SET007 Axioms: SET007+33.ax
%------------------------------------------------------------------------------
% File : SET007+33 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Semilattice Operations on Finite Subsets
% 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 : setwiseo [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 104 ( 23 unt; 0 def)
% Number of atoms : 792 ( 100 equ)
% Maximal formula atoms : 31 ( 7 avg)
% Number of connectives : 803 ( 115 ~; 12 |; 313 &)
% ( 12 <=>; 351 =>; 0 <=; 0 <~>)
% Maximal formula depth : 30 ( 9 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 22 ( 20 usr; 1 prp; 0-3 aty)
% Number of functors : 36 ( 36 usr; 6 con; 0-6 aty)
% Number of variables : 385 ( 378 !; 7 ?)
% 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_setwiseo,axiom,
! [A] :
? [B] :
( m1_subset_1(B,k5_finsub_1(A))
& v1_xboole_0(B)
& v1_finset_1(B) ) ).
fof(rc2_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_subset_1(B,k5_finsub_1(A))
& ~ v1_xboole_0(B)
& v1_finset_1(B) ) ) ).
fof(t1_setwiseo,axiom,
$true ).
fof(t2_setwiseo,axiom,
$true ).
fof(t3_setwiseo,axiom,
! [A,B,C] : r1_tarski(k1_tarski(A),k1_enumset1(A,B,C)) ).
fof(t4_setwiseo,axiom,
! [A,B,C] : r1_tarski(k2_tarski(A,B),k1_enumset1(A,B,C)) ).
fof(t5_setwiseo,axiom,
! [A,B,C] :
~ ( r1_tarski(A,k2_xboole_0(B,k1_tarski(C)))
& ~ r2_hidden(C,A)
& ~ r1_tarski(A,B) ) ).
fof(t6_setwiseo,axiom,
! [A,B,C] :
( r2_hidden(A,k2_xboole_0(B,k1_tarski(C)))
<=> ( r2_hidden(A,B)
| A = C ) ) ).
fof(t7_setwiseo,axiom,
$true ).
fof(t8_setwiseo,axiom,
! [A,B,C] :
( r1_tarski(k2_xboole_0(A,k1_tarski(B)),C)
<=> ( r2_hidden(B,C)
& r1_tarski(A,C) ) ) ).
fof(t9_setwiseo,axiom,
$true ).
fof(t10_setwiseo,axiom,
$true ).
fof(t11_setwiseo,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> k9_relat_1(C,k4_xboole_0(B,k10_relat_1(C,A))) = k4_xboole_0(k9_relat_1(C,B),A) ) ).
fof(t12_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,A)
& m2_relset_1(C,B,A) )
=> ! [D] :
( m1_subset_1(D,B)
=> r2_hidden(D,k3_funct_2(B,A,C,k1_tarski(k8_funct_2(B,A,C,D)))) ) ) ) ) ).
fof(t13_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,A)
& m2_relset_1(C,B,A) )
=> ! [D] :
( m1_subset_1(D,B)
=> k2_funct_2(B,A,C,k1_tarski(D)) = k1_tarski(k8_funct_2(B,A,C,D)) ) ) ) ) ).
fof(t14_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k5_finsub_1(A))
=> ! [C] :
( r2_hidden(C,B)
=> m1_subset_1(C,A) ) ) ) ).
fof(t15_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,k5_finsub_1(A))
=> ! [D,E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,B)
& m2_relset_1(E,A,B) )
=> ( ! [F] :
( m1_subset_1(F,A)
=> ( r2_hidden(F,C)
=> r2_hidden(k8_funct_2(A,B,E,F),D) ) )
=> r1_tarski(k2_funct_2(A,B,E,C),D) ) ) ) ) ) ).
fof(t16_setwiseo,axiom,
! [A,B] :
( m1_subset_1(B,k5_finsub_1(A))
=> ! [C] :
( r1_tarski(C,B)
=> m1_subset_1(C,k5_finsub_1(A)) ) ) ).
fof(t17_setwiseo,axiom,
$true ).
fof(t18_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k5_finsub_1(A))
=> ~ ( B != k1_xboole_0
& ! [C] :
( m1_subset_1(C,A)
=> ~ r2_hidden(C,B) ) ) ) ) ).
fof(t19_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,A)
& m2_relset_1(C,B,A) )
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(B))
=> ( k2_funct_2(B,A,C,D) = k1_xboole_0
=> D = k1_xboole_0 ) ) ) ) ) ).
fof(d1_setwiseo,axiom,
! [A] : k1_setwiseo(A) = k1_xboole_0 ).
fof(d2_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ( v1_setwiseo(B,A)
<=> ? [C] :
( m1_subset_1(C,A)
& r3_binop_1(A,C,B) ) ) ) ) ).
fof(t20_setwiseo,axiom,
$true ).
fof(t21_setwiseo,axiom,
$true ).
fof(t22_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ( v1_setwiseo(B,A)
<=> r3_binop_1(A,k3_binop_1(A,B),B) ) ) ) ).
fof(t23_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m2_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ( v1_setwiseo(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ( k2_binop_1(A,A,A,B,k3_binop_1(A,B),C) = C
& k2_binop_1(A,A,A,B,C,k3_binop_1(A,B)) = C ) ) ) ) ) ).
fof(d3_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(B,B),B)
& m2_relset_1(C,k2_zfmisc_1(B,B),B) )
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(A))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,B)
& m2_relset_1(E,A,B) )
=> ( ( v1_binop_1(C,B)
& v2_binop_1(C,B) )
=> ( ( D = k1_xboole_0
& ~ v1_setwiseo(C,B) )
| ! [F] :
( m1_subset_1(F,B)
=> ( F = k7_setwiseo(A,B,C,D,E)
<=> ? [G] :
( v1_funct_1(G)
& v1_funct_2(G,k5_finsub_1(A),B)
& m2_relset_1(G,k5_finsub_1(A),B)
& F = k8_funct_2(k5_finsub_1(A),B,G,D)
& ! [H] :
( m1_subset_1(H,B)
=> ( r3_binop_1(B,H,C)
=> k1_funct_1(G,k1_xboole_0) = H ) )
& ! [H] :
( m1_subset_1(H,A)
=> k8_funct_2(k5_finsub_1(A),B,G,k2_setwiseo(A,H)) = k8_funct_2(A,B,E,H) )
& ! [H] :
( m1_subset_1(H,k5_finsub_1(A))
=> ( r1_tarski(H,D)
=> ( H = k1_xboole_0
| ! [I] :
( m1_subset_1(I,A)
=> ( r2_hidden(I,k6_setwiseo(A,D,H))
=> k8_funct_2(k5_finsub_1(A),B,G,k5_setwiseo(A,H,k2_setwiseo(A,I))) = k2_binop_1(B,B,B,C,k8_funct_2(k5_finsub_1(A),B,G,H),k8_funct_2(A,B,E,I)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t24_setwiseo,axiom,
$true ).
fof(t25_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(B,B),B)
& m2_relset_1(C,k2_zfmisc_1(B,B),B) )
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(A))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,B)
& m2_relset_1(E,A,B) )
=> ( ( v3_binop_1(C,B)
& v1_binop_1(C,B)
& v2_binop_1(C,B) )
=> ( ( D = k1_xboole_0
& ~ v1_setwiseo(C,B) )
| ! [F] :
( m1_subset_1(F,B)
=> ( F = k7_setwiseo(A,B,C,D,E)
<=> ? [G] :
( v1_funct_1(G)
& v1_funct_2(G,k5_finsub_1(A),B)
& m2_relset_1(G,k5_finsub_1(A),B)
& F = k8_funct_2(k5_finsub_1(A),B,G,D)
& ! [H] :
( m1_subset_1(H,B)
=> ( r3_binop_1(B,H,C)
=> k1_funct_1(G,k1_xboole_0) = H ) )
& ! [H] :
( m1_subset_1(H,A)
=> k8_funct_2(k5_finsub_1(A),B,G,k2_setwiseo(A,H)) = k8_funct_2(A,B,E,H) )
& ! [H] :
( m1_subset_1(H,k5_finsub_1(A))
=> ( r1_tarski(H,D)
=> ( H = k1_xboole_0
| ! [I] :
( m1_subset_1(I,A)
=> ( r2_hidden(I,D)
=> k8_funct_2(k5_finsub_1(A),B,G,k5_setwiseo(A,H,k2_setwiseo(A,I))) = k2_binop_1(B,B,B,C,k8_funct_2(k5_finsub_1(A),B,G,H),k8_funct_2(A,B,E,I)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t26_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,B,A)
& m2_relset_1(D,B,A) )
=> ( ( v1_binop_1(C,A)
& v2_binop_1(C,A) )
=> ! [E] :
( m1_subset_1(E,B)
=> k7_setwiseo(B,A,C,k2_setwiseo(B,E),D) = k8_funct_2(B,A,D,E) ) ) ) ) ) ) ).
fof(t27_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,B,A)
& m2_relset_1(D,B,A) )
=> ( ( v3_binop_1(C,A)
& v1_binop_1(C,A)
& v2_binop_1(C,A) )
=> ! [E] :
( m1_subset_1(E,B)
=> ! [F] :
( m1_subset_1(F,B)
=> k7_setwiseo(B,A,C,k3_setwiseo(B,E,F),D) = k2_binop_1(A,A,A,C,k8_funct_2(B,A,D,E),k8_funct_2(B,A,D,F)) ) ) ) ) ) ) ) ).
fof(t28_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,B,A)
& m2_relset_1(D,B,A) )
=> ( ( v3_binop_1(C,A)
& v1_binop_1(C,A)
& v2_binop_1(C,A) )
=> ! [E] :
( m1_subset_1(E,B)
=> ! [F] :
( m1_subset_1(F,B)
=> ! [G] :
( m1_subset_1(G,B)
=> k7_setwiseo(B,A,C,k4_setwiseo(B,E,F,G),D) = k2_binop_1(A,A,A,C,k2_binop_1(A,A,A,C,k8_funct_2(B,A,D,E),k8_funct_2(B,A,D,F)),k8_funct_2(B,A,D,G)) ) ) ) ) ) ) ) ) ).
fof(t29_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(B))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,B,A)
& m2_relset_1(E,B,A) )
=> ( ( v3_binop_1(C,A)
& v1_binop_1(C,A)
& v2_binop_1(C,A) )
=> ( D = k1_xboole_0
| ! [F] :
( m1_subset_1(F,B)
=> k7_setwiseo(B,A,C,k5_setwiseo(B,D,k2_setwiseo(B,F)),E) = k2_binop_1(A,A,A,C,k7_setwiseo(B,A,C,D,E),k8_funct_2(B,A,E,F)) ) ) ) ) ) ) ) ) ).
fof(t30_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,B,A)
& m2_relset_1(D,B,A) )
=> ( ( v3_binop_1(C,A)
& v1_binop_1(C,A)
& v2_binop_1(C,A) )
=> ! [E] :
( m1_subset_1(E,k5_finsub_1(B))
=> ! [F] :
( m1_subset_1(F,k5_finsub_1(B))
=> ~ ( E != k1_xboole_0
& F != k1_xboole_0
& k7_setwiseo(B,A,C,k5_setwiseo(B,E,F),D) != k2_binop_1(A,A,A,C,k7_setwiseo(B,A,C,E,D),k7_setwiseo(B,A,C,F,D)) ) ) ) ) ) ) ) ) ).
fof(t31_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(B))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,B,A)
& m2_relset_1(E,B,A) )
=> ( ( v1_binop_1(C,A)
& v2_binop_1(C,A)
& v3_binop_1(C,A) )
=> ! [F] :
( m1_subset_1(F,B)
=> ( r2_hidden(F,D)
=> k2_binop_1(A,A,A,C,k8_funct_2(B,A,E,F),k7_setwiseo(B,A,C,D,E)) = k7_setwiseo(B,A,C,D,E) ) ) ) ) ) ) ) ) ).
fof(t32_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,B,A)
& m2_relset_1(D,B,A) )
=> ( ( v1_binop_1(C,A)
& v2_binop_1(C,A)
& v3_binop_1(C,A) )
=> ! [E] :
( m1_subset_1(E,k5_finsub_1(B))
=> ! [F] :
( m1_subset_1(F,k5_finsub_1(B))
=> ( r1_tarski(E,F)
=> ( E = k1_xboole_0
| k2_binop_1(A,A,A,C,k7_setwiseo(B,A,C,E,D),k7_setwiseo(B,A,C,F,D)) = k7_setwiseo(B,A,C,F,D) ) ) ) ) ) ) ) ) ) ).
fof(t33_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(B))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,B,A)
& m2_relset_1(E,B,A) )
=> ( ( v1_binop_1(C,A)
& v2_binop_1(C,A)
& v3_binop_1(C,A) )
=> ( D = k1_xboole_0
| ! [F] :
( m1_subset_1(F,A)
=> ( ! [G] :
( m1_subset_1(G,B)
=> ( r2_hidden(G,D)
=> k8_funct_2(B,A,E,G) = F ) )
=> k7_setwiseo(B,A,C,D,E) = F ) ) ) ) ) ) ) ) ) ).
fof(t34_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(B,B),B)
& m2_relset_1(C,k2_zfmisc_1(B,B),B) )
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(A))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,B)
& m2_relset_1(E,A,B) )
=> ( ( v1_binop_1(C,B)
& v2_binop_1(C,B)
& v3_binop_1(C,B) )
=> ! [F] :
( m1_subset_1(F,B)
=> ( k2_funct_2(A,B,E,D) = k2_setwiseo(B,F)
=> k7_setwiseo(A,B,C,D,E) = F ) ) ) ) ) ) ) ) ).
fof(t35_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(B,B),B)
& m2_relset_1(C,k2_zfmisc_1(B,B),B) )
=> ( ( v1_binop_1(C,B)
& v2_binop_1(C,B)
& v3_binop_1(C,B) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,B)
& m2_relset_1(D,A,B) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,B)
& m2_relset_1(E,A,B) )
=> ! [F] :
( m1_subset_1(F,k5_finsub_1(A))
=> ! [G] :
( m1_subset_1(G,k5_finsub_1(A))
=> ( k2_funct_2(A,B,D,F) = k2_funct_2(A,B,E,G)
=> ( F = k1_xboole_0
| k7_setwiseo(A,B,C,F,D) = k7_setwiseo(A,B,C,G,E) ) ) ) ) ) ) ) ) ) ) ).
fof(t36_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,A),A)
& m2_relset_1(D,k2_zfmisc_1(A,A),A) )
=> ( ( v3_binop_1(C,A)
& v1_binop_1(C,A)
& v2_binop_1(C,A)
& r6_binop_1(A,D,C) )
=> ! [E] :
( m1_subset_1(E,k5_finsub_1(B))
=> ( E != k1_xboole_0
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,B,A)
& m2_relset_1(F,B,A) )
=> ! [G] :
( m1_subset_1(G,A)
=> k2_binop_1(A,A,A,D,G,k7_setwiseo(B,A,C,E,F)) = k7_setwiseo(B,A,C,E,k8_funcop_1(A,B,D,G,F)) ) ) ) ) ) ) ) ) ) ).
fof(t37_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,A),A)
& m2_relset_1(D,k2_zfmisc_1(A,A),A) )
=> ( ( v3_binop_1(C,A)
& v1_binop_1(C,A)
& v2_binop_1(C,A)
& r6_binop_1(A,D,C) )
=> ! [E] :
( m1_subset_1(E,k5_finsub_1(B))
=> ( E != k1_xboole_0
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,B,A)
& m2_relset_1(F,B,A) )
=> ! [G] :
( m1_subset_1(G,A)
=> k2_binop_1(A,A,A,D,k7_setwiseo(B,A,C,E,F),G) = k7_setwiseo(B,A,C,E,k7_funcop_1(A,B,D,F,G)) ) ) ) ) ) ) ) ) ) ).
fof(t38_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,A),A)
& m2_relset_1(D,k2_zfmisc_1(A,A),A) )
=> ( ( v3_binop_1(D,A)
& v1_binop_1(D,A)
& v2_binop_1(D,A) )
=> ! [E] :
( m1_subset_1(E,k5_finsub_1(B))
=> ( E != k1_xboole_0
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,B,C)
& m2_relset_1(F,B,C) )
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,C,A)
& m2_relset_1(G,C,A) )
=> k7_setwiseo(C,A,D,k8_setwiseo(B,C,F,E),G) = k7_setwiseo(B,A,D,E,k7_funct_2(B,C,A,F,G)) ) ) ) ) ) ) ) ) ) ).
fof(t39_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(B,B),B)
& m2_relset_1(C,k2_zfmisc_1(B,B),B) )
=> ( ( v1_binop_1(C,B)
& v2_binop_1(C,B)
& v3_binop_1(C,B) )
=> ! [D] :
( ~ v1_xboole_0(D)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(D,D),D)
& m2_relset_1(E,k2_zfmisc_1(D,D),D) )
=> ( ( v1_binop_1(E,D)
& v2_binop_1(E,D)
& v3_binop_1(E,D) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,A,B)
& m2_relset_1(F,A,B) )
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,B,D)
& m2_relset_1(G,B,D) )
=> ( ! [H] :
( m1_subset_1(H,B)
=> ! [I] :
( m1_subset_1(I,B)
=> k8_funct_2(B,D,G,k2_binop_1(B,B,B,C,H,I)) = k2_binop_1(D,D,D,E,k8_funct_2(B,D,G,H),k8_funct_2(B,D,G,I)) ) )
=> ! [H] :
( m1_subset_1(H,k5_finsub_1(A))
=> ( H != k1_xboole_0
=> k8_funct_2(B,D,G,k7_setwiseo(A,B,C,H,F)) = k7_setwiseo(A,D,E,H,k7_funct_2(A,B,D,F,G)) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t40_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ( ( v1_binop_1(C,A)
& v2_binop_1(C,A)
& v1_setwiseo(C,A) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,B,A)
& m2_relset_1(D,B,A) )
=> k7_setwiseo(B,A,C,k1_setwiseo(B),D) = k3_binop_1(A,C) ) ) ) ) ) ).
fof(t41_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(B))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,B,A)
& m2_relset_1(E,B,A) )
=> ( ( v3_binop_1(C,A)
& v1_binop_1(C,A)
& v2_binop_1(C,A)
& v1_setwiseo(C,A) )
=> ! [F] :
( m1_subset_1(F,B)
=> k7_setwiseo(B,A,C,k5_setwiseo(B,D,k2_setwiseo(B,F)),E) = k2_binop_1(A,A,A,C,k7_setwiseo(B,A,C,D,E),k8_funct_2(B,A,E,F)) ) ) ) ) ) ) ) ).
fof(t42_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m2_relset_1(C,k2_zfmisc_1(A,A),A) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,B,A)
& m2_relset_1(D,B,A) )
=> ( ( v3_binop_1(C,A)
& v1_binop_1(C,A)
& v2_binop_1(C,A)
& v1_setwiseo(C,A) )
=> ! [E] :
( m1_subset_1(E,k5_finsub_1(B))
=> ! [F] :
( m1_subset_1(F,k5_finsub_1(B))
=> k7_setwiseo(B,A,C,k5_setwiseo(B,E,F),D) = k2_binop_1(A,A,A,C,k7_setwiseo(B,A,C,E,D),k7_setwiseo(B,A,C,F,D)) ) ) ) ) ) ) ) ).
fof(t43_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(B,B),B)
& m2_relset_1(C,k2_zfmisc_1(B,B),B) )
=> ( ( v1_binop_1(C,B)
& v2_binop_1(C,B)
& v3_binop_1(C,B)
& v1_setwiseo(C,B) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,B)
& m2_relset_1(D,A,B) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,B)
& m2_relset_1(E,A,B) )
=> ! [F] :
( m1_subset_1(F,k5_finsub_1(A))
=> ! [G] :
( m1_subset_1(G,k5_finsub_1(A))
=> ( k8_setwiseo(A,B,D,F) = k8_setwiseo(A,B,E,G)
=> k7_setwiseo(A,B,C,F,D) = k7_setwiseo(A,B,C,G,E) ) ) ) ) ) ) ) ) ) ).
fof(t44_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,A),A)
& m2_relset_1(D,k2_zfmisc_1(A,A),A) )
=> ( ( v3_binop_1(D,A)
& v1_binop_1(D,A)
& v2_binop_1(D,A)
& v1_setwiseo(D,A) )
=> ! [E] :
( m1_subset_1(E,k5_finsub_1(B))
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,B,C)
& m2_relset_1(F,B,C) )
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,C,A)
& m2_relset_1(G,C,A) )
=> k7_setwiseo(C,A,D,k8_setwiseo(B,C,F,E),G) = k7_setwiseo(B,A,D,E,k7_funct_2(B,C,A,F,G)) ) ) ) ) ) ) ) ) ).
fof(t45_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(B,B),B)
& m2_relset_1(C,k2_zfmisc_1(B,B),B) )
=> ( ( v1_binop_1(C,B)
& v2_binop_1(C,B)
& v3_binop_1(C,B)
& v1_setwiseo(C,B) )
=> ! [D] :
( ~ v1_xboole_0(D)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(D,D),D)
& m2_relset_1(E,k2_zfmisc_1(D,D),D) )
=> ( ( v1_binop_1(E,D)
& v2_binop_1(E,D)
& v3_binop_1(E,D)
& v1_setwiseo(E,D) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,A,B)
& m2_relset_1(F,A,B) )
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,B,D)
& m2_relset_1(G,B,D) )
=> ( ( k8_funct_2(B,D,G,k3_binop_1(B,C)) = k3_binop_1(D,E)
& ! [H] :
( m1_subset_1(H,B)
=> ! [I] :
( m1_subset_1(I,B)
=> k8_funct_2(B,D,G,k2_binop_1(B,B,B,C,H,I)) = k2_binop_1(D,D,D,E,k8_funct_2(B,D,G,H),k8_funct_2(B,D,G,I)) ) ) )
=> ! [H] :
( m1_subset_1(H,k5_finsub_1(A))
=> k8_funct_2(B,D,G,k7_setwiseo(A,B,C,H,F)) = k7_setwiseo(A,D,E,H,k7_funct_2(A,B,D,F,G)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d4_setwiseo,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k5_finsub_1(A))
& m2_relset_1(B,k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k5_finsub_1(A)) )
=> ( B = k9_setwiseo(A)
<=> ! [C] :
( m1_subset_1(C,k5_finsub_1(A))
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(A))
=> k2_binop_1(k5_finsub_1(A),k5_finsub_1(A),k5_finsub_1(A),B,C,D) = k5_setwiseo(A,C,D) ) ) ) ) ).
fof(t46_setwiseo,axiom,
$true ).
fof(t47_setwiseo,axiom,
$true ).
fof(t48_setwiseo,axiom,
$true ).
fof(t49_setwiseo,axiom,
! [A] : v3_binop_1(k9_setwiseo(A),k5_finsub_1(A)) ).
fof(t50_setwiseo,axiom,
! [A] : v1_binop_1(k9_setwiseo(A),k5_finsub_1(A)) ).
fof(t51_setwiseo,axiom,
! [A] : v2_binop_1(k9_setwiseo(A),k5_finsub_1(A)) ).
fof(t52_setwiseo,axiom,
! [A] : r3_binop_1(k5_finsub_1(A),k1_setwiseo(A),k9_setwiseo(A)) ).
fof(t53_setwiseo,axiom,
! [A] : v1_setwiseo(k9_setwiseo(A),k5_finsub_1(A)) ).
fof(t54_setwiseo,axiom,
! [A] : r3_binop_1(k5_finsub_1(A),k3_binop_1(k5_finsub_1(A),k9_setwiseo(A)),k9_setwiseo(A)) ).
fof(t55_setwiseo,axiom,
! [A] : k3_binop_1(k5_finsub_1(A),k9_setwiseo(A)) = k1_xboole_0 ).
fof(d5_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C] :
( m1_subset_1(C,k5_finsub_1(A))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,k5_finsub_1(B))
& m2_relset_1(D,A,k5_finsub_1(B)) )
=> k10_setwiseo(A,B,C,D) = k7_setwiseo(A,k5_finsub_1(B),k9_setwiseo(B),C,D) ) ) ) ).
fof(t56_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,k5_finsub_1(B))
& m2_relset_1(C,A,k5_finsub_1(B)) )
=> ! [D] :
( m1_subset_1(D,A)
=> k10_setwiseo(A,B,k2_setwiseo(A,D),C) = k8_funct_2(A,k5_finsub_1(B),C,D) ) ) ) ).
fof(t57_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,k5_finsub_1(B))
& m2_relset_1(C,A,k5_finsub_1(B)) )
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> k10_setwiseo(A,B,k3_setwiseo(A,D,E),C) = k5_setwiseo(B,k8_funct_2(A,k5_finsub_1(B),C,D),k8_funct_2(A,k5_finsub_1(B),C,E)) ) ) ) ) ).
fof(t58_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,k5_finsub_1(B))
& m2_relset_1(C,A,k5_finsub_1(B)) )
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> ! [F] :
( m1_subset_1(F,A)
=> k10_setwiseo(A,B,k4_setwiseo(A,D,E,F),C) = k5_setwiseo(B,k5_setwiseo(B,k8_funct_2(A,k5_finsub_1(B),C,D),k8_funct_2(A,k5_finsub_1(B),C,E)),k8_funct_2(A,k5_finsub_1(B),C,F)) ) ) ) ) ) ).
fof(t59_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,k5_finsub_1(B))
& m2_relset_1(C,A,k5_finsub_1(B)) )
=> k10_setwiseo(A,B,k1_setwiseo(A),C) = k1_xboole_0 ) ) ).
fof(t60_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,k5_finsub_1(B))
& m2_relset_1(C,A,k5_finsub_1(B)) )
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,k5_finsub_1(A))
=> k10_setwiseo(A,B,k5_setwiseo(A,E,k2_setwiseo(A,D)),C) = k5_setwiseo(B,k10_setwiseo(A,B,E,C),k8_funct_2(A,k5_finsub_1(B),C,D)) ) ) ) ) ).
fof(t61_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,k5_finsub_1(B))
& m2_relset_1(C,A,k5_finsub_1(B)) )
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(A))
=> k10_setwiseo(A,B,D,C) = k3_tarski(k8_setwiseo(A,k5_finsub_1(B),C,D)) ) ) ) ).
fof(t62_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,k5_finsub_1(B))
& m2_relset_1(C,A,k5_finsub_1(B)) )
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(A))
=> ! [E] :
( m1_subset_1(E,k5_finsub_1(A))
=> k10_setwiseo(A,B,k5_setwiseo(A,D,E),C) = k5_setwiseo(B,k10_setwiseo(A,B,D,C),k10_setwiseo(A,B,E,C)) ) ) ) ) ).
fof(t63_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C,D] :
( m1_subset_1(D,k5_finsub_1(A))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,B)
& m2_relset_1(E,A,B) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,B,k5_finsub_1(C))
& m2_relset_1(F,B,k5_finsub_1(C)) )
=> k10_setwiseo(B,C,k8_setwiseo(A,B,E,D),F) = k10_setwiseo(A,C,D,k7_funct_2(A,B,k5_finsub_1(C),E,F)) ) ) ) ) ) ).
fof(t64_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C,D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,A),A)
& m2_relset_1(D,k2_zfmisc_1(A,A),A) )
=> ( ( v1_binop_1(D,A)
& v2_binop_1(D,A)
& v3_binop_1(D,A) )
=> ! [E] :
( m1_subset_1(E,k5_finsub_1(B))
=> ( E != k1_xboole_0
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,B,k5_finsub_1(C))
& m2_relset_1(F,B,k5_finsub_1(C)) )
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,k5_finsub_1(C),A)
& m2_relset_1(G,k5_finsub_1(C),A) )
=> ( ! [H] :
( m1_subset_1(H,k5_finsub_1(C))
=> ! [I] :
( m1_subset_1(I,k5_finsub_1(C))
=> k8_funct_2(k5_finsub_1(C),A,G,k5_setwiseo(C,H,I)) = k2_binop_1(A,A,A,D,k8_funct_2(k5_finsub_1(C),A,G,H),k8_funct_2(k5_finsub_1(C),A,G,I)) ) )
=> k8_funct_2(k5_finsub_1(C),A,G,k10_setwiseo(B,C,E,F)) = k7_setwiseo(B,A,D,E,k7_funct_2(B,k5_finsub_1(C),A,F,G)) ) ) ) ) ) ) ) ) ) ).
fof(t65_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C,D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(B,B),B)
& m2_relset_1(D,k2_zfmisc_1(B,B),B) )
=> ( ( v1_binop_1(D,B)
& v2_binop_1(D,B)
& v3_binop_1(D,B)
& v1_setwiseo(D,B) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,k5_finsub_1(C))
& m2_relset_1(E,A,k5_finsub_1(C)) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k5_finsub_1(C),B)
& m2_relset_1(F,k5_finsub_1(C),B) )
=> ( ( k8_funct_2(k5_finsub_1(C),B,F,k1_setwiseo(C)) = k3_binop_1(B,D)
& ! [G] :
( m1_subset_1(G,k5_finsub_1(C))
=> ! [H] :
( m1_subset_1(H,k5_finsub_1(C))
=> k8_funct_2(k5_finsub_1(C),B,F,k5_setwiseo(C,G,H)) = k2_binop_1(B,B,B,D,k8_funct_2(k5_finsub_1(C),B,F,G),k8_funct_2(k5_finsub_1(C),B,F,H)) ) ) )
=> ! [G] :
( m1_subset_1(G,k5_finsub_1(A))
=> k8_funct_2(k5_finsub_1(C),B,F,k10_setwiseo(A,C,G,E)) = k7_setwiseo(A,B,D,G,k7_funct_2(A,k5_finsub_1(C),B,E,F)) ) ) ) ) ) ) ) ) ).
fof(d6_setwiseo,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k5_finsub_1(A))
& m2_relset_1(B,A,k5_finsub_1(A)) )
=> ( B = k11_setwiseo(A)
<=> ! [C] :
( r2_hidden(C,A)
=> k1_funct_1(B,C) = k1_tarski(C) ) ) ) ).
fof(t66_setwiseo,axiom,
$true ).
fof(t67_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k5_finsub_1(A))
& m2_relset_1(B,A,k5_finsub_1(A)) )
=> ( B = k11_setwiseo(A)
<=> ! [C] :
( m1_subset_1(C,A)
=> k8_funct_2(A,k5_finsub_1(A),B,C) = k2_setwiseo(A,C) ) ) ) ) ).
fof(t68_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C] :
( m1_subset_1(C,A)
=> ( r2_hidden(B,k8_funct_2(A,k5_finsub_1(A),k11_setwiseo(A),C))
<=> B = C ) ) ) ).
fof(t69_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ( ( r2_hidden(B,k8_funct_2(A,k5_finsub_1(A),k11_setwiseo(A),D))
& r2_hidden(C,k8_funct_2(A,k5_finsub_1(A),k11_setwiseo(A),D)) )
=> B = C ) ) ) ) ) ).
fof(t70_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,k5_finsub_1(B))
& m2_relset_1(C,A,k5_finsub_1(B)) )
=> ! [D] :
( m1_subset_1(D,k5_finsub_1(A))
=> ! [E] :
( r2_hidden(E,k10_setwiseo(A,B,D,C))
<=> ? [F] :
( m1_subset_1(F,A)
& r2_hidden(F,D)
& r2_hidden(E,k8_funct_2(A,k5_finsub_1(B),C,F)) ) ) ) ) ) ).
fof(t71_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k5_finsub_1(A))
=> k10_setwiseo(A,A,B,k11_setwiseo(A)) = B ) ) ).
fof(t72_setwiseo,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C,D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,k5_finsub_1(B))
& m2_relset_1(D,A,k5_finsub_1(B)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_finsub_1(B),k5_finsub_1(C))
& m2_relset_1(E,k5_finsub_1(B),k5_finsub_1(C)) )
=> ( ( k8_funct_2(k5_finsub_1(B),k5_finsub_1(C),E,k1_setwiseo(B)) = k1_setwiseo(C)
& ! [F] :
( m1_subset_1(F,k5_finsub_1(B))
=> ! [G] :
( m1_subset_1(G,k5_finsub_1(B))
=> k8_funct_2(k5_finsub_1(B),k5_finsub_1(C),E,k5_setwiseo(B,F,G)) = k5_setwiseo(C,k8_funct_2(k5_finsub_1(B),k5_finsub_1(C),E,F),k8_funct_2(k5_finsub_1(B),k5_finsub_1(C),E,G)) ) ) )
=> ! [F] :
( m1_subset_1(F,k5_finsub_1(A))
=> k8_funct_2(k5_finsub_1(B),k5_finsub_1(C),E,k10_setwiseo(A,B,F,D)) = k10_setwiseo(A,C,F,k7_funct_2(A,k5_finsub_1(B),k5_finsub_1(C),D,E)) ) ) ) ) ) ).
fof(s1_setwiseo,axiom,
? [A] :
( v1_funct_1(A)
& v1_funct_2(A,f1_s1_setwiseo,k5_finsub_1(f1_s1_setwiseo))
& m2_relset_1(A,f1_s1_setwiseo,k5_finsub_1(f1_s1_setwiseo))
& ! [B] :
( m1_subset_1(B,f1_s1_setwiseo)
=> ! [C] :
( m1_subset_1(C,f1_s1_setwiseo)
=> ( r2_hidden(C,k8_funct_2(f1_s1_setwiseo,k5_finsub_1(f1_s1_setwiseo),A,B))
<=> ( r2_hidden(C,f2_s1_setwiseo)
& p1_s1_setwiseo(C,B) ) ) ) ) ) ).
fof(s2_setwiseo,axiom,
( ( p1_s2_setwiseo(k1_setwiseo(f1_s2_setwiseo))
& ! [A] :
( m1_subset_1(A,k5_finsub_1(f1_s2_setwiseo))
=> ! [B] :
( m1_subset_1(B,f1_s2_setwiseo)
=> ( p1_s2_setwiseo(A)
=> ( r2_hidden(B,A)
| p1_s2_setwiseo(k5_setwiseo(f1_s2_setwiseo,A,k2_setwiseo(f1_s2_setwiseo,B))) ) ) ) ) )
=> ! [A] :
( m1_subset_1(A,k5_finsub_1(f1_s2_setwiseo))
=> p1_s2_setwiseo(A) ) ) ).
fof(s3_setwiseo,axiom,
( ( ! [A] :
( m1_subset_1(A,f1_s3_setwiseo)
=> p1_s3_setwiseo(k2_setwiseo(f1_s3_setwiseo,A)) )
& ! [A] :
( m1_subset_1(A,k5_finsub_1(f1_s3_setwiseo))
=> ! [B] :
( m1_subset_1(B,k5_finsub_1(f1_s3_setwiseo))
=> ( ( p1_s3_setwiseo(A)
& p1_s3_setwiseo(B) )
=> ( A = k1_xboole_0
| B = k1_xboole_0
| p1_s3_setwiseo(k5_setwiseo(f1_s3_setwiseo,A,B)) ) ) ) ) )
=> ! [A] :
( m1_subset_1(A,k5_finsub_1(f1_s3_setwiseo))
=> ( A != k1_xboole_0
=> p1_s3_setwiseo(A) ) ) ) ).
fof(s4_setwiseo,axiom,
( ( p1_s4_setwiseo(k1_setwiseo(f1_s4_setwiseo))
& ! [A] :
( m1_subset_1(A,k5_finsub_1(f1_s4_setwiseo))
=> ! [B] :
( m1_subset_1(B,f1_s4_setwiseo)
=> ( p1_s4_setwiseo(A)
=> p1_s4_setwiseo(k5_setwiseo(f1_s4_setwiseo,A,k2_setwiseo(f1_s4_setwiseo,B))) ) ) ) )
=> ! [A] :
( m1_subset_1(A,k5_finsub_1(f1_s4_setwiseo))
=> p1_s4_setwiseo(A) ) ) ).
fof(dt_k1_setwiseo,axiom,
! [A] :
( v1_xboole_0(k1_setwiseo(A))
& m1_subset_1(k1_setwiseo(A),k5_finsub_1(A)) ) ).
fof(dt_k2_setwiseo,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A) )
=> m1_subset_1(k2_setwiseo(A,B),k5_finsub_1(A)) ) ).
fof(redefinition_k2_setwiseo,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A) )
=> k2_setwiseo(A,B) = k1_tarski(B) ) ).
fof(dt_k3_setwiseo,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> m1_subset_1(k3_setwiseo(A,B,C),k5_finsub_1(A)) ) ).
fof(commutativity_k3_setwiseo,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> k3_setwiseo(A,B,C) = k3_setwiseo(A,C,B) ) ).
fof(redefinition_k3_setwiseo,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> k3_setwiseo(A,B,C) = k2_tarski(B,C) ) ).
fof(dt_k4_setwiseo,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A)
& m1_subset_1(D,A) )
=> m1_subset_1(k4_setwiseo(A,B,C,D),k5_finsub_1(A)) ) ).
fof(redefinition_k4_setwiseo,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A)
& m1_subset_1(D,A) )
=> k4_setwiseo(A,B,C,D) = k1_enumset1(B,C,D) ) ).
fof(dt_k5_setwiseo,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k5_finsub_1(A))
& m1_subset_1(C,k5_finsub_1(A)) )
=> m1_subset_1(k5_setwiseo(A,B,C),k5_finsub_1(A)) ) ).
fof(commutativity_k5_setwiseo,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k5_finsub_1(A))
& m1_subset_1(C,k5_finsub_1(A)) )
=> k5_setwiseo(A,B,C) = k5_setwiseo(A,C,B) ) ).
fof(idempotence_k5_setwiseo,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k5_finsub_1(A))
& m1_subset_1(C,k5_finsub_1(A)) )
=> k5_setwiseo(A,B,B) = B ) ).
fof(redefinition_k5_setwiseo,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k5_finsub_1(A))
& m1_subset_1(C,k5_finsub_1(A)) )
=> k5_setwiseo(A,B,C) = k2_xboole_0(B,C) ) ).
fof(dt_k6_setwiseo,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k5_finsub_1(A))
& m1_subset_1(C,k5_finsub_1(A)) )
=> m1_subset_1(k6_setwiseo(A,B,C),k5_finsub_1(A)) ) ).
fof(redefinition_k6_setwiseo,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k5_finsub_1(A))
& m1_subset_1(C,k5_finsub_1(A)) )
=> k6_setwiseo(A,B,C) = k4_xboole_0(B,C) ) ).
fof(dt_k7_setwiseo,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(B,B),B)
& m1_relset_1(C,k2_zfmisc_1(B,B),B)
& m1_subset_1(D,k5_finsub_1(A))
& v1_funct_1(E)
& v1_funct_2(E,A,B)
& m1_relset_1(E,A,B) )
=> m1_subset_1(k7_setwiseo(A,B,C,D,E),B) ) ).
fof(dt_k8_setwiseo,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& v1_funct_1(C)
& v1_funct_2(C,A,B)
& m1_relset_1(C,A,B)
& m1_subset_1(D,k5_finsub_1(A)) )
=> m1_subset_1(k8_setwiseo(A,B,C,D),k5_finsub_1(B)) ) ).
fof(redefinition_k8_setwiseo,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& v1_funct_1(C)
& v1_funct_2(C,A,B)
& m1_relset_1(C,A,B)
& m1_subset_1(D,k5_finsub_1(A)) )
=> k8_setwiseo(A,B,C,D) = k9_relat_1(C,D) ) ).
fof(dt_k9_setwiseo,axiom,
! [A] :
( v1_funct_1(k9_setwiseo(A))
& v1_funct_2(k9_setwiseo(A),k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k5_finsub_1(A))
& m2_relset_1(k9_setwiseo(A),k2_zfmisc_1(k5_finsub_1(A),k5_finsub_1(A)),k5_finsub_1(A)) ) ).
fof(dt_k10_setwiseo,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(C,k5_finsub_1(A))
& v1_funct_1(D)
& v1_funct_2(D,A,k5_finsub_1(B))
& m1_relset_1(D,A,k5_finsub_1(B)) )
=> m1_subset_1(k10_setwiseo(A,B,C,D),k5_finsub_1(B)) ) ).
fof(dt_k11_setwiseo,axiom,
! [A] :
( v1_funct_1(k11_setwiseo(A))
& v1_funct_2(k11_setwiseo(A),A,k5_finsub_1(A))
& m2_relset_1(k11_setwiseo(A),A,k5_finsub_1(A)) ) ).
%------------------------------------------------------------------------------