## SET007 Axioms: SET007+5.ax

```%------------------------------------------------------------------------------
% File     : SET007+5 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Enumerated Sets
% 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    : enumset1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  137 ( 131 unit)
%            Number of atoms       :  176 ( 115 equality)
%            Maximal formula depth :   22 (   4 average)
%            Number of connectives :   78 (  39 ~  ;   0  |;  27  &)
%                                         (  12 <=>;   0 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    3 (   1 propositional; 0-2 arity)
%            Number of functors    :    9 (   0 constant; 1-8 arity)
%            Number of variables   :  373 (   0 singleton; 373 !;   0 ?)
%            Maximal term depth    :    3 (   2 average)
% 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(d1_enumset1,axiom,(
! [A,B,C,D] :
( D = k1_enumset1(A,B,C)
<=> ! [E] :
( r2_hidden(E,D)
<=> ~ ( E != A
& E != B
& E != C ) ) ) )).

fof(d2_enumset1,axiom,(
! [A,B,C,D,E] :
( E = k2_enumset1(A,B,C,D)
<=> ! [F] :
( r2_hidden(F,E)
<=> ~ ( F != A
& F != B
& F != C
& F != D ) ) ) )).

fof(d3_enumset1,axiom,(
! [A,B,C,D,E,F] :
( F = k3_enumset1(A,B,C,D,E)
<=> ! [G] :
( r2_hidden(G,F)
<=> ~ ( G != A
& G != B
& G != C
& G != D
& G != E ) ) ) )).

fof(d4_enumset1,axiom,(
! [A,B,C,D,E,F,G] :
( G = k4_enumset1(A,B,C,D,E,F)
<=> ! [H] :
( r2_hidden(H,G)
<=> ~ ( H != A
& H != B
& H != C
& H != D
& H != E
& H != F ) ) ) )).

fof(d5_enumset1,axiom,(
! [A,B,C,D,E,F,G,H] :
( H = k5_enumset1(A,B,C,D,E,F,G)
<=> ! [I] :
( r2_hidden(I,H)
<=> ~ ( I != A
& I != B
& I != C
& I != D
& I != E
& I != F
& I != G ) ) ) )).

fof(d6_enumset1,axiom,(
! [A,B,C,D,E,F,G,H,I] :
( I = k6_enumset1(A,B,C,D,E,F,G,H)
<=> ! [J] :
( r2_hidden(J,I)
<=> ~ ( J != A
& J != B
& J != C
& J != D
& J != E
& J != F
& J != G
& J != H ) ) ) )).

fof(t1_enumset1,axiom,(
\$true )).

fof(t2_enumset1,axiom,(
\$true )).

fof(t3_enumset1,axiom,(
\$true )).

fof(t4_enumset1,axiom,(
\$true )).

fof(t5_enumset1,axiom,(
\$true )).

fof(t6_enumset1,axiom,(
\$true )).

fof(t7_enumset1,axiom,(
\$true )).

fof(t8_enumset1,axiom,(
\$true )).

fof(t9_enumset1,axiom,(
\$true )).

fof(t10_enumset1,axiom,(
\$true )).

fof(t11_enumset1,axiom,(
\$true )).

fof(t12_enumset1,axiom,(
\$true )).

fof(t13_enumset1,axiom,(
\$true )).

fof(t14_enumset1,axiom,(
\$true )).

fof(t15_enumset1,axiom,(
\$true )).

fof(t16_enumset1,axiom,(
\$true )).

fof(t17_enumset1,axiom,(
\$true )).

fof(t18_enumset1,axiom,(
\$true )).

fof(t19_enumset1,axiom,(
\$true )).

fof(t20_enumset1,axiom,(
\$true )).

fof(t21_enumset1,axiom,(
\$true )).

fof(t22_enumset1,axiom,(
\$true )).

fof(t23_enumset1,axiom,(
\$true )).

fof(t24_enumset1,axiom,(
\$true )).

fof(t25_enumset1,axiom,(
\$true )).

fof(t26_enumset1,axiom,(
\$true )).

fof(t27_enumset1,axiom,(
\$true )).

fof(t28_enumset1,axiom,(
\$true )).

fof(t29_enumset1,axiom,(
\$true )).

fof(t30_enumset1,axiom,(
\$true )).

fof(t31_enumset1,axiom,(
\$true )).

fof(t32_enumset1,axiom,(
\$true )).

fof(t33_enumset1,axiom,(
\$true )).

fof(t34_enumset1,axiom,(
\$true )).

fof(t35_enumset1,axiom,(
\$true )).

fof(t36_enumset1,axiom,(
\$true )).

fof(t37_enumset1,axiom,(
\$true )).

fof(t38_enumset1,axiom,(
\$true )).

fof(t39_enumset1,axiom,(
\$true )).

fof(t40_enumset1,axiom,(
\$true )).

fof(t41_enumset1,axiom,(
! [A,B] : k2_tarski(A,B) = k2_xboole_0(k1_tarski(A),k1_tarski(B)) )).

fof(t42_enumset1,axiom,(
! [A,B,C] : k1_enumset1(A,B,C) = k2_xboole_0(k1_tarski(A),k2_tarski(B,C)) )).

fof(t43_enumset1,axiom,(
! [A,B,C] : k1_enumset1(A,B,C) = k2_xboole_0(k2_tarski(A,B),k1_tarski(C)) )).

fof(t44_enumset1,axiom,(
! [A,B,C,D] : k2_enumset1(A,B,C,D) = k2_xboole_0(k1_tarski(A),k1_enumset1(B,C,D)) )).

fof(t45_enumset1,axiom,(
! [A,B,C,D] : k2_enumset1(A,B,C,D) = k2_xboole_0(k2_tarski(A,B),k2_tarski(C,D)) )).

fof(t46_enumset1,axiom,(
! [A,B,C,D] : k2_enumset1(A,B,C,D) = k2_xboole_0(k1_enumset1(A,B,C),k1_tarski(D)) )).

fof(t47_enumset1,axiom,(
! [A,B,C,D,E] : k3_enumset1(A,B,C,D,E) = k2_xboole_0(k1_tarski(A),k2_enumset1(B,C,D,E)) )).

fof(t48_enumset1,axiom,(
! [A,B,C,D,E] : k3_enumset1(A,B,C,D,E) = k2_xboole_0(k2_tarski(A,B),k1_enumset1(C,D,E)) )).

fof(t49_enumset1,axiom,(
! [A,B,C,D,E] : k3_enumset1(A,B,C,D,E) = k2_xboole_0(k1_enumset1(A,B,C),k2_tarski(D,E)) )).

fof(t50_enumset1,axiom,(
! [A,B,C,D,E] : k3_enumset1(A,B,C,D,E) = k2_xboole_0(k2_enumset1(A,B,C,D),k1_tarski(E)) )).

fof(t51_enumset1,axiom,(
! [A,B,C,D,E,F] : k4_enumset1(A,B,C,D,E,F) = k2_xboole_0(k1_tarski(A),k3_enumset1(B,C,D,E,F)) )).

fof(t52_enumset1,axiom,(
! [A,B,C,D,E,F] : k4_enumset1(A,B,C,D,E,F) = k2_xboole_0(k2_tarski(A,B),k2_enumset1(C,D,E,F)) )).

fof(t53_enumset1,axiom,(
! [A,B,C,D,E,F] : k4_enumset1(A,B,C,D,E,F) = k2_xboole_0(k1_enumset1(A,B,C),k1_enumset1(D,E,F)) )).

fof(t54_enumset1,axiom,(
! [A,B,C,D,E,F] : k4_enumset1(A,B,C,D,E,F) = k2_xboole_0(k2_enumset1(A,B,C,D),k2_tarski(E,F)) )).

fof(t55_enumset1,axiom,(
! [A,B,C,D,E,F] : k4_enumset1(A,B,C,D,E,F) = k2_xboole_0(k3_enumset1(A,B,C,D,E),k1_tarski(F)) )).

fof(t56_enumset1,axiom,(
! [A,B,C,D,E,F,G] : k5_enumset1(A,B,C,D,E,F,G) = k2_xboole_0(k1_tarski(A),k4_enumset1(B,C,D,E,F,G)) )).

fof(t57_enumset1,axiom,(
! [A,B,C,D,E,F,G] : k5_enumset1(A,B,C,D,E,F,G) = k2_xboole_0(k2_tarski(A,B),k3_enumset1(C,D,E,F,G)) )).

fof(t58_enumset1,axiom,(
! [A,B,C,D,E,F,G] : k5_enumset1(A,B,C,D,E,F,G) = k2_xboole_0(k1_enumset1(A,B,C),k2_enumset1(D,E,F,G)) )).

fof(t59_enumset1,axiom,(
! [A,B,C,D,E,F,G] : k5_enumset1(A,B,C,D,E,F,G) = k2_xboole_0(k2_enumset1(A,B,C,D),k1_enumset1(E,F,G)) )).

fof(t60_enumset1,axiom,(
! [A,B,C,D,E,F,G] : k5_enumset1(A,B,C,D,E,F,G) = k2_xboole_0(k3_enumset1(A,B,C,D,E),k2_tarski(F,G)) )).

fof(t61_enumset1,axiom,(
! [A,B,C,D,E,F,G] : k5_enumset1(A,B,C,D,E,F,G) = k2_xboole_0(k4_enumset1(A,B,C,D,E,F),k1_tarski(G)) )).

fof(t62_enumset1,axiom,(
! [A,B,C,D,E,F,G,H] : k6_enumset1(A,B,C,D,E,F,G,H) = k2_xboole_0(k1_tarski(A),k5_enumset1(B,C,D,E,F,G,H)) )).

fof(t63_enumset1,axiom,(
! [A,B,C,D,E,F,G,H] : k6_enumset1(A,B,C,D,E,F,G,H) = k2_xboole_0(k2_tarski(A,B),k4_enumset1(C,D,E,F,G,H)) )).

fof(t64_enumset1,axiom,(
! [A,B,C,D,E,F,G,H] : k6_enumset1(A,B,C,D,E,F,G,H) = k2_xboole_0(k1_enumset1(A,B,C),k3_enumset1(D,E,F,G,H)) )).

fof(t65_enumset1,axiom,(
! [A,B,C,D,E,F,G,H] : k6_enumset1(A,B,C,D,E,F,G,H) = k2_xboole_0(k2_enumset1(A,B,C,D),k2_enumset1(E,F,G,H)) )).

fof(t66_enumset1,axiom,(
! [A,B,C,D,E,F,G,H] : k6_enumset1(A,B,C,D,E,F,G,H) = k2_xboole_0(k3_enumset1(A,B,C,D,E),k1_enumset1(F,G,H)) )).

fof(t67_enumset1,axiom,(
! [A,B,C,D,E,F,G,H] : k6_enumset1(A,B,C,D,E,F,G,H) = k2_xboole_0(k4_enumset1(A,B,C,D,E,F),k2_tarski(G,H)) )).

fof(t68_enumset1,axiom,(
! [A,B,C,D,E,F,G,H] : k6_enumset1(A,B,C,D,E,F,G,H) = k2_xboole_0(k5_enumset1(A,B,C,D,E,F,G),k1_tarski(H)) )).

fof(t69_enumset1,axiom,(
! [A] : k2_tarski(A,A) = k1_tarski(A) )).

fof(t70_enumset1,axiom,(
! [A,B] : k1_enumset1(A,A,B) = k2_tarski(A,B) )).

fof(t71_enumset1,axiom,(
! [A,B,C] : k2_enumset1(A,A,B,C) = k1_enumset1(A,B,C) )).

fof(t72_enumset1,axiom,(
! [A,B,C,D] : k3_enumset1(A,A,B,C,D) = k2_enumset1(A,B,C,D) )).

fof(t73_enumset1,axiom,(
! [A,B,C,D,E] : k4_enumset1(A,A,B,C,D,E) = k3_enumset1(A,B,C,D,E) )).

fof(t74_enumset1,axiom,(
! [A,B,C,D,E,F] : k5_enumset1(A,A,B,C,D,E,F) = k4_enumset1(A,B,C,D,E,F) )).

fof(t75_enumset1,axiom,(
! [A,B,C,D,E,F,G] : k6_enumset1(A,A,B,C,D,E,F,G) = k5_enumset1(A,B,C,D,E,F,G) )).

fof(t76_enumset1,axiom,(
! [A] : k1_enumset1(A,A,A) = k1_tarski(A) )).

fof(t77_enumset1,axiom,(
! [A,B] : k2_enumset1(A,A,A,B) = k2_tarski(A,B) )).

fof(t78_enumset1,axiom,(
! [A,B,C] : k3_enumset1(A,A,A,B,C) = k1_enumset1(A,B,C) )).

fof(t79_enumset1,axiom,(
! [A,B,C,D] : k4_enumset1(A,A,A,B,C,D) = k2_enumset1(A,B,C,D) )).

fof(t80_enumset1,axiom,(
! [A,B,C,D,E] : k5_enumset1(A,A,A,B,C,D,E) = k3_enumset1(A,B,C,D,E) )).

fof(t81_enumset1,axiom,(
! [A,B,C,D,E,F] : k6_enumset1(A,A,A,B,C,D,E,F) = k4_enumset1(A,B,C,D,E,F) )).

fof(t82_enumset1,axiom,(
! [A] : k2_enumset1(A,A,A,A) = k1_tarski(A) )).

fof(t83_enumset1,axiom,(
! [A,B] : k3_enumset1(A,A,A,A,B) = k2_tarski(A,B) )).

fof(t84_enumset1,axiom,(
! [A,B,C] : k4_enumset1(A,A,A,A,B,C) = k1_enumset1(A,B,C) )).

fof(t85_enumset1,axiom,(
! [A,B,C,D] : k5_enumset1(A,A,A,A,B,C,D) = k2_enumset1(A,B,C,D) )).

fof(t86_enumset1,axiom,(
! [A,B,C,D,E] : k6_enumset1(A,A,A,A,B,C,D,E) = k3_enumset1(A,B,C,D,E) )).

fof(t87_enumset1,axiom,(
! [A] : k3_enumset1(A,A,A,A,A) = k1_tarski(A) )).

fof(t88_enumset1,axiom,(
! [A,B] : k4_enumset1(A,A,A,A,A,B) = k2_tarski(A,B) )).

fof(t89_enumset1,axiom,(
! [A,B,C] : k5_enumset1(A,A,A,A,A,B,C) = k1_enumset1(A,B,C) )).

fof(t90_enumset1,axiom,(
! [A,B,C,D] : k6_enumset1(A,A,A,A,A,B,C,D) = k2_enumset1(A,B,C,D) )).

fof(t91_enumset1,axiom,(
! [A] : k4_enumset1(A,A,A,A,A,A) = k1_tarski(A) )).

fof(t92_enumset1,axiom,(
! [A,B] : k5_enumset1(A,A,A,A,A,A,B) = k2_tarski(A,B) )).

fof(t93_enumset1,axiom,(
! [A,B,C] : k6_enumset1(A,A,A,A,A,A,B,C) = k1_enumset1(A,B,C) )).

fof(t94_enumset1,axiom,(
! [A] : k5_enumset1(A,A,A,A,A,A,A) = k1_tarski(A) )).

fof(t95_enumset1,axiom,(
! [A,B] : k6_enumset1(A,A,A,A,A,A,A,B) = k2_tarski(A,B) )).

fof(t96_enumset1,axiom,(
! [A] : k6_enumset1(A,A,A,A,A,A,A,A) = k1_tarski(A) )).

fof(t97_enumset1,axiom,(
\$true )).

fof(t98_enumset1,axiom,(
! [A,B,C] : k1_enumset1(A,B,C) = k1_enumset1(A,C,B) )).

fof(t99_enumset1,axiom,(
! [A,B,C] : k1_enumset1(A,B,C) = k1_enumset1(B,A,C) )).

fof(t100_enumset1,axiom,(
! [A,B,C] : k1_enumset1(A,B,C) = k1_enumset1(B,C,A) )).

fof(t101_enumset1,axiom,(
\$true )).

fof(t102_enumset1,axiom,(
! [A,B,C] : k1_enumset1(A,B,C) = k1_enumset1(C,B,A) )).

fof(t103_enumset1,axiom,(
! [A,B,C,D] : k2_enumset1(A,B,C,D) = k2_enumset1(A,B,D,C) )).

fof(t104_enumset1,axiom,(
! [A,B,C,D] : k2_enumset1(A,B,C,D) = k2_enumset1(A,C,B,D) )).

fof(t105_enumset1,axiom,(
! [A,B,C,D] : k2_enumset1(A,B,C,D) = k2_enumset1(A,C,D,B) )).

fof(t106_enumset1,axiom,(
\$true )).

fof(t107_enumset1,axiom,(
! [A,B,C,D] : k2_enumset1(A,B,C,D) = k2_enumset1(A,D,C,B) )).

fof(t108_enumset1,axiom,(
! [A,B,C,D] : k2_enumset1(A,B,C,D) = k2_enumset1(B,A,C,D) )).

fof(t109_enumset1,axiom,(
! [A,B,C,D] : k2_enumset1(A,B,C,D) = k2_enumset1(B,A,D,C) )).

fof(t110_enumset1,axiom,(
! [A,B,C,D] : k2_enumset1(A,B,C,D) = k2_enumset1(B,C,A,D) )).

fof(t111_enumset1,axiom,(
! [A,B,C,D] : k2_enumset1(A,B,C,D) = k2_enumset1(B,C,D,A) )).

fof(t112_enumset1,axiom,(
! [A,B,C,D] : k2_enumset1(A,B,C,D) = k2_enumset1(B,D,A,C) )).

fof(t113_enumset1,axiom,(
! [A,B,C,D] : k2_enumset1(A,B,C,D) = k2_enumset1(B,D,C,A) )).

fof(t114_enumset1,axiom,(
\$true )).

fof(t115_enumset1,axiom,(
\$true )).

fof(t116_enumset1,axiom,(
! [A,B,C,D] : k2_enumset1(A,B,C,D) = k2_enumset1(C,B,A,D) )).

fof(t117_enumset1,axiom,(
! [A,B,C,D] : k2_enumset1(A,B,C,D) = k2_enumset1(C,B,D,A) )).

fof(t118_enumset1,axiom,(
! [A,B,C,D] : k2_enumset1(A,B,C,D) = k2_enumset1(C,D,A,B) )).

fof(t119_enumset1,axiom,(
! [A,B,C,D] : k2_enumset1(A,B,C,D) = k2_enumset1(C,D,B,A) )).

fof(t120_enumset1,axiom,(
\$true )).

fof(t121_enumset1,axiom,(
\$true )).

fof(t122_enumset1,axiom,(
\$true )).

fof(t123_enumset1,axiom,(
! [A,B,C,D] : k2_enumset1(A,B,C,D) = k2_enumset1(D,B,C,A) )).

fof(t124_enumset1,axiom,(
\$true )).

fof(t125_enumset1,axiom,(
! [A,B,C,D] : k2_enumset1(A,B,C,D) = k2_enumset1(D,C,B,A) )).

fof(dt_k1_enumset1,axiom,(
\$true )).

fof(dt_k2_enumset1,axiom,(
\$true )).

fof(dt_k3_enumset1,axiom,(
\$true )).

fof(dt_k4_enumset1,axiom,(
\$true )).

fof(dt_k5_enumset1,axiom,(
\$true )).

fof(dt_k6_enumset1,axiom,(
\$true )).
%------------------------------------------------------------------------------
```