SET007 Axioms: SET007+396.ax
%------------------------------------------------------------------------------
% File : SET007+396 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Products of Many Sorted Algebras
% 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 : pralg_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 73 ( 14 unt; 0 def)
% Number of atoms : 465 ( 62 equ)
% Maximal formula atoms : 26 ( 6 avg)
% Number of connectives : 478 ( 86 ~; 0 |; 215 &)
% ( 17 <=>; 160 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 8 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 27 ( 25 usr; 1 prp; 0-4 aty)
% Number of functors : 53 ( 53 usr; 1 con; 0-9 aty)
% Number of variables : 221 ( 216 !; 5 ?)
% 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_pralg_2,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_fraenkel(A)
& v1_pralg_2(A) ) ).
fof(fc1_pralg_2,axiom,
! [A,B] :
( ( v2_relat_1(B)
& m1_pboole(B,A) )
=> ( ~ v1_xboole_0(k4_card_3(B))
& v1_fraenkel(k4_card_3(B))
& v1_pralg_2(k4_card_3(B)) ) ) ).
fof(fc2_pralg_2,axiom,
! [A,B,C] :
( ( v2_relat_1(B)
& m1_pboole(B,A)
& v2_relat_1(C)
& m1_pboole(C,A) )
=> ( v1_relat_1(k11_pboole(A,B,C))
& v2_relat_1(k11_pboole(A,B,C))
& v1_funct_1(k11_pboole(A,B,C)) ) ) ).
fof(fc3_pralg_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ~ v1_xboole_0(k7_pralg_2(A,B)) ) ).
fof(fc4_pralg_2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& ~ v3_struct_0(B)
& l1_msualg_1(B)
& m2_pralg_2(C,A,B) )
=> ~ v1_xboole_0(k8_pralg_2(A,B,C)) ) ).
fof(fc5_pralg_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& l3_msualg_1(B,A)
& v5_msualg_1(C,A)
& l3_msualg_1(C,A) )
=> v4_msualg_1(k9_pralg_2(A,B,C),A) ) ).
fof(fc6_pralg_2,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& l1_msualg_1(B)
& m1_subset_1(C,u1_struct_0(B))
& m2_pralg_2(D,A,B) )
=> ( v1_relat_1(k10_pralg_2(A,B,C,D))
& v2_relat_1(k10_pralg_2(A,B,C,D))
& v1_funct_1(k10_pralg_2(A,B,C,D)) ) ) ).
fof(fc7_pralg_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& l1_msualg_1(B)
& m2_pralg_2(C,A,B) )
=> ( v1_relat_1(k11_pralg_2(A,B,C))
& v2_relat_1(k11_pralg_2(A,B,C))
& ~ v3_relat_1(k11_pralg_2(A,B,C))
& v1_funct_1(k11_pralg_2(A,B,C)) ) ) ).
fof(fc8_pralg_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B)
& m2_pralg_2(C,A,B) )
=> v4_msualg_1(k15_pralg_2(A,B,C),B) ) ).
fof(d1_pralg_2,axiom,
! [A] :
( v1_pralg_2(A)
<=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( r2_hidden(B,A)
& r2_hidden(C,A) )
=> k1_relat_1(B) = k1_relat_1(C) ) ) ) ) ).
fof(t1_pralg_2,axiom,
( v1_fraenkel(k1_tarski(k1_xboole_0))
& v1_pralg_2(k1_tarski(k1_xboole_0)) ) ).
fof(d2_pralg_2,axiom,
! [A] :
( ( v1_fraenkel(A)
& v1_pralg_2(A) )
=> ! [B] :
( ( A != k1_xboole_0
=> ( B = k1_pralg_2(A)
<=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r2_hidden(C,A)
=> B = k1_relat_1(C) ) ) ) )
& ( A = k1_xboole_0
=> ( B = k1_pralg_2(A)
<=> B = k1_xboole_0 ) ) ) ) ).
fof(t2_pralg_2,axiom,
! [A] :
( ( v1_fraenkel(A)
& v1_pralg_2(A) )
=> ( A = k1_tarski(k1_xboole_0)
=> k1_pralg_2(A) = k1_xboole_0 ) ) ).
fof(d3_pralg_2,axiom,
$true ).
fof(d4_pralg_2,axiom,
$true ).
fof(d5_pralg_2,axiom,
$true ).
fof(d6_pralg_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( B = k2_pralg_2(A)
<=> ( ! [C] :
( r2_hidden(C,k1_relat_1(B))
<=> ? [D] :
( v1_relat_1(D)
& v1_funct_1(D)
& r2_hidden(D,k1_relat_1(A))
& C = k10_funct_6(D) ) )
& ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r2_hidden(C,k1_relat_1(B))
=> k1_funct_1(B,C) = k1_funct_1(A,k10_funct_6(C)) ) ) ) ) ) ) ).
fof(t3_pralg_2,axiom,
$true ).
fof(t4_pralg_2,axiom,
$true ).
fof(t5_pralg_2,axiom,
$true ).
fof(t6_pralg_2,axiom,
$true ).
fof(t7_pralg_2,axiom,
$true ).
fof(t8_pralg_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( k1_relat_1(A) = k1_tarski(k1_xboole_0)
=> k2_pralg_2(A) = A ) ) ).
fof(d7_pralg_2,axiom,
$true ).
fof(d8_pralg_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_funcop_1(A) )
=> ! [B] :
( ( v1_funcop_1(B)
& m1_pboole(B,k4_card_3(k2_funct_6(A))) )
=> ( B = k3_pralg_2(A)
<=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( r2_hidden(C,k4_card_3(k2_funct_6(A)))
=> k1_funct_1(B,C) = k15_pralg_1(A,C) ) ) ) ) ) ).
fof(t9_pralg_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C] :
( m1_pboole(C,A)
=> ! [D] :
( m1_pboole(D,A)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,B,A)
& m2_relset_1(E,B,A) )
=> r6_pboole(B,k7_pboole(B,A,E,k11_pboole(A,C,D)),k11_pboole(B,k7_pboole(B,A,E,C),k7_pboole(B,A,E,D))) ) ) ) ) ).
fof(d9_pralg_2,axiom,
$true ).
fof(d10_pralg_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C] :
( ( v2_relat_1(C)
& m1_pboole(C,A) )
=> ! [D] :
( ( v2_relat_1(D)
& m1_pboole(D,A) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,B,k3_finseq_2(A))
& m2_relset_1(E,B,k3_finseq_2(A)) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,B,A)
& m2_relset_1(F,B,A) )
=> ! [G,H] :
( ( v1_funct_1(H)
& v1_funct_2(H,k1_funct_1(k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,C)),G),k1_funct_1(k8_pboole(B,A,F,C),G))
& m2_relset_1(H,k1_funct_1(k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,C)),G),k1_funct_1(k8_pboole(B,A,F,C),G)) )
=> ! [I] :
( ( v1_funct_1(I)
& v1_funct_2(I,k1_funct_1(k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,D)),G),k1_funct_1(k8_pboole(B,A,F,D),G))
& m2_relset_1(I,k1_funct_1(k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,D)),G),k1_funct_1(k8_pboole(B,A,F,D),G)) )
=> ( r2_hidden(G,B)
=> ! [J] :
( ( v1_funct_1(J)
& v1_funct_2(J,k1_funct_1(k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,k11_pboole(A,C,D))),G),k1_funct_1(k8_pboole(B,A,F,k11_pboole(A,C,D)),G))
& m2_relset_1(J,k1_funct_1(k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,k11_pboole(A,C,D))),G),k1_funct_1(k8_pboole(B,A,F,k11_pboole(A,C,D)),G)) )
=> ( J = k4_pralg_2(A,B,C,D,E,F,G,H,I)
<=> ! [K] :
( ( v1_relat_1(K)
& v1_funct_1(K) )
=> ( r2_hidden(K,k1_funct_1(k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,k11_pboole(A,C,D))),G))
=> k1_funct_1(J,K) = k4_tarski(k1_funct_1(H,k15_mcart_1(K)),k1_funct_1(I,k16_mcart_1(K))) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d11_pralg_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C] :
( ( v2_relat_1(C)
& m1_pboole(C,A) )
=> ! [D] :
( ( v2_relat_1(D)
& m1_pboole(D,A) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,B,k3_finseq_2(A))
& m2_relset_1(E,B,k3_finseq_2(A)) )
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,B,A)
& m2_relset_1(F,B,A) )
=> ! [G] :
( m3_pboole(G,B,k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,C)),k8_pboole(B,A,F,C))
=> ! [H] :
( m3_pboole(H,B,k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,D)),k8_pboole(B,A,F,D))
=> ! [I] :
( m3_pboole(I,B,k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,k11_pboole(A,C,D))),k8_pboole(B,A,F,k11_pboole(A,C,D)))
=> ( I = k5_pralg_2(A,B,C,D,E,F,G,H)
<=> ! [J] :
( r2_hidden(J,B)
=> ! [K] :
( ( v1_funct_1(K)
& v1_funct_2(K,k1_funct_1(k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,C)),J),k1_funct_1(k8_pboole(B,A,F,C),J))
& m2_relset_1(K,k1_funct_1(k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,C)),J),k1_funct_1(k8_pboole(B,A,F,C),J)) )
=> ! [L] :
( ( v1_funct_1(L)
& v1_funct_2(L,k1_funct_1(k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,D)),J),k1_funct_1(k8_pboole(B,A,F,D),J))
& m2_relset_1(L,k1_funct_1(k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,D)),J),k1_funct_1(k8_pboole(B,A,F,D),J)) )
=> ( ( K = k1_funct_1(G,J)
& L = k1_funct_1(H,J) )
=> k1_funct_1(I,J) = k4_pralg_2(A,B,C,D,E,F,J,K,L) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d12_pralg_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l1_msualg_1(B) )
=> ! [C] :
( m1_pboole(C,A)
=> ( m2_pralg_2(C,A,B)
<=> ! [D] :
( r2_hidden(D,A)
=> ( v5_msualg_1(k1_funct_1(C,D),B)
& l3_msualg_1(k1_funct_1(C,D),B) ) ) ) ) ) ).
fof(d13_pralg_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A) )
=> ! [B] :
( l3_msualg_1(B,A)
=> k7_pralg_2(A,B) = k3_tarski(k2_relat_1(u4_msualg_1(A,B))) ) ) ).
fof(t10_pralg_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( l3_msualg_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_msualg_1(A))
=> ( k3_msualg_1(A,C,B) = k4_card_3(k5_relat_1(k1_msualg_1(A,C),u4_msualg_1(A,B)))
& k1_relat_1(k5_relat_1(k1_msualg_1(A,C),u4_msualg_1(A,B))) = k1_relat_1(k1_msualg_1(A,C))
& k4_msualg_1(A,C,B) = k1_funct_1(u4_msualg_1(A,B),k2_msualg_1(A,C)) ) ) ) ) ).
fof(t11_pralg_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( l3_msualg_1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_msualg_1(A))
=> ( k1_msualg_1(A,C) = k1_xboole_0
=> k3_msualg_1(A,C,B) = k1_tarski(k1_xboole_0) ) ) ) ) ).
fof(d15_pralg_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( ( v5_msualg_1(C,A)
& l3_msualg_1(C,A) )
=> k9_pralg_2(A,B,C) = g3_msualg_1(A,k11_pboole(u1_struct_0(A),u4_msualg_1(A,B),u4_msualg_1(A,C)),k5_pralg_2(u1_struct_0(A),u1_msualg_1(A),u4_msualg_1(A,B),u4_msualg_1(A,C),u2_msualg_1(A),u3_msualg_1(A),u5_msualg_1(A,B),u5_msualg_1(A,C))) ) ) ) ).
fof(d16_pralg_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l1_msualg_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m2_pralg_2(D,A,B)
=> ! [E] :
( m1_pboole(E,A)
=> ( ( A != k1_xboole_0
=> ( E = k10_pralg_2(A,B,C,D)
<=> ! [F] :
~ ( r2_hidden(F,A)
& ! [G] :
( l3_msualg_1(G,B)
=> ~ ( G = k1_funct_1(D,F)
& k1_funct_1(E,F) = k1_funct_1(u4_msualg_1(B,G),C) ) ) ) ) )
& ( A = k1_xboole_0
=> ( E = k10_pralg_2(A,B,C,D)
<=> E = k1_xboole_0 ) ) ) ) ) ) ) ).
fof(d17_pralg_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l1_msualg_1(B) )
=> ! [C] :
( m2_pralg_2(C,A,B)
=> ! [D] :
( m1_pboole(D,u1_struct_0(B))
=> ( D = k11_pralg_2(A,B,C)
<=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> k1_funct_1(D,E) = k4_card_3(k10_pralg_2(A,B,E,C)) ) ) ) ) ) ).
fof(d18_pralg_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l1_msualg_1(B) )
=> ! [C] :
( m2_pralg_2(C,A,B)
=> ! [D] :
( ( v1_funcop_1(D)
& m1_pboole(D,A) )
=> ( ( A != k1_xboole_0
=> ( D = k12_pralg_2(A,B,C)
<=> ! [E] :
~ ( r2_hidden(E,A)
& ! [F] :
( l3_msualg_1(F,B)
=> ~ ( F = k1_funct_1(C,E)
& k1_funct_1(D,E) = u5_msualg_1(B,F) ) ) ) ) )
& ( A = k1_xboole_0
=> ( D = k12_pralg_2(A,B,C)
<=> D = k1_xboole_0 ) ) ) ) ) ) ).
fof(t12_pralg_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l1_msualg_1(B) )
=> ! [C] :
( m2_pralg_2(C,A,B)
=> k1_relat_1(k4_funct_5(k12_pralg_2(A,B,C))) = k2_zfmisc_1(A,u1_msualg_1(B)) ) ) ).
fof(t13_pralg_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m2_pralg_2(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_msualg_1(B))
=> r2_hidden(k10_funct_6(k12_pralg_2(A,B,C)),k1_funct_2(u1_msualg_1(B),k1_funct_2(A,k2_relat_1(k4_funct_5(k12_pralg_2(A,B,C)))))) ) ) ) ) ).
fof(d19_pralg_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m2_pralg_2(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_msualg_1(B))
=> k13_pralg_2(A,B,C,D) = k1_funct_1(k10_funct_6(k12_pralg_2(A,B,C)),D) ) ) ) ).
fof(t14_pralg_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( ~ v3_struct_0(C)
& ~ v2_msualg_1(C)
& l1_msualg_1(C) )
=> ! [D] :
( m2_pralg_2(D,A,C)
=> ! [E] :
( m1_subset_1(E,u1_msualg_1(C))
=> k1_funct_1(k13_pralg_2(A,C,D,E),B) = k5_msualg_1(C,E,k6_pralg_2(A,C,D,B)) ) ) ) ) ) ).
fof(t15_pralg_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m2_pralg_2(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_msualg_1(B))
=> ! [E] :
( r2_hidden(E,k2_relat_1(k3_pralg_2(k13_pralg_2(A,B,C,D))))
=> ( v1_relat_1(E)
& v1_funct_1(E) ) ) ) ) ) ) ).
fof(t16_pralg_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m2_pralg_2(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_msualg_1(B))
=> ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E) )
=> ( r2_hidden(E,k2_relat_1(k3_pralg_2(k13_pralg_2(A,B,C,D))))
=> ( k1_relat_1(E) = A
& ! [F] :
( m1_subset_1(F,A)
=> r2_hidden(k1_funct_1(E,F),k4_msualg_1(B,D,k6_pralg_2(A,B,C,F))) ) ) ) ) ) ) ) ) ).
fof(t17_pralg_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m2_pralg_2(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_msualg_1(B))
=> ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E) )
=> ( r2_hidden(E,k1_relat_1(k3_pralg_2(k13_pralg_2(A,B,C,D))))
=> ( k1_relat_1(E) = A
& ! [F] :
( m1_subset_1(F,A)
=> r2_hidden(k1_funct_1(E,F),k3_msualg_1(B,D,k6_pralg_2(A,B,C,F))) )
& r1_tarski(k2_relat_1(E),k1_fraenkel(k1_relat_1(k1_msualg_1(B,D)),k8_pralg_2(A,B,C))) ) ) ) ) ) ) ) ).
fof(t18_pralg_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m2_pralg_2(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_msualg_1(B))
=> ( k1_relat_1(k2_funct_6(k13_pralg_2(A,B,C,D))) = A
& ! [E] :
( m1_subset_1(E,A)
=> k1_funct_1(k2_funct_6(k13_pralg_2(A,B,C,D)),E) = k3_msualg_1(B,D,k6_pralg_2(A,B,C,E)) ) ) ) ) ) ) ).
fof(d20_pralg_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m2_pralg_2(C,A,B)
=> ! [D] :
( m3_pboole(D,u1_msualg_1(B),k8_pboole(u1_msualg_1(B),k3_finseq_2(u1_struct_0(B)),u2_msualg_1(B),k6_pboole(u1_struct_0(B),k11_pralg_2(A,B,C))),k8_pboole(u1_msualg_1(B),u1_struct_0(B),u3_msualg_1(B),k11_pralg_2(A,B,C)))
=> ( ( A != k1_xboole_0
=> ( D = k14_pralg_2(A,B,C)
<=> ! [E] :
( m1_subset_1(E,u1_msualg_1(B))
=> k1_funct_1(D,E) = k1_cqc_lang(k1_msualg_1(B,E),k1_xboole_0,k10_funct_6(k13_pralg_2(A,B,C,E)),k2_pralg_2(k3_pralg_2(k13_pralg_2(A,B,C,E)))) ) ) )
& ( A = k1_xboole_0
=> D = k14_pralg_2(A,B,C) ) ) ) ) ) ).
fof(d21_pralg_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m2_pralg_2(C,A,B)
=> k15_pralg_2(A,B,C) = g3_msualg_1(B,k11_pralg_2(A,B,C),k14_pralg_2(A,B,C)) ) ) ).
fof(t19_pralg_2,axiom,
$true ).
fof(t20_pralg_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B) )
=> ! [C] :
( m2_pralg_2(C,A,B)
=> ( r6_pboole(u1_struct_0(B),u4_msualg_1(B,k15_pralg_2(A,B,C)),k11_pralg_2(A,B,C))
& r6_pboole(u1_msualg_1(B),u5_msualg_1(B,k15_pralg_2(A,B,C)),k14_pralg_2(A,B,C)) ) ) ) ).
fof(dt_m1_pralg_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_fraenkel(A)
& v1_pralg_2(A) )
=> ! [B] :
( m1_pralg_2(B,A)
=> m1_pboole(B,k1_pralg_2(A)) ) ) ).
fof(existence_m1_pralg_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_fraenkel(A)
& v1_pralg_2(A) )
=> ? [B] : m1_pralg_2(B,A) ) ).
fof(redefinition_m1_pralg_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_fraenkel(A)
& v1_pralg_2(A) )
=> ! [B] :
( m1_pralg_2(B,A)
<=> m1_subset_1(B,A) ) ) ).
fof(dt_m2_pralg_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l1_msualg_1(B) )
=> ! [C] :
( m2_pralg_2(C,A,B)
=> m1_pboole(C,A) ) ) ).
fof(existence_m2_pralg_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(B)
& l1_msualg_1(B) )
=> ? [C] : m2_pralg_2(C,A,B) ) ).
fof(dt_k1_pralg_2,axiom,
$true ).
fof(dt_k2_pralg_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v1_relat_1(k2_pralg_2(A))
& v1_funct_1(k2_pralg_2(A)) ) ) ).
fof(dt_k3_pralg_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_funcop_1(A) )
=> ( v1_funcop_1(k3_pralg_2(A))
& m1_pboole(k3_pralg_2(A),k4_card_3(k2_funct_6(A))) ) ) ).
fof(redefinition_k3_pralg_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_funcop_1(A) )
=> k3_pralg_2(A) = k7_funct_6(A) ) ).
fof(dt_k4_pralg_2,axiom,
! [A,B,C,D,E,F,G,H,I] :
( ( ~ v1_xboole_0(A)
& v2_relat_1(C)
& m1_pboole(C,A)
& v2_relat_1(D)
& m1_pboole(D,A)
& v1_funct_1(E)
& v1_funct_2(E,B,k3_finseq_2(A))
& m1_relset_1(E,B,k3_finseq_2(A))
& v1_funct_1(F)
& v1_funct_2(F,B,A)
& m1_relset_1(F,B,A)
& v1_funct_1(H)
& v1_funct_2(H,k1_funct_1(k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,C)),G),k1_funct_1(k8_pboole(B,A,F,C),G))
& m1_relset_1(H,k1_funct_1(k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,C)),G),k1_funct_1(k8_pboole(B,A,F,C),G))
& v1_funct_1(I)
& v1_funct_2(I,k1_funct_1(k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,D)),G),k1_funct_1(k8_pboole(B,A,F,D),G))
& m1_relset_1(I,k1_funct_1(k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,D)),G),k1_funct_1(k8_pboole(B,A,F,D),G)) )
=> ( v1_funct_1(k4_pralg_2(A,B,C,D,E,F,G,H,I))
& v1_funct_2(k4_pralg_2(A,B,C,D,E,F,G,H,I),k1_funct_1(k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,k11_pboole(A,C,D))),G),k1_funct_1(k8_pboole(B,A,F,k11_pboole(A,C,D)),G))
& m2_relset_1(k4_pralg_2(A,B,C,D,E,F,G,H,I),k1_funct_1(k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,k11_pboole(A,C,D))),G),k1_funct_1(k8_pboole(B,A,F,k11_pboole(A,C,D)),G)) ) ) ).
fof(dt_k5_pralg_2,axiom,
! [A,B,C,D,E,F,G,H] :
( ( ~ v1_xboole_0(A)
& v2_relat_1(C)
& m1_pboole(C,A)
& v2_relat_1(D)
& m1_pboole(D,A)
& v1_funct_1(E)
& v1_funct_2(E,B,k3_finseq_2(A))
& m1_relset_1(E,B,k3_finseq_2(A))
& v1_funct_1(F)
& v1_funct_2(F,B,A)
& m1_relset_1(F,B,A)
& m3_pboole(G,B,k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,C)),k8_pboole(B,A,F,C))
& m3_pboole(H,B,k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,D)),k8_pboole(B,A,F,D)) )
=> m3_pboole(k5_pralg_2(A,B,C,D,E,F,G,H),B,k8_pboole(B,k3_finseq_2(A),E,k6_pboole(A,k11_pboole(A,C,D))),k8_pboole(B,A,F,k11_pboole(A,C,D))) ) ).
fof(dt_k6_pralg_2,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v3_struct_0(B)
& l1_msualg_1(B)
& m2_pralg_2(C,A,B)
& m1_subset_1(D,A) )
=> ( v5_msualg_1(k6_pralg_2(A,B,C,D),B)
& l3_msualg_1(k6_pralg_2(A,B,C,D),B) ) ) ).
fof(redefinition_k6_pralg_2,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v3_struct_0(B)
& l1_msualg_1(B)
& m2_pralg_2(C,A,B)
& m1_subset_1(D,A) )
=> k6_pralg_2(A,B,C,D) = k1_funct_1(C,D) ) ).
fof(dt_k7_pralg_2,axiom,
$true ).
fof(dt_k8_pralg_2,axiom,
$true ).
fof(dt_k9_pralg_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& l3_msualg_1(B,A)
& v5_msualg_1(C,A)
& l3_msualg_1(C,A) )
=> l3_msualg_1(k9_pralg_2(A,B,C),A) ) ).
fof(dt_k10_pralg_2,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& l1_msualg_1(B)
& m1_subset_1(C,u1_struct_0(B))
& m2_pralg_2(D,A,B) )
=> m1_pboole(k10_pralg_2(A,B,C,D),A) ) ).
fof(dt_k11_pralg_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& l1_msualg_1(B)
& m2_pralg_2(C,A,B) )
=> m1_pboole(k11_pralg_2(A,B,C),u1_struct_0(B)) ) ).
fof(dt_k12_pralg_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& l1_msualg_1(B)
& m2_pralg_2(C,A,B) )
=> ( v1_funcop_1(k12_pralg_2(A,B,C))
& m1_pboole(k12_pralg_2(A,B,C),A) ) ) ).
fof(dt_k13_pralg_2,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B)
& m2_pralg_2(C,A,B)
& m1_subset_1(D,u1_msualg_1(B)) )
=> ( v1_funcop_1(k13_pralg_2(A,B,C,D))
& m1_pboole(k13_pralg_2(A,B,C,D),A) ) ) ).
fof(dt_k14_pralg_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B)
& m2_pralg_2(C,A,B) )
=> m3_pboole(k14_pralg_2(A,B,C),u1_msualg_1(B),k8_pboole(u1_msualg_1(B),k3_finseq_2(u1_struct_0(B)),u2_msualg_1(B),k6_pboole(u1_struct_0(B),k11_pralg_2(A,B,C))),k8_pboole(u1_msualg_1(B),u1_struct_0(B),u3_msualg_1(B),k11_pralg_2(A,B,C))) ) ).
fof(dt_k15_pralg_2,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B)
& m2_pralg_2(C,A,B) )
=> l3_msualg_1(k15_pralg_2(A,B,C),B) ) ).
fof(d14_pralg_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_msualg_1(B) )
=> ! [C] :
( m2_pralg_2(C,A,B)
=> k8_pralg_2(A,B,C) = k3_tarski(a_3_0_pralg_2(A,B,C)) ) ) ) ).
fof(fraenkel_a_3_0_pralg_2,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& ~ v3_struct_0(C)
& l1_msualg_1(C)
& m2_pralg_2(D,B,C) )
=> ( r2_hidden(A,a_3_0_pralg_2(B,C,D))
<=> ? [E] :
( m1_subset_1(E,B)
& A = k7_pralg_2(C,k6_pralg_2(B,C,D,E)) ) ) ) ).
%------------------------------------------------------------------------------