_tarski(B)))
<=> r2_hidden(k4_tarski(B,A),C) ) ) ).
fof(t10_relset_2,axiom,
! [A,B] :
( v1_relat_1(B)
=> ! [C] :
( v1_relat_1(C)
=> k9_relat_1(k2_xboole_0(B,C),k1_tarski(A)) = k2_xboole_0(k9_relat_1(B,k1_tarski(A)),k9_relat_1(C,k1_tarski(A))) ) ) ).
fof(t11_relset_2,axiom,
! [A,B] :
( v1_relat_1(B)
=> ! [C] :
( v1_relat_1(C)
=> k9_relat_1(k3_xboole_0(B,C),k1_tarski(A)) = k3_xboole_0(k9_relat_1(B,k1_tarski(A)),k9_relat_1(C,k1_tarski(A))) ) ) ).
fof(t12_relset_2,axiom,
! [A,B] :
( v1_relat_1(B)
=> ! [C] :
( v1_relat_1(C)
=> k9_relat_1(k4_xboole_0(B,C),k1_tarski(A)) = k4_xboole_0(k9_relat_1(B,k1_tarski(A)),k9_relat_1(C,k1_tarski(A))) ) ) ).
fof(t13_relset_2,axiom,
! [A,B] :
( v1_relat_1(B)
=> ! [C] :
( v1_relat_1(C)
=> r1_tarski(k9_relat_1(k3_xboole_0(B,C),k3_pua2mss1(A)),k3_xboole_0(k9_relat_1(B,k3_pua2mss1(A)),k9_relat_1(C,k3_pua2mss1(A)))) ) ) ).
fof(d2_relset_2,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
=> ! [D] :
( m1_subset_1(D,A)
=> k3_relset_2(A,B,C,D) = k11_relset_1(A,B,C,k1_tarski(D)) ) ) ).
fof(d3_relset_2,axiom,
! [A,B] :
( v1_relat_1(B)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( C = k5_relset_2(A,B)
<=> ( k1_relat_1(C) = k1_zfmisc_1(A)
& ! [D] :
( r1_tarski(D,A)
=> k1_funct_1(C,D) = k9_relat_1(B,D) ) ) ) ) ) ).
fof(t19_relset_2,axiom,
! [A,B,C,D] :
( m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(B,C)))
=> ( r2_hidden(A,k1_relat_1(k5_relset_2(B,D)))
=> k1_funct_1(k5_relset_2(B,D),A) = k9_relat_1(D,A) ) ) ).
fof(t20_relset_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> r1_tarski(k2_relat_1(k5_relset_2(A,C)),k1_zfmisc_1(k2_relat_1(C))) ) ).
fof(t21_relset_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> ( v1_funct_1(k5_relset_2(A,C))
& v1_funct_2(k5_relset_2(A,C),k1_zfmisc_1(A),k1_zfmisc_1(k2_relat_1(C)))
& m2_relset_1(k5_relset_2(A,C),k1_zfmisc_1(A),k1_zfmisc_1(k2_relat_1(C))) ) ) ).
fof(t22_relset_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> r1_tarski(k5_setfam_1(B,k4_relset_2(k1_zfmisc_1(A),B,k6_relset_2(B,A,C),A)),k9_relat_1(C,k3_tarski(A))) ) ).
fof(d4_relset_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> k7_relset_2(A,B,C,D) = k8_setfam_1(B,k4_relset_2(k1_zfmisc_1(A),B,k6_relset_2(B,A,D),k3_pua2mss1(C))) ) ) ).
fof(t23_relset_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> ( k4_relset_2(k1_zfmisc_1(A),B,k6_relset_2(B,A,D),k3_pua2mss1(C)) = k1_xboole_0
<=> C = k1_xboole_0 ) ) ) ).
fof(t24_relset_2,axiom,
! [A,B,C,D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> ( r2_hidden(C,k8_relset_2(A,B,D,E))
=> ! [F] :
( r2_hidden(F,D)
=> r2_hidden(C,k9_relat_1(E,k1_tarski(F))) ) ) ) ) ).
fof(t25_relset_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C] :
( m1_subset_1(C,k1_zfmisc_1(B))
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k2_zfmisc_1(B,A)))
=> ( r2_hidden(D,k8_relset_2(B,A,C,E))
<=> ! [F] :
( r2_hidden(F,C)
=> r2_hidden(D,k9_relat_1(E,k1_tarski(F))) ) ) ) ) ) ) ).
fof(t26_relset_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> ( k4_relset_2(k1_zfmisc_1(A),B,k6_relset_2(B,A,E),k3_pua2mss1(C)) = k1_xboole_0
=> k8_relset_2(A,B,k4_subset_1(A,C,D),E) = k8_relset_2(A,B,D,E) ) ) ) ) ).
fof(t27_relset_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> k8_relset_2(A,B,k4_subset_1(A,C,D),E) = k5_subset_1(B,k8_relset_2(A,B,C,E),k8_relset_2(A,B,D,E)) ) ) ) ).
fof(t29_relset_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> ( C = k1_xboole_0
=> k8_relset_2(A,B,C,D) = B ) ) ) ).
fof(t30_relset_2,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( k5_setfam_1(A,B) = k1_xboole_0
<=> ! [C] :
( r2_hidden(C,B)
=> C = k1_xboole_0 ) ) ) ).
fof(t32_relset_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> ( r1_tarski(C,D)
=> r1_tarski(k8_relset_2(A,B,D,E),k8_relset_2(A,B,C,E)) ) ) ) ) ).
fof(t33_relset_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> r1_tarski(k4_subset_1(B,k8_relset_2(A,B,C,E),k8_relset_2(A,B,D,E)),k8_relset_2(A,B,k5_subset_1(A,C,D),E)) ) ) ) ).
fof(t34_relset_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> k8_relset_2(A,B,C,k5_subset_1(k2_zfmisc_1(A,B),D,E)) = k5_subset_1(B,k8_relset_2(A,B,C,D),k8_relset_2(A,B,C,E)) ) ) ) ).
fof(t37_relset_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> ( D = k1_xboole_0
=> ( C = k1_xboole_0
| k8_relset_2(A,B,C,D) = k1_xboole_0 ) ) ) ) ).
fof(t38_relset_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> ( D = k2_zfmisc_1(A,B)
=> k8_relset_2(A,B,C,D) = B ) ) ) ).
fof(t40_relset_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> ( r1_tarski(D,E)
=> r1_tarski(k8_relset_2(A,B,C,D),k8_relset_2(A,B,C,E)) ) ) ) ) ).
fof(t41_relset_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> r1_tarski(k4_subset_1(B,k8_relset_2(A,B,C,D),k8_relset_2(A,B,C,E)),k8_relset_2(A,B,C,k4_subset_1(k2_zfmisc_1(A,B),D,E))) ) ) ) ).
fof(t42_relset_2,axiom,
! [A,B,C,D,E] :
( m1_subset_1(E,k1_zfmisc_1(k2_zfmisc_1(C,D)))
=> ( r2_hidden(A,k9_relat_1(k3_subset_1(k2_zfmisc_1(C,D),E),k1_tarski(B)))
<=> ( ~ r2_hidden(k4_tarski(B,A),E)
& r2_hidden(B,C)
& r2_hidden(A,D) ) ) ) ).
fof(t43_relset_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> ( C != k1_xboole_0
=> r1_tarski(k8_relset_2(A,B,C,D),k9_relat_1(D,C)) ) ) ) ).
fof(t44_relset_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> ! [D,E] :
( ~ ( ~ r1_xboole_0(D,k9_relat_1(k4_relat_1(C),E))
& ! [F,G] :
~ ( r2_hidden(F,D)
& r2_hidden(G,E)
& r2_hidden(F,k9_relat_1(k4_relat_1(C),k1_tarski(G))) ) )
& ~ ( ? [F,G] :
( r2_hidden(F,D)
& r2_hidden(G,E)
& r2_hidden(F,k9_relat_1(k4_relat_1(C),k1_tarski(G))) )
& r1_xboole_0(D,k9_relat_1(k4_relat_1(C),E)) ) ) ) ).
fof(t45_relset_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> ! [D,E] :
( ~ ( ? [F,G] :
( r2_hidden(F,D)
& r2_hidden(G,E)
& r2_hidden(F,k9_relat_1(k4_relat_1(C),k1_tarski(G))) )
& r1_xboole_0(E,k9_relat_1(C,D)) )
& ~ ( ~ r1_xboole_0(E,k9_relat_1(C,D))
& ! [F,G] :
~ ( r2_hidden(F,D)
& r2_hidden(G,E)
& r2_hidden(F,k9_relat_1(k4_relat_1(C),k1_tarski(G))) ) ) ) ) ).
fof(t46_relset_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(B))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> ( r1_xboole_0(C,k9_relat_1(k4_relat_1(E),D))
<=> r1_xboole_0(D,k9_relat_1(E,C)) ) ) ) ) ).
fof(t47_relset_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> ! [D] : k9_relat_1(C,D) = k9_relat_1(C,k3_xboole_0(D,k1_funct_5(C))) ) ).
fof(t48_relset_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> ! [D] : k9_relat_1(k4_relat_1(C),D) = k9_relat_1(k4_relat_1(C),k3_xboole_0(D,k2_funct_5(C))) ) ).
fof(t49_relset_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(A,B)))
=> k3_subset_1(B,k8_relset_2(A,B,C,D)) = k9_relat_1(k3_subset_1(k2_zfmisc_1(A,B),D),C) ) ) ).
fof(t50_relset_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m2_relset_1(D,A,B)
=> k3_subset_1(B,k10_relset_1(A,B,D,C)) = k8_relset_2(A,B,C,k3_subset_1(k2_zfmisc_1(A,B),D)) ) ) ).
fof(t51_relset_2,axiom,
! [A,B,C] :
( m2_relset_1(C,B,A)
=> ( k1_funct_5(C) = k10_relset_1(A,B,k6_relset_1(B,A,C),A)
& k2_funct_5(C) = k10_relset_1(B,A,C,B) ) ) ).
fof(t52_relset_2,axiom,
! [A,B,C,D] :
( m2_relset_1(D,A,B)
=> ! [E] :
( m2_relset_1(E,B,C)
=> ( k1_funct_5(k9_relset_2(A,B,C,D,E)) = k10_relset_1(B,A,k6_relset_1(A,B,D),k1_funct_5(E))
& r1_tarski(k1_funct_5(k9_relset_2(A,B,C,D,E)),k1_funct_5(D)) ) ) ) ).
fof(t53_relset_2,axiom,
! [A,B,C,D] :
( m2_relset_1(D,A,B)
=> ! [E] :
( m2_relset_1(E,B,C)
=> ( k2_funct_5(k9_relset_2(A,B,C,D,E)) = k10_relset_1(B,C,E,k2_funct_5(D))
& r1_tarski(k2_funct_5(k9_relset_2(A,B,C,D,E)),k2_funct_5(E)) ) ) ) ).
fof(t54_relset_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m2_relset_1(D,A,B)
=> ( r1_tarski(C,k1_funct_5(D))
<=> r1_tarski(C,k10_relset_1(A,A,k9_relset_2(A,B,A,D,k6_relset_1(A,B,D)),C)) ) ) ) ).
fof(t55_relset_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(B))
=> ! [D] :
( m2_relset_1(D,A,B)
=> ( r1_tarski(C,k2_funct_5(D))
<=> r1_tarski(C,k10_relset_1(B,B,k9_relset_2(B,A,B,k6_relset_1(A,B,D),D),C)) ) ) ) ).
fof(t56_relset_2,axiom,
! [A,B,C] :
( m2_relset_1(C,B,A)
=> ( k1_funct_5(C) = k10_relset_1(A,B,k6_relset_1(B,A,C),A)
& k10_relset_1(A,B,k6_relset_1(B,A,C),k10_relset_1(B,A,C,B)) = k10_relset_1(A,B,k6_relset_1(B,A,C),k2_funct_5(C)) ) ) ).
fof(t57_relset_2,axiom,
! [A,B,C] :
( m2_relset_1(C,B,A)
=> k10_relset_1(A,B,k6_relset_1(B,A,C),A) = k10_relset_1(B,B,k9_relset_2(B,A,B,C,k6_relset_1(B,A,C)),B) ) ).
fof(t58_relset_2,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
=> k10_relset_1(A,B,C,A) = k10_relset_1(B,B,k9_relset_2(B,A,B,k6_relset_1(A,B,C),C),B) ) ).
fof(t59_relset_2,axiom,
! [A,B,C,D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ! [E] :
( m2_relset_1(E,A,B)
=> ! [F] :
( m2_relset_1(F,B,C)
=> k8_relset_2(B,C,k10_relset_1(A,B,E,D),F) = k8_relset_2(A,C,D,k3_subset_1(k2_zfmisc_1(A,C),k9_relset_2(A,B,C,E,k3_subset_1(k2_zfmisc_1(B,C),F)))) ) ) ) ).
fof(t60_relset_2,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
=> k4_relat_1(k3_subset_1(k2_zfmisc_1(A,B),C)) = k3_subset_1(k2_zfmisc_1(B,A),k6_relset_1(A,B,C)) ) ).
fof(t61_relset_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(B))
=> ! [E] :
( m2_relset_1(E,A,B)
=> ( r1_tarski(C,k8_relset_2(B,A,D,k6_relset_1(A,B,E)))
<=> r1_tarski(D,k8_relset_2(A,B,C,E)) ) ) ) ) ).
fof(t62_relset_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(B))
=> ! [E] :
( m2_relset_1(E,A,B)
=> ( r1_tarski(k10_relset_1(A,B,E,k3_subset_1(A,C)),k3_subset_1(B,D))
<=> r1_tarski(k10_relset_1(B,A,k6_relset_1(A,B,E),D),C) ) ) ) ) ).
fof(t63_relset_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(B))
=> ! [E] :
( m2_relset_1(E,A,B)
=> ( r1_tarski(C,k8_relset_2(B,A,k8_relset_2(A,B,C,E),k6_relset_1(A,B,E)))
& r1_tarski(D,k8_relset_2(A,B,k8_relset_2(B,A,D,k6_relset_1(A,B,E)),E)) ) ) ) ) ).
fof(t64_relset_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(B))
=> ! [E] :
( m2_relset_1(E,A,B)
=> ( k8_relset_2(A,B,C,E) = k8_relset_2(A,B,k8_relset_2(B,A,k8_relset_2(A,B,C,E),k6_relset_1(A,B,E)),E)
& k8_relset_2(B,A,D,k6_relset_1(A,B,E)) = k8_relset_2(B,A,k8_relset_2(A,B,k8_relset_2(B,A,D,k6_relset_1(A,B,E)),E),k6_relset_1(A,B,E)) ) ) ) ) ).
fof(t65_relset_2,axiom,
! [A,B,C] :
( m2_relset_1(C,A,B)
=> k9_relset_2(A,A,B,k6_partfun1(A),C) = k9_relset_2(A,B,B,C,k6_partfun1(B)) ) ).
fof(dt_k1_relset_2,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> m1_subset_1(k1_relset_2(A,B,C),k1_zfmisc_1(k1_zfmisc_1(A))) ) ).
fof(commutativity_k1_relset_2,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> k1_relset_2(A,B,C) = k1_relset_2(A,C,B) ) ).
fof(idempotence_k1_relset_2,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> k1_relset_2(A,B,B) = B ) ).
fof(redefinition_k1_relset_2,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> k1_relset_2(A,B,C) = k3_xboole_0(B,C) ) ).
fof(dt_k2_relset_2,axiom,
! [A,B,C,D] :
( ( m1_relset_1(C,A,B)
& m1_subset_1(D,A) )
=> m1_subset_1(k2_relset_2(A,B,C,D),k1_zfmisc_1(B)) ) ).
fof(dt_k3_relset_2,axiom,
! [A,B,C,D] :
( ( m1_relset_1(C,A,B)
& m1_subset_1(D,A) )
=> m1_subset_1(k3_relset_2(A,B,C,D),k1_zfmisc_1(A)) ) ).
fof(dt_k4_relset_2,axiom,
! [A,B,C,D] :
( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,k1_zfmisc_1(B))))
=> m1_subset_1(k4_relset_2(A,B,C,D),k1_zfmisc_1(k1_zfmisc_1(B))) ) ).
fof(redefinition_k4_relset_2,axiom,
! [A,B,C,D] :
( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,k1_zfmisc_1(B))))
=> k4_relset_2(A,B,C,D) = k9_relat_1(C,D) ) ).
fof(dt_k5_relset_2,axiom,
! [A,B] :
( v1_relat_1(B)
=> ( v1_relat_1(k5_relset_2(A,B))
& v1_funct_1(k5_relset_2(A,B)) ) ) ).
fof(dt_k6_relset_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(B,A)))
=> ( v1_funct_1(k6_relset_2(A,B,C))
& v1_funct_2(k6_relset_2(A,B,C),k1_zfmisc_1(B),k1_zfmisc_1(A))
& m2_relset_1(k6_relset_2(A,B,C),k1_zfmisc_1(B),k1_zfmisc_1(A)) ) ) ).
fof(redefinition_k6_relset_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(B,A)))
=> k6_relset_2(A,B,C) = k5_relset_2(B,C) ) ).
fof(dt_k7_relset_2,axiom,
$true ).
fof(dt_k8_relset_2,axiom,
! [A,B,C,D] :
( ( m1_subset_1(C,k1_zfmisc_1(A))
& m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(A,B))) )
=> m1_subset_1(k8_relset_2(A,B,C,D),k1_zfmisc_1(B)) ) ).
fof(redefinition_k8_relset_2,axiom,
! [A,B,C,D] :
( ( m1_subset_1(C,k1_zfmisc_1(A))
& m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(A,B))) )
=> k8_relset_2(A,B,C,D) = k7_relset_2(A,B,C,D) ) ).
fof(dt_k9_relset_2,axiom,
! [A,B,C,D,E] :
( ( m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(A,B)))
& m1_subset_1(E,k1_zfmisc_1(k2_zfmisc_1(B,C))) )
=> m2_relset_1(k9_relset_2(A,B,C,D,E),A,C) ) ).
fof(redefinition_k9_relset_2,axiom,
! [A,B,C,D,E] :
( ( m1_subset_1(D,k1_zfmisc_1(k2_zfmisc_1(A,B)))
& m1_subset_1(E,k1_zfmisc_1(k2_zfmisc_1(B,C))) )
=> k9_relset_2(A,B,C,D,E) = k5_relat_1(D,E) ) ).
fof(t14_relset_2,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ! [C] :
( v1_relat_1(C)
=> k9_relat_1(C,k5_setfam_1(A,B)) = k3_tarski(a_3_0_relset_2(A,B,C)) ) ) ).
fof(t15_relset_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> B = k3_tarski(a_2_0_relset_2(A,B)) ) ) ).
fof(t16_relset_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> m1_subset_1(a_2_0_relset_2(A,B),k1_zfmisc_1(k1_zfmisc_1(A))) ) ) ).
fof(t17_relset_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m2_relset_1(D,A,B)
=> k10_relset_1(A,B,D,C) = k3_tarski(a_4_0_relset_2(A,B,C,D)) ) ) ) ).
fof(t18_relset_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m2_relset_1(D,A,B)
=> m1_subset_1(a_4_0_relset_2(A,B,C,D),k1_zfmisc_1(k1_zfmisc_1(B))) ) ) ) ).
fof(t28_relset_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ! [D] :
( m2_relset_1(D,A,B)
=> m1_subset_1(a_4_1_relset_2(A,B,C,D),k1_zfmisc_1(k1_zfmisc_1(B))) ) ) ) ).
fof(t31_relset_2,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m2_relset_1(C,A,B)
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(B)))
=> ( E = a_4_2_relset_2(A,B,C,D)
=> k8_relset_2(A,B,k5_setfam_1(A,D),C) = k8_setfam_1(B,E) ) ) ) ) ) ).
fof(t35_relset_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(k2_zfmisc_1(A,B))))
=> k9_relat_1(k5_setfam_1(k2_zfmisc_1(A,B),D),C) = k3_tarski(a_4_3_relset_2(A,B,C,D)) ) ) ).
fof(t36_relset_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(k2_zfmisc_1(A,B))))
=> ! [D,E,F] :
( m1_subset_1(F,k1_zfmisc_1(D))
=> m1_subset_1(a_6_0_relset_2(A,B,C,D,E,F),k1_zfmisc_1(k1_zfmisc_1(E))) ) ) ).
fof(t39_relset_2,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(B))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(k2_zfmisc_1(B,A))))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( E = a_4_4_relset_2(A,B,C,D)
=> k8_relset_2(B,A,C,k8_setfam_1(k2_zfmisc_1(B,A),D)) = k8_setfam_1(A,E) ) ) ) ) ).
fof(fraenkel_a_3_0_relset_2,axiom,
! [A,B,C,D] :
( ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(B)))
& v1_relat_1(D) )
=> ( r2_hidden(A,a_3_0_relset_2(B,C,D))
<=> ? [E] :
( m1_subset_1(E,k1_zfmisc_1(B))
& A = k9_relat_1(D,E)
& r2_hidden(E,C) ) ) ) ).
fof(fraenkel_a_2_0_relset_2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(C,k1_zfmisc_1(B)) )
=> ( r2_hidden(A,a_2_0_relset_2(B,C))
<=> ? [D] :
( m1_subset_1(D,B)
& A = k1_tarski(D)
& r2_hidden(D,C) ) ) ) ).
fof(fraenkel_a_4_0_relset_2,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(D,k1_zfmisc_1(B))
& m2_relset_1(E,B,C) )
=> ( r2_hidden(A,a_4_0_relset_2(B,C,D,E))
<=> ? [F] :
( m1_subset_1(F,B)
& A = k2_relset_2(B,C,E,F)
& r2_hidden(F,D) ) ) ) ).
fof(fraenkel_a_4_1_relset_2,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(B)))
& m2_relset_1(E,B,C) )
=> ( r2_hidden(A,a_4_1_relset_2(B,C,D,E))
<=> ? [F] :
( m1_subset_1(F,k1_zfmisc_1(B))
& A = k8_relset_2(B,C,F,E)
& r2_hidden(F,D) ) ) ) ).
fof(fraenkel_a_4_2_relset_2,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(C)
& m2_relset_1(D,B,C)
& m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(B))) )
=> ( r2_hidden(A,a_4_2_relset_2(B,C,D,E))
<=> ? [F] :
( m1_subset_1(F,k1_zfmisc_1(B))
& A = k8_relset_2(B,C,F,D)
& r2_hidden(F,E) ) ) ) ).
fof(fraenkel_a_4_3_relset_2,axiom,
! [A,B,C,D,E] :
( ( m1_subset_1(D,k1_zfmisc_1(B))
& m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(k2_zfmisc_1(B,C)))) )
=> ( r2_hidden(A,a_4_3_relset_2(B,C,D,E))
<=> ? [F] :
( m1_subset_1(F,k1_zfmisc_1(k2_zfmisc_1(B,C)))
& A = k9_relat_1(F,D)
& r2_hidden(F,E) ) ) ) ).
fof(fraenkel_a_6_0_relset_2,axiom,
! [A,B,C,D,E,F,G] :
( ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(k2_zfmisc_1(B,C))))
& m1_subset_1(G,k1_zfmisc_1(E)) )
=> ( r2_hidden(A,a_6_0_relset_2(B,C,D,E,F,G))
<=> ? [H] :
( m1_subset_1(H,k1_zfmisc_1(k2_zfmisc_1(E,F)))
& A = k8_relset_2(E,F,G,H)
& r2_hidden(H,D) ) ) ) ).
fof(fraenkel_a_4_4_relset_2,axiom,
! [A,B,C,D,E] :
( ( m1_subset_1(D,k1_zfmisc_1(C))
& m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(k2_zfmisc_1(C,B)))) )
=> ( r2_hidden(A,a_4_4_relset_2(B,C,D,E))
<=> ? [F] :
( m1_subset_1(F,k1_zfmisc_1(k2_zfmisc_1(C,B)))
& A = k8_relset_2(C,B,D,F)
& r2_hidden(F,E) ) ) ) ).
%------------------------------------------------------------------------------