SET007 Axioms: SET007+18.ax
%------------------------------------------------------------------------------
% File : SET007+18 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Tuples, Projections and Cartesian Products
% 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 : mcart_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 129 ( 32 unt; 0 def)
% Number of atoms : 587 ( 333 equ)
% Maximal formula atoms : 15 ( 4 avg)
% Number of connectives : 656 ( 198 ~; 44 |; 247 &)
% ( 17 <=>; 150 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 10 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 8 ( 6 usr; 1 prp; 0-2 aty)
% Number of functors : 32 ( 32 usr; 1 con; 0-8 aty)
% Number of variables : 625 ( 590 !; 35 ?)
% 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_mcart_1,axiom,
! [A] :
~ ( A != k1_xboole_0
& ! [B] :
~ ( r2_hidden(B,A)
& r1_xboole_0(B,A) ) ) ).
fof(t2_mcart_1,axiom,
! [A] :
~ ( A != k1_xboole_0
& ! [B] :
~ ( r2_hidden(B,A)
& ! [C] :
( r2_hidden(C,B)
=> r1_xboole_0(C,A) ) ) ) ).
fof(t3_mcart_1,axiom,
! [A] :
~ ( A != k1_xboole_0
& ! [B] :
~ ( r2_hidden(B,A)
& ! [C,D] :
( ( r2_hidden(C,D)
& r2_hidden(D,B) )
=> r1_xboole_0(C,A) ) ) ) ).
fof(t4_mcart_1,axiom,
! [A] :
~ ( A != k1_xboole_0
& ! [B] :
~ ( r2_hidden(B,A)
& ! [C,D,E] :
( ( r2_hidden(C,D)
& r2_hidden(D,E)
& r2_hidden(E,B) )
=> r1_xboole_0(C,A) ) ) ) ).
fof(t5_mcart_1,axiom,
! [A] :
~ ( A != k1_xboole_0
& ! [B] :
~ ( r2_hidden(B,A)
& ! [C,D,E,F] :
( ( r2_hidden(C,D)
& r2_hidden(D,E)
& r2_hidden(E,F)
& r2_hidden(F,B) )
=> r1_xboole_0(C,A) ) ) ) ).
fof(t6_mcart_1,axiom,
! [A] :
~ ( A != k1_xboole_0
& ! [B] :
~ ( r2_hidden(B,A)
& ! [C,D,E,F,G] :
( ( r2_hidden(C,D)
& r2_hidden(D,E)
& r2_hidden(E,F)
& r2_hidden(F,G)
& r2_hidden(G,B) )
=> r1_xboole_0(C,A) ) ) ) ).
fof(d1_mcart_1,axiom,
! [A] :
( ? [B,C] : A = k4_tarski(B,C)
=> ! [B] :
( B = k1_mcart_1(A)
<=> ! [C,D] :
( A = k4_tarski(C,D)
=> B = C ) ) ) ).
fof(d2_mcart_1,axiom,
! [A] :
( ? [B,C] : A = k4_tarski(B,C)
=> ! [B] :
( B = k2_mcart_1(A)
<=> ! [C,D] :
( A = k4_tarski(C,D)
=> B = D ) ) ) ).
fof(t7_mcart_1,axiom,
! [A,B] :
( k1_mcart_1(k4_tarski(A,B)) = A
& k2_mcart_1(k4_tarski(A,B)) = B ) ).
fof(t8_mcart_1,axiom,
! [A] :
( ? [B,C] : A = k4_tarski(B,C)
=> k4_tarski(k1_mcart_1(A),k2_mcart_1(A)) = A ) ).
fof(t9_mcart_1,axiom,
! [A] :
~ ( A != k1_xboole_0
& ! [B] :
~ ( r2_hidden(B,A)
& ! [C,D] :
~ ( ( r2_hidden(C,A)
| r2_hidden(D,A) )
& B = k4_tarski(C,D) ) ) ) ).
fof(t10_mcart_1,axiom,
! [A,B,C] :
( r2_hidden(A,k2_zfmisc_1(B,C))
=> ( r2_hidden(k1_mcart_1(A),B)
& r2_hidden(k2_mcart_1(A),C) ) ) ).
fof(t11_mcart_1,axiom,
! [A,B,C] :
( ( r2_hidden(k1_mcart_1(A),B)
& r2_hidden(k2_mcart_1(A),C) )
=> ( ! [D,E] : A != k4_tarski(D,E)
| r2_hidden(A,k2_zfmisc_1(B,C)) ) ) ).
fof(t12_mcart_1,axiom,
! [A,B,C] :
( r2_hidden(A,k2_zfmisc_1(k1_tarski(B),C))
=> ( k1_mcart_1(A) = B
& r2_hidden(k2_mcart_1(A),C) ) ) ).
fof(t13_mcart_1,axiom,
! [A,B,C] :
( r2_hidden(A,k2_zfmisc_1(B,k1_tarski(C)))
=> ( r2_hidden(k1_mcart_1(A),B)
& k2_mcart_1(A) = C ) ) ).
fof(t14_mcart_1,axiom,
! [A,B,C] :
( r2_hidden(A,k2_zfmisc_1(k1_tarski(B),k1_tarski(C)))
=> ( k1_mcart_1(A) = B
& k2_mcart_1(A) = C ) ) ).
fof(t15_mcart_1,axiom,
! [A,B,C,D] :
( r2_hidden(A,k2_zfmisc_1(k2_tarski(B,C),D))
=> ( ( k1_mcart_1(A) = B
| k1_mcart_1(A) = C )
& r2_hidden(k2_mcart_1(A),D) ) ) ).
fof(t16_mcart_1,axiom,
! [A,B,C,D] :
( r2_hidden(A,k2_zfmisc_1(B,k2_tarski(C,D)))
=> ( r2_hidden(k1_mcart_1(A),B)
& ( k2_mcart_1(A) = C
| k2_mcart_1(A) = D ) ) ) ).
fof(t17_mcart_1,axiom,
! [A,B,C,D] :
( r2_hidden(A,k2_zfmisc_1(k2_tarski(B,C),k1_tarski(D)))
=> ( ( k1_mcart_1(A) = B
| k1_mcart_1(A) = C )
& k2_mcart_1(A) = D ) ) ).
fof(t18_mcart_1,axiom,
! [A,B,C,D] :
( r2_hidden(A,k2_zfmisc_1(k1_tarski(B),k2_tarski(C,D)))
=> ( k1_mcart_1(A) = B
& ( k2_mcart_1(A) = C
| k2_mcart_1(A) = D ) ) ) ).
fof(t19_mcart_1,axiom,
! [A,B,C,D,E] :
( r2_hidden(A,k2_zfmisc_1(k2_tarski(B,C),k2_tarski(D,E)))
=> ( ( k1_mcart_1(A) = B
| k1_mcart_1(A) = C )
& ( k2_mcart_1(A) = D
| k2_mcart_1(A) = E ) ) ) ).
fof(t20_mcart_1,axiom,
! [A] :
( ? [B,C] : A = k4_tarski(B,C)
=> ( A != k1_mcart_1(A)
& A != k2_mcart_1(A) ) ) ).
fof(t21_mcart_1,axiom,
$true ).
fof(t22_mcart_1,axiom,
$true ).
fof(t23_mcart_1,axiom,
! [A,B,C] :
( r2_hidden(A,k2_zfmisc_1(B,C))
=> A = k4_tarski(k1_mcart_1(A),k2_mcart_1(A)) ) ).
fof(t24_mcart_1,axiom,
! [A,B] :
~ ( A != k1_xboole_0
& B != k1_xboole_0
& ~ ! [C] :
( m1_subset_1(C,k2_zfmisc_1(A,B))
=> C = k4_tarski(k1_mcart_1(C),k2_mcart_1(C)) ) ) ).
fof(t25_mcart_1,axiom,
! [A,B,C,D] : k2_zfmisc_1(k2_tarski(A,B),k2_tarski(C,D)) = k2_enumset1(k4_tarski(A,C),k4_tarski(A,D),k4_tarski(B,C),k4_tarski(B,D)) ).
fof(t26_mcart_1,axiom,
! [A,B] :
~ ( A != k1_xboole_0
& B != k1_xboole_0
& ~ ! [C] :
( m1_subset_1(C,k2_zfmisc_1(A,B))
=> ( C != k1_mcart_1(C)
& C != k2_mcart_1(C) ) ) ) ).
fof(d3_mcart_1,axiom,
! [A,B,C] : k3_mcart_1(A,B,C) = k4_tarski(k4_tarski(A,B),C) ).
fof(t27_mcart_1,axiom,
$true ).
fof(t28_mcart_1,axiom,
! [A,B,C,D,E,F] :
( k3_mcart_1(A,B,C) = k3_mcart_1(D,E,F)
=> ( A = D
& B = E
& C = F ) ) ).
fof(t29_mcart_1,axiom,
! [A] :
~ ( A != k1_xboole_0
& ! [B] :
~ ( r2_hidden(B,A)
& ! [C,D,E] :
~ ( ( r2_hidden(C,A)
| r2_hidden(D,A) )
& B = k3_mcart_1(C,D,E) ) ) ) ).
fof(d4_mcart_1,axiom,
! [A,B,C,D] : k4_mcart_1(A,B,C,D) = k4_tarski(k3_mcart_1(A,B,C),D) ).
fof(t30_mcart_1,axiom,
$true ).
fof(t31_mcart_1,axiom,
! [A,B,C,D] : k4_mcart_1(A,B,C,D) = k4_tarski(k4_tarski(k4_tarski(A,B),C),D) ).
fof(t32_mcart_1,axiom,
! [A,B,C,D] : k4_mcart_1(A,B,C,D) = k3_mcart_1(k4_tarski(A,B),C,D) ).
fof(t33_mcart_1,axiom,
! [A,B,C,D,E,F,G,H] :
( k4_mcart_1(A,B,C,D) = k4_mcart_1(E,F,G,H)
=> ( A = E
& B = F
& C = G
& D = H ) ) ).
fof(t34_mcart_1,axiom,
! [A] :
~ ( A != k1_xboole_0
& ! [B] :
~ ( r2_hidden(B,A)
& ! [C,D,E,F] :
~ ( ( r2_hidden(C,A)
| r2_hidden(D,A) )
& B = k4_mcart_1(C,D,E,F) ) ) ) ).
fof(t35_mcart_1,axiom,
! [A,B,C] :
( ( A != k1_xboole_0
& B != k1_xboole_0
& C != k1_xboole_0 )
<=> k3_zfmisc_1(A,B,C) != k1_xboole_0 ) ).
fof(t36_mcart_1,axiom,
! [A,B,C,D,E,F] :
( k3_zfmisc_1(A,B,C) = k3_zfmisc_1(D,E,F)
=> ( A = k1_xboole_0
| B = k1_xboole_0
| C = k1_xboole_0
| ( A = D
& B = E
& C = F ) ) ) ).
fof(t37_mcart_1,axiom,
! [A,B,C,D,E,F] :
( k3_zfmisc_1(A,B,C) = k3_zfmisc_1(D,E,F)
=> ( k3_zfmisc_1(A,B,C) = k1_xboole_0
| ( A = D
& B = E
& C = F ) ) ) ).
fof(t38_mcart_1,axiom,
! [A,B] :
( k3_zfmisc_1(A,A,A) = k3_zfmisc_1(B,B,B)
=> A = B ) ).
fof(t39_mcart_1,axiom,
! [A,B,C] : k3_zfmisc_1(k1_tarski(A),k1_tarski(B),k1_tarski(C)) = k1_tarski(k3_mcart_1(A,B,C)) ).
fof(t40_mcart_1,axiom,
! [A,B,C,D] : k3_zfmisc_1(k2_tarski(A,B),k1_tarski(C),k1_tarski(D)) = k2_tarski(k3_mcart_1(A,C,D),k3_mcart_1(B,C,D)) ).
fof(t41_mcart_1,axiom,
! [A,B,C,D] : k3_zfmisc_1(k1_tarski(A),k2_tarski(B,C),k1_tarski(D)) = k2_tarski(k3_mcart_1(A,B,D),k3_mcart_1(A,C,D)) ).
fof(t42_mcart_1,axiom,
! [A,B,C,D] : k3_zfmisc_1(k1_tarski(A),k1_tarski(B),k2_tarski(C,D)) = k2_tarski(k3_mcart_1(A,B,C),k3_mcart_1(A,B,D)) ).
fof(t43_mcart_1,axiom,
! [A,B,C,D,E] : k3_zfmisc_1(k2_tarski(A,B),k2_tarski(C,D),k1_tarski(E)) = k2_enumset1(k3_mcart_1(A,C,E),k3_mcart_1(B,C,E),k3_mcart_1(A,D,E),k3_mcart_1(B,D,E)) ).
fof(t44_mcart_1,axiom,
! [A,B,C,D,E] : k3_zfmisc_1(k2_tarski(A,B),k1_tarski(C),k2_tarski(D,E)) = k2_enumset1(k3_mcart_1(A,C,D),k3_mcart_1(B,C,D),k3_mcart_1(A,C,E),k3_mcart_1(B,C,E)) ).
fof(t45_mcart_1,axiom,
! [A,B,C,D,E] : k3_zfmisc_1(k1_tarski(A),k2_tarski(B,C),k2_tarski(D,E)) = k2_enumset1(k3_mcart_1(A,B,D),k3_mcart_1(A,C,D),k3_mcart_1(A,B,E),k3_mcart_1(A,C,E)) ).
fof(t46_mcart_1,axiom,
! [A,B,C,D,E,F] : k3_zfmisc_1(k2_tarski(A,B),k2_tarski(C,D),k2_tarski(E,F)) = k6_enumset1(k3_mcart_1(A,C,E),k3_mcart_1(A,D,E),k3_mcart_1(A,C,F),k3_mcart_1(A,D,F),k3_mcart_1(B,C,E),k3_mcart_1(B,D,E),k3_mcart_1(B,C,F),k3_mcart_1(B,D,F)) ).
fof(d5_mcart_1,axiom,
! [A,B,C] :
~ ( A != k1_xboole_0
& B != k1_xboole_0
& C != k1_xboole_0
& ~ ! [D] :
( m1_subset_1(D,k3_zfmisc_1(A,B,C))
=> ! [E] :
( m1_subset_1(E,A)
=> ( E = k5_mcart_1(A,B,C,D)
<=> ! [F,G,H] :
( D = k3_mcart_1(F,G,H)
=> E = F ) ) ) ) ) ).
fof(d6_mcart_1,axiom,
! [A,B,C] :
~ ( A != k1_xboole_0
& B != k1_xboole_0
& C != k1_xboole_0
& ~ ! [D] :
( m1_subset_1(D,k3_zfmisc_1(A,B,C))
=> ! [E] :
( m1_subset_1(E,B)
=> ( E = k6_mcart_1(A,B,C,D)
<=> ! [F,G,H] :
( D = k3_mcart_1(F,G,H)
=> E = G ) ) ) ) ) ).
fof(d7_mcart_1,axiom,
! [A,B,C] :
~ ( A != k1_xboole_0
& B != k1_xboole_0
& C != k1_xboole_0
& ~ ! [D] :
( m1_subset_1(D,k3_zfmisc_1(A,B,C))
=> ! [E] :
( m1_subset_1(E,C)
=> ( E = k7_mcart_1(A,B,C,D)
<=> ! [F,G,H] :
( D = k3_mcart_1(F,G,H)
=> E = H ) ) ) ) ) ).
fof(t47_mcart_1,axiom,
! [A,B,C] :
~ ( A != k1_xboole_0
& B != k1_xboole_0
& C != k1_xboole_0
& ? [D] :
( m1_subset_1(D,k3_zfmisc_1(A,B,C))
& ? [E,F,G] :
( D = k3_mcart_1(E,F,G)
& ~ ( k5_mcart_1(A,B,C,D) = E
& k6_mcart_1(A,B,C,D) = F
& k7_mcart_1(A,B,C,D) = G ) ) ) ) ).
fof(t48_mcart_1,axiom,
! [A,B,C] :
~ ( A != k1_xboole_0
& B != k1_xboole_0
& C != k1_xboole_0
& ~ ! [D] :
( m1_subset_1(D,k3_zfmisc_1(A,B,C))
=> D = k3_mcart_1(k5_mcart_1(A,B,C,D),k6_mcart_1(A,B,C,D),k7_mcart_1(A,B,C,D)) ) ) ).
fof(t49_mcart_1,axiom,
! [A,B,C] :
( ~ ( ~ r1_tarski(A,k3_zfmisc_1(A,B,C))
& ~ r1_tarski(A,k3_zfmisc_1(B,C,A))
& ~ r1_tarski(A,k3_zfmisc_1(C,A,B)) )
=> A = k1_xboole_0 ) ).
fof(t50_mcart_1,axiom,
! [A,B,C] :
~ ( A != k1_xboole_0
& B != k1_xboole_0
& C != k1_xboole_0
& ~ ! [D] :
( m1_subset_1(D,k3_zfmisc_1(A,B,C))
=> ( k5_mcart_1(A,B,C,D) = k1_mcart_1(k1_mcart_1(D))
& k6_mcart_1(A,B,C,D) = k2_mcart_1(k1_mcart_1(D))
& k7_mcart_1(A,B,C,D) = k2_mcart_1(D) ) ) ) ).
fof(t51_mcart_1,axiom,
! [A,B,C] :
~ ( A != k1_xboole_0
& B != k1_xboole_0
& C != k1_xboole_0
& ~ ! [D] :
( m1_subset_1(D,k3_zfmisc_1(A,B,C))
=> ( D != k5_mcart_1(A,B,C,D)
& D != k6_mcart_1(A,B,C,D)
& D != k7_mcart_1(A,B,C,D) ) ) ) ).
fof(t52_mcart_1,axiom,
! [A,B,C,D,E,F] :
( ~ r1_xboole_0(k3_zfmisc_1(A,B,C),k3_zfmisc_1(D,E,F))
=> ( ~ r1_xboole_0(A,D)
& ~ r1_xboole_0(B,E)
& ~ r1_xboole_0(C,F) ) ) ).
fof(t53_mcart_1,axiom,
! [A,B,C,D] : k4_zfmisc_1(A,B,C,D) = k2_zfmisc_1(k2_zfmisc_1(k2_zfmisc_1(A,B),C),D) ).
fof(t54_mcart_1,axiom,
! [A,B,C,D] : k3_zfmisc_1(k2_zfmisc_1(A,B),C,D) = k4_zfmisc_1(A,B,C,D) ).
fof(t55_mcart_1,axiom,
! [A,B,C,D] :
( ( A != k1_xboole_0
& B != k1_xboole_0
& C != k1_xboole_0
& D != k1_xboole_0 )
<=> k4_zfmisc_1(A,B,C,D) != k1_xboole_0 ) ).
fof(t56_mcart_1,axiom,
! [A,B,C,D,E,F,G,H] :
( k4_zfmisc_1(A,B,C,D) = k4_zfmisc_1(E,F,G,H)
=> ( A = k1_xboole_0
| B = k1_xboole_0
| C = k1_xboole_0
| D = k1_xboole_0
| ( A = E
& B = F
& C = G
& D = H ) ) ) ).
fof(t57_mcart_1,axiom,
! [A,B,C,D,E,F,G,H] :
( k4_zfmisc_1(A,B,C,D) = k4_zfmisc_1(E,F,G,H)
=> ( k4_zfmisc_1(A,B,C,D) = k1_xboole_0
| ( A = E
& B = F
& C = G
& D = H ) ) ) ).
fof(t58_mcart_1,axiom,
! [A,B] :
( k4_zfmisc_1(A,A,A,A) = k4_zfmisc_1(B,B,B,B)
=> A = B ) ).
fof(d8_mcart_1,axiom,
! [A,B,C,D] :
~ ( A != k1_xboole_0
& B != k1_xboole_0
& C != k1_xboole_0
& D != k1_xboole_0
& ~ ! [E] :
( m1_subset_1(E,k4_zfmisc_1(A,B,C,D))
=> ! [F] :
( m1_subset_1(F,A)
=> ( F = k8_mcart_1(A,B,C,D,E)
<=> ! [G,H,I,J] :
( E = k4_mcart_1(G,H,I,J)
=> F = G ) ) ) ) ) ).
fof(d9_mcart_1,axiom,
! [A,B,C,D] :
~ ( A != k1_xboole_0
& B != k1_xboole_0
& C != k1_xboole_0
& D != k1_xboole_0
& ~ ! [E] :
( m1_subset_1(E,k4_zfmisc_1(A,B,C,D))
=> ! [F] :
( m1_subset_1(F,B)
=> ( F = k9_mcart_1(A,B,C,D,E)
<=> ! [G,H,I,J] :
( E = k4_mcart_1(G,H,I,J)
=> F = H ) ) ) ) ) ).
fof(d10_mcart_1,axiom,
! [A,B,C,D] :
~ ( A != k1_xboole_0
& B != k1_xboole_0
& C != k1_xboole_0
& D != k1_xboole_0
& ~ ! [E] :
( m1_subset_1(E,k4_zfmisc_1(A,B,C,D))
=> ! [F] :
( m1_subset_1(F,C)
=> ( F = k10_mcart_1(A,B,C,D,E)
<=> ! [G,H,I,J] :
( E = k4_mcart_1(G,H,I,J)
=> F = I ) ) ) ) ) ).
fof(d11_mcart_1,axiom,
! [A,B,C,D] :
~ ( A != k1_xboole_0
& B != k1_xboole_0
& C != k1_xboole_0
& D != k1_xboole_0
& ~ ! [E] :
( m1_subset_1(E,k4_zfmisc_1(A,B,C,D))
=> ! [F] :
( m1_subset_1(F,D)
=> ( F = k11_mcart_1(A,B,C,D,E)
<=> ! [G,H,I,J] :
( E = k4_mcart_1(G,H,I,J)
=> F = J ) ) ) ) ) ).
fof(t59_mcart_1,axiom,
! [A,B,C,D] :
~ ( A != k1_xboole_0
& B != k1_xboole_0
& C != k1_xboole_0
& D != k1_xboole_0
& ? [E] :
( m1_subset_1(E,k4_zfmisc_1(A,B,C,D))
& ? [F,G,H,I] :
( E = k4_mcart_1(F,G,H,I)
& ~ ( k8_mcart_1(A,B,C,D,E) = F
& k9_mcart_1(A,B,C,D,E) = G
& k10_mcart_1(A,B,C,D,E) = H
& k11_mcart_1(A,B,C,D,E) = I ) ) ) ) ).
fof(t60_mcart_1,axiom,
! [A,B,C,D] :
~ ( A != k1_xboole_0
& B != k1_xboole_0
& C != k1_xboole_0
& D != k1_xboole_0
& ~ ! [E] :
( m1_subset_1(E,k4_zfmisc_1(A,B,C,D))
=> E = k4_mcart_1(k8_mcart_1(A,B,C,D,E),k9_mcart_1(A,B,C,D,E),k10_mcart_1(A,B,C,D,E),k11_mcart_1(A,B,C,D,E)) ) ) ).
fof(t61_mcart_1,axiom,
! [A,B,C,D] :
~ ( A != k1_xboole_0
& B != k1_xboole_0
& C != k1_xboole_0
& D != k1_xboole_0
& ~ ! [E] :
( m1_subset_1(E,k4_zfmisc_1(A,B,C,D))
=> ( k8_mcart_1(A,B,C,D,E) = k1_mcart_1(k1_mcart_1(k1_mcart_1(E)))
& k9_mcart_1(A,B,C,D,E) = k2_mcart_1(k1_mcart_1(k1_mcart_1(E)))
& k10_mcart_1(A,B,C,D,E) = k2_mcart_1(k1_mcart_1(E))
& k11_mcart_1(A,B,C,D,E) = k2_mcart_1(E) ) ) ) ).
fof(t62_mcart_1,axiom,
! [A,B,C,D] :
~ ( A != k1_xboole_0
& B != k1_xboole_0
& C != k1_xboole_0
& D != k1_xboole_0
& ~ ! [E] :
( m1_subset_1(E,k4_zfmisc_1(A,B,C,D))
=> ( E != k8_mcart_1(A,B,C,D,E)
& E != k9_mcart_1(A,B,C,D,E)
& E != k10_mcart_1(A,B,C,D,E)
& E != k11_mcart_1(A,B,C,D,E) ) ) ) ).
fof(t63_mcart_1,axiom,
! [A,B,C,D] :
( ~ ( ~ r1_tarski(A,k4_zfmisc_1(A,B,C,D))
& ~ r1_tarski(A,k4_zfmisc_1(B,C,D,A))
& ~ r1_tarski(A,k4_zfmisc_1(C,D,A,B))
& ~ r1_tarski(A,k4_zfmisc_1(D,A,B,C)) )
=> A = k1_xboole_0 ) ).
fof(t64_mcart_1,axiom,
! [A,B,C,D,E,F,G,H] :
( ~ r1_xboole_0(k4_zfmisc_1(A,B,C,D),k4_zfmisc_1(E,F,G,H))
=> ( ~ r1_xboole_0(A,E)
& ~ r1_xboole_0(B,F)
& ~ r1_xboole_0(C,G)
& ~ r1_xboole_0(D,H) ) ) ).
fof(t65_mcart_1,axiom,
! [A,B,C,D] : k4_zfmisc_1(k1_tarski(A),k1_tarski(B),k1_tarski(C),k1_tarski(D)) = k1_tarski(k4_mcart_1(A,B,C,D)) ).
fof(t66_mcart_1,axiom,
! [A,B] :
( k2_zfmisc_1(A,B) != k1_xboole_0
=> ! [C] :
( m1_subset_1(C,k2_zfmisc_1(A,B))
=> ( C != k1_mcart_1(C)
& C != k2_mcart_1(C) ) ) ) ).
fof(t67_mcart_1,axiom,
! [A,B,C] :
( r2_hidden(A,k2_zfmisc_1(B,C))
=> ( A != k1_mcart_1(A)
& A != k2_mcart_1(A) ) ) ).
fof(t68_mcart_1,axiom,
! [A,B,C,D] :
( m1_subset_1(D,k3_zfmisc_1(A,B,C))
=> ~ ( A != k1_xboole_0
& B != k1_xboole_0
& C != k1_xboole_0
& ? [E,F,G] :
( D = k3_mcart_1(E,F,G)
& ~ ( k5_mcart_1(A,B,C,D) = E
& k6_mcart_1(A,B,C,D) = F
& k7_mcart_1(A,B,C,D) = G ) ) ) ) ).
fof(t69_mcart_1,axiom,
! [A,B,C,D,E] :
( m1_subset_1(E,k3_zfmisc_1(A,B,C))
=> ( ! [F] :
( m1_subset_1(F,A)
=> ! [G] :
( m1_subset_1(G,B)
=> ! [H] :
( m1_subset_1(H,C)
=> ( E = k3_mcart_1(F,G,H)
=> D = F ) ) ) )
=> ( A = k1_xboole_0
| B = k1_xboole_0
| C = k1_xboole_0
| D = k5_mcart_1(A,B,C,E) ) ) ) ).
fof(t70_mcart_1,axiom,
! [A,B,C,D,E] :
( m1_subset_1(E,k3_zfmisc_1(A,B,C))
=> ( ! [F] :
( m1_subset_1(F,A)
=> ! [G] :
( m1_subset_1(G,B)
=> ! [H] :
( m1_subset_1(H,C)
=> ( E = k3_mcart_1(F,G,H)
=> D = G ) ) ) )
=> ( A = k1_xboole_0
| B = k1_xboole_0
| C = k1_xboole_0
| D = k6_mcart_1(A,B,C,E) ) ) ) ).
fof(t71_mcart_1,axiom,
! [A,B,C,D,E] :
( m1_subset_1(E,k3_zfmisc_1(A,B,C))
=> ( ! [F] :
( m1_subset_1(F,A)
=> ! [G] :
( m1_subset_1(G,B)
=> ! [H] :
( m1_subset_1(H,C)
=> ( E = k3_mcart_1(F,G,H)
=> D = H ) ) ) )
=> ( A = k1_xboole_0
| B = k1_xboole_0
| C = k1_xboole_0
| D = k7_mcart_1(A,B,C,E) ) ) ) ).
fof(t72_mcart_1,axiom,
! [A,B,C,D] :
~ ( r2_hidden(A,k3_zfmisc_1(B,C,D))
& ! [E,F,G] :
~ ( r2_hidden(E,B)
& r2_hidden(F,C)
& r2_hidden(G,D)
& A = k3_mcart_1(E,F,G) ) ) ).
fof(t73_mcart_1,axiom,
! [A,B,C,D,E,F] :
( r2_hidden(k3_mcart_1(A,B,C),k3_zfmisc_1(D,E,F))
<=> ( r2_hidden(A,D)
& r2_hidden(B,E)
& r2_hidden(C,F) ) ) ).
fof(t74_mcart_1,axiom,
! [A,B,C,D] :
( ! [E] :
( r2_hidden(E,A)
<=> ? [F,G,H] :
( r2_hidden(F,B)
& r2_hidden(G,C)
& r2_hidden(H,D)
& E = k3_mcart_1(F,G,H) ) )
=> A = k3_zfmisc_1(B,C,D) ) ).
fof(t75_mcart_1,axiom,
! [A,B,C,D,E,F] :
~ ( A != k1_xboole_0
& B != k1_xboole_0
& C != k1_xboole_0
& D != k1_xboole_0
& E != k1_xboole_0
& F != k1_xboole_0
& ? [G] :
( m1_subset_1(G,k3_zfmisc_1(A,B,C))
& ? [H] :
( m1_subset_1(H,k3_zfmisc_1(D,E,F))
& G = H
& ~ ( k5_mcart_1(A,B,C,G) = k5_mcart_1(D,E,F,H)
& k6_mcart_1(A,B,C,G) = k6_mcart_1(D,E,F,H)
& k7_mcart_1(A,B,C,G) = k7_mcart_1(D,E,F,H) ) ) ) ) ).
fof(t76_mcart_1,axiom,
! [A,B,C,D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(B))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(C))
=> ! [G] :
( m1_subset_1(G,k3_zfmisc_1(A,B,C))
=> ( r2_hidden(G,k3_zfmisc_1(D,E,F))
=> ( r2_hidden(k5_mcart_1(A,B,C,G),D)
& r2_hidden(k6_mcart_1(A,B,C,G),E)
& r2_hidden(k7_mcart_1(A,B,C,G),F) ) ) ) ) ) ) ).
fof(t77_mcart_1,axiom,
! [A,B,C,D,E,F] :
( ( r1_tarski(A,B)
& r1_tarski(C,D)
& r1_tarski(E,F) )
=> r1_tarski(k3_zfmisc_1(A,C,E),k3_zfmisc_1(B,D,F)) ) ).
fof(t78_mcart_1,axiom,
! [A,B,C,D,E] :
( m1_subset_1(E,k4_zfmisc_1(A,B,C,D))
=> ~ ( A != k1_xboole_0
& B != k1_xboole_0
& C != k1_xboole_0
& D != k1_xboole_0
& ? [F,G,H,I] :
( E = k4_mcart_1(F,G,H,I)
& ~ ( k8_mcart_1(A,B,C,D,E) = F
& k9_mcart_1(A,B,C,D,E) = G
& k10_mcart_1(A,B,C,D,E) = H
& k11_mcart_1(A,B,C,D,E) = I ) ) ) ) ).
fof(t79_mcart_1,axiom,
! [A,B,C,D,E,F] :
( m1_subset_1(F,k4_zfmisc_1(A,B,C,D))
=> ( ! [G] :
( m1_subset_1(G,A)
=> ! [H] :
( m1_subset_1(H,B)
=> ! [I] :
( m1_subset_1(I,C)
=> ! [J] :
( m1_subset_1(J,D)
=> ( F = k4_mcart_1(G,H,I,J)
=> E = G ) ) ) ) )
=> ( A = k1_xboole_0
| B = k1_xboole_0
| C = k1_xboole_0
| D = k1_xboole_0
| E = k8_mcart_1(A,B,C,D,F) ) ) ) ).
fof(t80_mcart_1,axiom,
! [A,B,C,D,E,F] :
( m1_subset_1(F,k4_zfmisc_1(A,B,C,D))
=> ( ! [G] :
( m1_subset_1(G,A)
=> ! [H] :
( m1_subset_1(H,B)
=> ! [I] :
( m1_subset_1(I,C)
=> ! [J] :
( m1_subset_1(J,D)
=> ( F = k4_mcart_1(G,H,I,J)
=> E = H ) ) ) ) )
=> ( A = k1_xboole_0
| B = k1_xboole_0
| C = k1_xboole_0
| D = k1_xboole_0
| E = k9_mcart_1(A,B,C,D,F) ) ) ) ).
fof(t81_mcart_1,axiom,
! [A,B,C,D,E,F] :
( m1_subset_1(F,k4_zfmisc_1(A,B,C,D))
=> ( ! [G] :
( m1_subset_1(G,A)
=> ! [H] :
( m1_subset_1(H,B)
=> ! [I] :
( m1_subset_1(I,C)
=> ! [J] :
( m1_subset_1(J,D)
=> ( F = k4_mcart_1(G,H,I,J)
=> E = I ) ) ) ) )
=> ( A = k1_xboole_0
| B = k1_xboole_0
| C = k1_xboole_0
| D = k1_xboole_0
| E = k10_mcart_1(A,B,C,D,F) ) ) ) ).
fof(t82_mcart_1,axiom,
! [A,B,C,D,E,F] :
( m1_subset_1(F,k4_zfmisc_1(A,B,C,D))
=> ( ! [G] :
( m1_subset_1(G,A)
=> ! [H] :
( m1_subset_1(H,B)
=> ! [I] :
( m1_subset_1(I,C)
=> ! [J] :
( m1_subset_1(J,D)
=> ( F = k4_mcart_1(G,H,I,J)
=> E = J ) ) ) ) )
=> ( A = k1_xboole_0
| B = k1_xboole_0
| C = k1_xboole_0
| D = k1_xboole_0
| E = k11_mcart_1(A,B,C,D,F) ) ) ) ).
fof(t83_mcart_1,axiom,
! [A,B,C,D,E] :
~ ( r2_hidden(A,k4_zfmisc_1(B,C,D,E))
& ! [F,G,H,I] :
~ ( r2_hidden(F,B)
& r2_hidden(G,C)
& r2_hidden(H,D)
& r2_hidden(I,E)
& A = k4_mcart_1(F,G,H,I) ) ) ).
fof(t84_mcart_1,axiom,
! [A,B,C,D,E,F,G,H] :
( r2_hidden(k4_mcart_1(A,B,C,D),k4_zfmisc_1(E,F,G,H))
<=> ( r2_hidden(A,E)
& r2_hidden(B,F)
& r2_hidden(C,G)
& r2_hidden(D,H) ) ) ).
fof(t85_mcart_1,axiom,
! [A,B,C,D,E] :
( ! [F] :
( r2_hidden(F,A)
<=> ? [G,H,I,J] :
( r2_hidden(G,B)
& r2_hidden(H,C)
& r2_hidden(I,D)
& r2_hidden(J,E)
& F = k4_mcart_1(G,H,I,J) ) )
=> A = k4_zfmisc_1(B,C,D,E) ) ).
fof(t86_mcart_1,axiom,
! [A,B,C,D,E,F,G,H] :
~ ( A != k1_xboole_0
& B != k1_xboole_0
& C != k1_xboole_0
& D != k1_xboole_0
& E != k1_xboole_0
& F != k1_xboole_0
& G != k1_xboole_0
& H != k1_xboole_0
& ? [I] :
( m1_subset_1(I,k4_zfmisc_1(A,B,C,D))
& ? [J] :
( m1_subset_1(J,k4_zfmisc_1(E,F,G,H))
& I = J
& ~ ( k8_mcart_1(A,B,C,D,I) = k8_mcart_1(E,F,G,H,J)
& k9_mcart_1(A,B,C,D,I) = k9_mcart_1(E,F,G,H,J)
& k10_mcart_1(A,B,C,D,I) = k10_mcart_1(E,F,G,H,J)
& k11_mcart_1(A,B,C,D,I) = k11_mcart_1(E,F,G,H,J) ) ) ) ) ).
fof(t87_mcart_1,axiom,
! [A,B,C,D,E] :
( m1_subset_1(E,k1_zfmisc_1(A))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(B))
=> ! [G] :
( m1_subset_1(G,k1_zfmisc_1(C))
=> ! [H] :
( m1_subset_1(H,k1_zfmisc_1(D))
=> ! [I] :
( m1_subset_1(I,k4_zfmisc_1(A,B,C,D))
=> ( r2_hidden(I,k4_zfmisc_1(E,F,G,H))
=> ( r2_hidden(k8_mcart_1(A,B,C,D,I),E)
& r2_hidden(k9_mcart_1(A,B,C,D,I),F)
& r2_hidden(k10_mcart_1(A,B,C,D,I),G)
& r2_hidden(k11_mcart_1(A,B,C,D,I),H) ) ) ) ) ) ) ) ).
fof(t88_mcart_1,axiom,
! [A,B,C,D,E,F,G,H] :
( ( r1_tarski(A,B)
& r1_tarski(C,D)
& r1_tarski(E,F)
& r1_tarski(G,H) )
=> r1_tarski(k4_zfmisc_1(A,C,E,G),k4_zfmisc_1(B,D,F,H)) ) ).
fof(d12_mcart_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( B = k15_mcart_1(A)
<=> ( k1_relat_1(B) = k1_relat_1(A)
& ! [C] :
( r2_hidden(C,k1_relat_1(A))
=> k1_funct_1(B,C) = k1_mcart_1(k1_funct_1(A,C)) ) ) ) ) ) ).
fof(d13_mcart_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( B = k16_mcart_1(A)
<=> ( k1_relat_1(B) = k1_relat_1(A)
& ! [C] :
( r2_hidden(C,k1_relat_1(A))
=> k1_funct_1(B,C) = k2_mcart_1(k1_funct_1(A,C)) ) ) ) ) ) ).
fof(d14_mcart_1,axiom,
! [A] : k17_mcart_1(A) = k1_mcart_1(k1_mcart_1(A)) ).
fof(d15_mcart_1,axiom,
! [A] : k18_mcart_1(A) = k2_mcart_1(k1_mcart_1(A)) ).
fof(d16_mcart_1,axiom,
! [A] : k19_mcart_1(A) = k1_mcart_1(k2_mcart_1(A)) ).
fof(d17_mcart_1,axiom,
! [A] : k20_mcart_1(A) = k2_mcart_1(k2_mcart_1(A)) ).
fof(t89_mcart_1,axiom,
! [A,B,C,D,E,F] :
( k17_mcart_1(k4_tarski(k4_tarski(A,B),C)) = A
& k18_mcart_1(k4_tarski(k4_tarski(A,B),C)) = B
& k19_mcart_1(k4_tarski(F,k4_tarski(D,E))) = D
& k20_mcart_1(k4_tarski(F,k4_tarski(D,E))) = E ) ).
fof(dt_k1_mcart_1,axiom,
$true ).
fof(dt_k2_mcart_1,axiom,
$true ).
fof(dt_k3_mcart_1,axiom,
$true ).
fof(dt_k4_mcart_1,axiom,
$true ).
fof(dt_k5_mcart_1,axiom,
! [A,B,C,D] :
( m1_subset_1(D,k3_zfmisc_1(A,B,C))
=> m1_subset_1(k5_mcart_1(A,B,C,D),A) ) ).
fof(dt_k6_mcart_1,axiom,
! [A,B,C,D] :
( m1_subset_1(D,k3_zfmisc_1(A,B,C))
=> m1_subset_1(k6_mcart_1(A,B,C,D),B) ) ).
fof(dt_k7_mcart_1,axiom,
! [A,B,C,D] :
( m1_subset_1(D,k3_zfmisc_1(A,B,C))
=> m1_subset_1(k7_mcart_1(A,B,C,D),C) ) ).
fof(dt_k8_mcart_1,axiom,
! [A,B,C,D,E] :
( m1_subset_1(E,k4_zfmisc_1(A,B,C,D))
=> m1_subset_1(k8_mcart_1(A,B,C,D,E),A) ) ).
fof(dt_k9_mcart_1,axiom,
! [A,B,C,D,E] :
( m1_subset_1(E,k4_zfmisc_1(A,B,C,D))
=> m1_subset_1(k9_mcart_1(A,B,C,D,E),B) ) ).
fof(dt_k10_mcart_1,axiom,
! [A,B,C,D,E] :
( m1_subset_1(E,k4_zfmisc_1(A,B,C,D))
=> m1_subset_1(k10_mcart_1(A,B,C,D,E),C) ) ).
fof(dt_k11_mcart_1,axiom,
! [A,B,C,D,E] :
( m1_subset_1(E,k4_zfmisc_1(A,B,C,D))
=> m1_subset_1(k11_mcart_1(A,B,C,D,E),D) ) ).
fof(dt_k12_mcart_1,axiom,
! [A,B,C,D] :
( ( m1_subset_1(C,k1_zfmisc_1(A))
& m1_subset_1(D,k1_zfmisc_1(B)) )
=> m1_subset_1(k12_mcart_1(A,B,C,D),k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ).
fof(redefinition_k12_mcart_1,axiom,
! [A,B,C,D] :
( ( m1_subset_1(C,k1_zfmisc_1(A))
& m1_subset_1(D,k1_zfmisc_1(B)) )
=> k12_mcart_1(A,B,C,D) = k2_zfmisc_1(C,D) ) ).
fof(dt_k13_mcart_1,axiom,
! [A,B,C,D,E,F] :
( ( m1_subset_1(D,k1_zfmisc_1(A))
& m1_subset_1(E,k1_zfmisc_1(B))
& m1_subset_1(F,k1_zfmisc_1(C)) )
=> m1_subset_1(k13_mcart_1(A,B,C,D,E,F),k1_zfmisc_1(k3_zfmisc_1(A,B,C))) ) ).
fof(redefinition_k13_mcart_1,axiom,
! [A,B,C,D,E,F] :
( ( m1_subset_1(D,k1_zfmisc_1(A))
& m1_subset_1(E,k1_zfmisc_1(B))
& m1_subset_1(F,k1_zfmisc_1(C)) )
=> k13_mcart_1(A,B,C,D,E,F) = k3_zfmisc_1(D,E,F) ) ).
fof(dt_k14_mcart_1,axiom,
! [A,B,C,D,E,F,G,H] :
( ( m1_subset_1(E,k1_zfmisc_1(A))
& m1_subset_1(F,k1_zfmisc_1(B))
& m1_subset_1(G,k1_zfmisc_1(C))
& m1_subset_1(H,k1_zfmisc_1(D)) )
=> m1_subset_1(k14_mcart_1(A,B,C,D,E,F,G,H),k1_zfmisc_1(k4_zfmisc_1(A,B,C,D))) ) ).
fof(redefinition_k14_mcart_1,axiom,
! [A,B,C,D,E,F,G,H] :
( ( m1_subset_1(E,k1_zfmisc_1(A))
& m1_subset_1(F,k1_zfmisc_1(B))
& m1_subset_1(G,k1_zfmisc_1(C))
& m1_subset_1(H,k1_zfmisc_1(D)) )
=> k14_mcart_1(A,B,C,D,E,F,G,H) = k4_zfmisc_1(E,F,G,H) ) ).
fof(dt_k15_mcart_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v1_relat_1(k15_mcart_1(A))
& v1_funct_1(k15_mcart_1(A)) ) ) ).
fof(dt_k16_mcart_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v1_relat_1(k16_mcart_1(A))
& v1_funct_1(k16_mcart_1(A)) ) ) ).
fof(dt_k17_mcart_1,axiom,
$true ).
fof(dt_k18_mcart_1,axiom,
$true ).
fof(dt_k19_mcart_1,axiom,
$true ).
fof(dt_k20_mcart_1,axiom,
$true ).
%------------------------------------------------------------------------------