%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET082-6 : TPTP v9.0.0. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n007.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:25:09 AM UTC 2025
% Result : Unsatisfiable 0.20s 0.42s
% Output : Proof 0.20s
% Verified :
% SZS Type : Refutation
% Derivation depth : 20
% Number of leaves : 45
% Syntax : Number of formulae : 121 ( 46 unt; 0 typ; 0 def)
% Number of atoms : 403 ( 122 equ)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 461 ( 189 ~; 223 |; 0 &)
% ( 49 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 4 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of FOOLs : 10 ( 10 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 6 ( 4 usr; 1 prp; 0-2 aty)
% Number of functors : 9 ( 9 usr; 4 con; 0-2 aty)
% Number of variables : 188 ( 170 !; 0 ?; 188 :)
% Comments :
%------------------------------------------------------------------------------
tff(member_type,type,
member: ( $i * $i ) > $o ).
tff(universal_class_type,type,
universal_class: $i ).
tff(unordered_pair_type,type,
unordered_pair: ( $i * $i ) > $i ).
tff(singleton_type,type,
singleton: $i > $i ).
tff(x_type,type,
x: $i ).
tff(image_type,type,
image: ( $i * $i ) > $i ).
tff(choice_type,type,
choice: $i ).
tff(sum_class_type,type,
sum_class: $i > $i ).
tff(apply_type,type,
apply: ( $i * $i ) > $i ).
tff(null_class_type,type,
null_class: $i ).
tff(function_type,type,
function: $i > $o ).
tff(1,plain,
^ [X: $i] :
refl(( ( unordered_pair(X,X) = singleton(X) )
<=> ( unordered_pair(X,X) = singleton(X) ) )),
inference(bind,[status(th)],[]) ).
tff(2,plain,
( ! [X: $i] : ( unordered_pair(X,X) = singleton(X) )
<=> ! [X: $i] : ( unordered_pair(X,X) = singleton(X) ) ),
inference(quant_intro,[status(thm)],[1]) ).
tff(3,plain,
( ! [X: $i] : ( unordered_pair(X,X) = singleton(X) )
<=> ! [X: $i] : ( unordered_pair(X,X) = singleton(X) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(4,axiom,
! [X: $i] : ( unordered_pair(X,X) = singleton(X) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax',singleton_set) ).
tff(5,plain,
! [X: $i] : ( unordered_pair(X,X) = singleton(X) ),
inference(modus_ponens,[status(thm)],[4,3]) ).
tff(6,plain,
! [X: $i] : ( unordered_pair(X,X) = singleton(X) ),
inference(skolemize,[status(sab)],[5]) ).
tff(7,plain,
! [X: $i] : ( unordered_pair(X,X) = singleton(X) ),
inference(modus_ponens,[status(thm)],[6,2]) ).
tff(8,plain,
( ~ ! [X: $i] : ( unordered_pair(X,X) = singleton(X) )
| ( unordered_pair(singleton(x),singleton(x)) = singleton(singleton(x)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(9,plain,
unordered_pair(singleton(x),singleton(x)) = singleton(singleton(x)),
inference(unit_resolution,[status(thm)],[8,7]) ).
tff(10,plain,
( member(unordered_pair(singleton(x),singleton(x)),universal_class)
<=> member(singleton(singleton(x)),universal_class) ),
inference(monotonicity,[status(thm)],[9]) ).
tff(11,plain,
( member(singleton(singleton(x)),universal_class)
<=> member(unordered_pair(singleton(x),singleton(x)),universal_class) ),
inference(symmetry,[status(thm)],[10]) ).
tff(12,plain,
( ~ member(singleton(singleton(x)),universal_class)
<=> ~ member(unordered_pair(singleton(x),singleton(x)),universal_class) ),
inference(monotonicity,[status(thm)],[11]) ).
tff(13,plain,
^ [Y: $i,Xf: $i] :
refl(( ( sum_class(image(Xf,singleton(Y))) = apply(Xf,Y) )
<=> ( sum_class(image(Xf,singleton(Y))) = apply(Xf,Y) ) )),
inference(bind,[status(th)],[]) ).
tff(14,plain,
( ! [Y: $i,Xf: $i] : ( sum_class(image(Xf,singleton(Y))) = apply(Xf,Y) )
<=> ! [Y: $i,Xf: $i] : ( sum_class(image(Xf,singleton(Y))) = apply(Xf,Y) ) ),
inference(quant_intro,[status(thm)],[13]) ).
tff(15,plain,
( ! [Y: $i,Xf: $i] : ( sum_class(image(Xf,singleton(Y))) = apply(Xf,Y) )
<=> ! [Y: $i,Xf: $i] : ( sum_class(image(Xf,singleton(Y))) = apply(Xf,Y) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(16,axiom,
! [Y: $i,Xf: $i] : ( sum_class(image(Xf,singleton(Y))) = apply(Xf,Y) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax',apply) ).
tff(17,plain,
! [Y: $i,Xf: $i] : ( sum_class(image(Xf,singleton(Y))) = apply(Xf,Y) ),
inference(modus_ponens,[status(thm)],[16,15]) ).
tff(18,plain,
! [Y: $i,Xf: $i] : ( sum_class(image(Xf,singleton(Y))) = apply(Xf,Y) ),
inference(skolemize,[status(sab)],[17]) ).
tff(19,plain,
! [Y: $i,Xf: $i] : ( sum_class(image(Xf,singleton(Y))) = apply(Xf,Y) ),
inference(modus_ponens,[status(thm)],[18,14]) ).
tff(20,plain,
( ~ ! [Y: $i,Xf: $i] : ( sum_class(image(Xf,singleton(Y))) = apply(Xf,Y) )
| ( sum_class(image(choice,singleton(singleton(x)))) = apply(choice,singleton(x)) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(21,plain,
sum_class(image(choice,singleton(singleton(x)))) = apply(choice,singleton(x)),
inference(unit_resolution,[status(thm)],[20,19]) ).
tff(22,plain,
apply(choice,singleton(x)) = sum_class(image(choice,singleton(singleton(x)))),
inference(symmetry,[status(thm)],[21]) ).
tff(23,plain,
( ~ ! [X: $i] : ( unordered_pair(X,X) = singleton(X) )
| ( unordered_pair(x,x) = singleton(x) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(24,plain,
unordered_pair(x,x) = singleton(x),
inference(unit_resolution,[status(thm)],[23,7]) ).
tff(25,plain,
( member(apply(choice,singleton(x)),unordered_pair(x,x))
<=> member(apply(choice,singleton(x)),singleton(x)) ),
inference(monotonicity,[status(thm)],[24]) ).
tff(26,plain,
( member(apply(choice,singleton(x)),singleton(x))
<=> member(apply(choice,singleton(x)),unordered_pair(x,x)) ),
inference(symmetry,[status(thm)],[25]) ).
tff(27,plain,
singleton(x) = unordered_pair(x,x),
inference(symmetry,[status(thm)],[24]) ).
tff(28,plain,
( member(singleton(x),universal_class)
<=> member(unordered_pair(x,x),universal_class) ),
inference(monotonicity,[status(thm)],[27]) ).
tff(29,plain,
( member(unordered_pair(x,x),universal_class)
<=> member(singleton(x),universal_class) ),
inference(symmetry,[status(thm)],[28]) ).
tff(30,plain,
^ [Y: $i,X: $i] :
refl(( member(unordered_pair(X,Y),universal_class)
<=> member(unordered_pair(X,Y),universal_class) )),
inference(bind,[status(th)],[]) ).
tff(31,plain,
( ! [Y: $i,X: $i] : member(unordered_pair(X,Y),universal_class)
<=> ! [Y: $i,X: $i] : member(unordered_pair(X,Y),universal_class) ),
inference(quant_intro,[status(thm)],[30]) ).
tff(32,plain,
( ! [Y: $i,X: $i] : member(unordered_pair(X,Y),universal_class)
<=> ! [Y: $i,X: $i] : member(unordered_pair(X,Y),universal_class) ),
inference(rewrite,[status(thm)],[]) ).
tff(33,axiom,
! [Y: $i,X: $i] : member(unordered_pair(X,Y),universal_class),
file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax',unordered_pairs_in_universal) ).
tff(34,plain,
! [Y: $i,X: $i] : member(unordered_pair(X,Y),universal_class),
inference(modus_ponens,[status(thm)],[33,32]) ).
tff(35,plain,
! [Y: $i,X: $i] : member(unordered_pair(X,Y),universal_class),
inference(skolemize,[status(sab)],[34]) ).
tff(36,plain,
! [Y: $i,X: $i] : member(unordered_pair(X,Y),universal_class),
inference(modus_ponens,[status(thm)],[35,31]) ).
tff(37,plain,
( ~ ! [Y: $i,X: $i] : member(unordered_pair(X,Y),universal_class)
| member(unordered_pair(x,x),universal_class) ),
inference(quant_inst,[status(thm)],[]) ).
tff(38,plain,
member(unordered_pair(x,x),universal_class),
inference(unit_resolution,[status(thm)],[37,36]) ).
tff(39,plain,
member(singleton(x),universal_class),
inference(modus_ponens,[status(thm)],[38,29]) ).
tff(40,plain,
( ( singleton(x) != null_class )
<=> ( singleton(x) != null_class ) ),
inference(rewrite,[status(thm)],[]) ).
tff(41,axiom,
singleton(x) != null_class,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_singleton_is_null_class_2) ).
tff(42,plain,
singleton(x) != null_class,
inference(modus_ponens,[status(thm)],[41,40]) ).
tff(43,plain,
^ [Y: $i] :
refl(( ( ~ member(Y,universal_class)
| ( Y = null_class )
| member(apply(choice,Y),Y) )
<=> ( ~ member(Y,universal_class)
| ( Y = null_class )
| member(apply(choice,Y),Y) ) )),
inference(bind,[status(th)],[]) ).
tff(44,plain,
( ! [Y: $i] :
( ~ member(Y,universal_class)
| ( Y = null_class )
| member(apply(choice,Y),Y) )
<=> ! [Y: $i] :
( ~ member(Y,universal_class)
| ( Y = null_class )
| member(apply(choice,Y),Y) ) ),
inference(quant_intro,[status(thm)],[43]) ).
tff(45,plain,
( ! [Y: $i] :
( ~ member(Y,universal_class)
| ( Y = null_class )
| member(apply(choice,Y),Y) )
<=> ! [Y: $i] :
( ~ member(Y,universal_class)
| ( Y = null_class )
| member(apply(choice,Y),Y) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(46,plain,
^ [Y: $i] :
rewrite(( ( ~ member(Y,universal_class)
| ( Y = null_class )
| member(apply(choice,Y),Y) )
<=> ( ~ member(Y,universal_class)
| ( Y = null_class )
| member(apply(choice,Y),Y) ) )),
inference(bind,[status(th)],[]) ).
tff(47,plain,
( ! [Y: $i] :
( ~ member(Y,universal_class)
| ( Y = null_class )
| member(apply(choice,Y),Y) )
<=> ! [Y: $i] :
( ~ member(Y,universal_class)
| ( Y = null_class )
| member(apply(choice,Y),Y) ) ),
inference(quant_intro,[status(thm)],[46]) ).
tff(48,axiom,
! [Y: $i] :
( ~ member(Y,universal_class)
| ( Y = null_class )
| member(apply(choice,Y),Y) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax',choice2) ).
tff(49,plain,
! [Y: $i] :
( ~ member(Y,universal_class)
| ( Y = null_class )
| member(apply(choice,Y),Y) ),
inference(modus_ponens,[status(thm)],[48,47]) ).
tff(50,plain,
! [Y: $i] :
( ~ member(Y,universal_class)
| ( Y = null_class )
| member(apply(choice,Y),Y) ),
inference(modus_ponens,[status(thm)],[49,45]) ).
tff(51,plain,
! [Y: $i] :
( ~ member(Y,universal_class)
| ( Y = null_class )
| member(apply(choice,Y),Y) ),
inference(skolemize,[status(sab)],[50]) ).
tff(52,plain,
! [Y: $i] :
( ~ member(Y,universal_class)
| ( Y = null_class )
| member(apply(choice,Y),Y) ),
inference(modus_ponens,[status(thm)],[51,44]) ).
tff(53,plain,
( ( ~ ! [Y: $i] :
( ~ member(Y,universal_class)
| ( Y = null_class )
| member(apply(choice,Y),Y) )
| ~ member(singleton(x),universal_class)
| ( singleton(x) = null_class )
| member(apply(choice,singleton(x)),singleton(x)) )
<=> ( ~ ! [Y: $i] :
( ~ member(Y,universal_class)
| ( Y = null_class )
| member(apply(choice,Y),Y) )
| ~ member(singleton(x),universal_class)
| ( singleton(x) = null_class )
| member(apply(choice,singleton(x)),singleton(x)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(54,plain,
( ~ ! [Y: $i] :
( ~ member(Y,universal_class)
| ( Y = null_class )
| member(apply(choice,Y),Y) )
| ~ member(singleton(x),universal_class)
| ( singleton(x) = null_class )
| member(apply(choice,singleton(x)),singleton(x)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(55,plain,
( ~ ! [Y: $i] :
( ~ member(Y,universal_class)
| ( Y = null_class )
| member(apply(choice,Y),Y) )
| ~ member(singleton(x),universal_class)
| ( singleton(x) = null_class )
| member(apply(choice,singleton(x)),singleton(x)) ),
inference(modus_ponens,[status(thm)],[54,53]) ).
tff(56,plain,
( ~ member(singleton(x),universal_class)
| member(apply(choice,singleton(x)),singleton(x)) ),
inference(unit_resolution,[status(thm)],[55,52,42]) ).
tff(57,plain,
member(apply(choice,singleton(x)),singleton(x)),
inference(unit_resolution,[status(thm)],[56,39]) ).
tff(58,plain,
member(apply(choice,singleton(x)),unordered_pair(x,x)),
inference(modus_ponens,[status(thm)],[57,26]) ).
tff(59,plain,
^ [Y: $i,U: $i,X: $i] :
refl(( ( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) )
<=> ( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) ) )),
inference(bind,[status(th)],[]) ).
tff(60,plain,
( ! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) )
<=> ! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) ) ),
inference(quant_intro,[status(thm)],[59]) ).
tff(61,plain,
( ! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) )
<=> ! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(62,plain,
^ [Y: $i,U: $i,X: $i] :
rewrite(( ( ~ member(U,unordered_pair(X,Y))
| ( U = X )
| ( U = Y ) )
<=> ( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) ) )),
inference(bind,[status(th)],[]) ).
tff(63,plain,
( ! [Y: $i,U: $i,X: $i] :
( ~ member(U,unordered_pair(X,Y))
| ( U = X )
| ( U = Y ) )
<=> ! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) ) ),
inference(quant_intro,[status(thm)],[62]) ).
tff(64,axiom,
! [Y: $i,U: $i,X: $i] :
( ~ member(U,unordered_pair(X,Y))
| ( U = X )
| ( U = Y ) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax',unordered_pair_member) ).
tff(65,plain,
! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) ),
inference(modus_ponens,[status(thm)],[64,63]) ).
tff(66,plain,
! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) ),
inference(modus_ponens,[status(thm)],[65,61]) ).
tff(67,plain,
! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) ),
inference(skolemize,[status(sab)],[66]) ).
tff(68,plain,
! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) ),
inference(modus_ponens,[status(thm)],[67,60]) ).
tff(69,plain,
( ( ~ ! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) )
| ( apply(choice,singleton(x)) = x )
| ~ member(apply(choice,singleton(x)),unordered_pair(x,x)) )
<=> ( ~ ! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) )
| ( apply(choice,singleton(x)) = x )
| ~ member(apply(choice,singleton(x)),unordered_pair(x,x)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(70,plain,
( ( ( apply(choice,singleton(x)) = x )
| ( apply(choice,singleton(x)) = x )
| ~ member(apply(choice,singleton(x)),unordered_pair(x,x)) )
<=> ( ( apply(choice,singleton(x)) = x )
| ~ member(apply(choice,singleton(x)),unordered_pair(x,x)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(71,plain,
( ( ~ ! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) )
| ( apply(choice,singleton(x)) = x )
| ( apply(choice,singleton(x)) = x )
| ~ member(apply(choice,singleton(x)),unordered_pair(x,x)) )
<=> ( ~ ! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) )
| ( apply(choice,singleton(x)) = x )
| ~ member(apply(choice,singleton(x)),unordered_pair(x,x)) ) ),
inference(monotonicity,[status(thm)],[70]) ).
tff(72,plain,
( ( ~ ! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) )
| ( apply(choice,singleton(x)) = x )
| ( apply(choice,singleton(x)) = x )
| ~ member(apply(choice,singleton(x)),unordered_pair(x,x)) )
<=> ( ~ ! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) )
| ( apply(choice,singleton(x)) = x )
| ~ member(apply(choice,singleton(x)),unordered_pair(x,x)) ) ),
inference(transitivity,[status(thm)],[71,69]) ).
tff(73,plain,
( ~ ! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) )
| ( apply(choice,singleton(x)) = x )
| ( apply(choice,singleton(x)) = x )
| ~ member(apply(choice,singleton(x)),unordered_pair(x,x)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(74,plain,
( ~ ! [Y: $i,U: $i,X: $i] :
( ( U = Y )
| ( U = X )
| ~ member(U,unordered_pair(X,Y)) )
| ( apply(choice,singleton(x)) = x )
| ~ member(apply(choice,singleton(x)),unordered_pair(x,x)) ),
inference(modus_ponens,[status(thm)],[73,72]) ).
tff(75,plain,
( ( apply(choice,singleton(x)) = x )
| ~ member(apply(choice,singleton(x)),unordered_pair(x,x)) ),
inference(unit_resolution,[status(thm)],[74,68]) ).
tff(76,plain,
apply(choice,singleton(x)) = x,
inference(unit_resolution,[status(thm)],[75,58]) ).
tff(77,plain,
x = apply(choice,singleton(x)),
inference(symmetry,[status(thm)],[76]) ).
tff(78,plain,
x = sum_class(image(choice,singleton(singleton(x)))),
inference(transitivity,[status(thm)],[77,22]) ).
tff(79,plain,
( member(x,universal_class)
<=> member(sum_class(image(choice,singleton(singleton(x)))),universal_class) ),
inference(monotonicity,[status(thm)],[78]) ).
tff(80,plain,
( ~ member(x,universal_class)
<=> ~ member(sum_class(image(choice,singleton(singleton(x)))),universal_class) ),
inference(monotonicity,[status(thm)],[79]) ).
tff(81,plain,
( ~ member(x,universal_class)
<=> ~ member(x,universal_class) ),
inference(rewrite,[status(thm)],[]) ).
tff(82,axiom,
~ member(x,universal_class),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_singleton_is_null_class_1) ).
tff(83,plain,
~ member(x,universal_class),
inference(modus_ponens,[status(thm)],[82,81]) ).
tff(84,plain,
~ member(sum_class(image(choice,singleton(singleton(x)))),universal_class),
inference(modus_ponens,[status(thm)],[83,80]) ).
tff(85,plain,
^ [X: $i] :
refl(( ( ~ member(X,universal_class)
| member(sum_class(X),universal_class) )
<=> ( ~ member(X,universal_class)
| member(sum_class(X),universal_class) ) )),
inference(bind,[status(th)],[]) ).
tff(86,plain,
( ! [X: $i] :
( ~ member(X,universal_class)
| member(sum_class(X),universal_class) )
<=> ! [X: $i] :
( ~ member(X,universal_class)
| member(sum_class(X),universal_class) ) ),
inference(quant_intro,[status(thm)],[85]) ).
tff(87,plain,
( ! [X: $i] :
( ~ member(X,universal_class)
| member(sum_class(X),universal_class) )
<=> ! [X: $i] :
( ~ member(X,universal_class)
| member(sum_class(X),universal_class) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(88,axiom,
! [X: $i] :
( ~ member(X,universal_class)
| member(sum_class(X),universal_class) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax',sum_class2) ).
tff(89,plain,
! [X: $i] :
( ~ member(X,universal_class)
| member(sum_class(X),universal_class) ),
inference(modus_ponens,[status(thm)],[88,87]) ).
tff(90,plain,
! [X: $i] :
( ~ member(X,universal_class)
| member(sum_class(X),universal_class) ),
inference(skolemize,[status(sab)],[89]) ).
tff(91,plain,
! [X: $i] :
( ~ member(X,universal_class)
| member(sum_class(X),universal_class) ),
inference(modus_ponens,[status(thm)],[90,86]) ).
tff(92,plain,
( ( ~ ! [X: $i] :
( ~ member(X,universal_class)
| member(sum_class(X),universal_class) )
| ~ member(image(choice,singleton(singleton(x))),universal_class)
| member(sum_class(image(choice,singleton(singleton(x)))),universal_class) )
<=> ( ~ ! [X: $i] :
( ~ member(X,universal_class)
| member(sum_class(X),universal_class) )
| ~ member(image(choice,singleton(singleton(x))),universal_class)
| member(sum_class(image(choice,singleton(singleton(x)))),universal_class) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(93,plain,
( ~ ! [X: $i] :
( ~ member(X,universal_class)
| member(sum_class(X),universal_class) )
| ~ member(image(choice,singleton(singleton(x))),universal_class)
| member(sum_class(image(choice,singleton(singleton(x)))),universal_class) ),
inference(quant_inst,[status(thm)],[]) ).
tff(94,plain,
( ~ ! [X: $i] :
( ~ member(X,universal_class)
| member(sum_class(X),universal_class) )
| ~ member(image(choice,singleton(singleton(x))),universal_class)
| member(sum_class(image(choice,singleton(singleton(x)))),universal_class) ),
inference(modus_ponens,[status(thm)],[93,92]) ).
tff(95,plain,
( ~ member(image(choice,singleton(singleton(x))),universal_class)
| member(sum_class(image(choice,singleton(singleton(x)))),universal_class) ),
inference(unit_resolution,[status(thm)],[94,91]) ).
tff(96,plain,
~ member(image(choice,singleton(singleton(x))),universal_class),
inference(unit_resolution,[status(thm)],[95,84]) ).
tff(97,plain,
( function(choice)
<=> function(choice) ),
inference(rewrite,[status(thm)],[]) ).
tff(98,axiom,
function(choice),
file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax',choice1) ).
tff(99,plain,
function(choice),
inference(modus_ponens,[status(thm)],[98,97]) ).
tff(100,plain,
^ [Xf: $i,X: $i] :
refl(( ( ~ member(X,universal_class)
| member(image(Xf,X),universal_class)
| ~ function(Xf) )
<=> ( ~ member(X,universal_class)
| member(image(Xf,X),universal_class)
| ~ function(Xf) ) )),
inference(bind,[status(th)],[]) ).
tff(101,plain,
( ! [Xf: $i,X: $i] :
( ~ member(X,universal_class)
| member(image(Xf,X),universal_class)
| ~ function(Xf) )
<=> ! [Xf: $i,X: $i] :
( ~ member(X,universal_class)
| member(image(Xf,X),universal_class)
| ~ function(Xf) ) ),
inference(quant_intro,[status(thm)],[100]) ).
tff(102,plain,
( ! [Xf: $i,X: $i] :
( ~ member(X,universal_class)
| member(image(Xf,X),universal_class)
| ~ function(Xf) )
<=> ! [Xf: $i,X: $i] :
( ~ member(X,universal_class)
| member(image(Xf,X),universal_class)
| ~ function(Xf) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(103,plain,
^ [Xf: $i,X: $i] :
rewrite(( ( ~ function(Xf)
| ~ member(X,universal_class)
| member(image(Xf,X),universal_class) )
<=> ( ~ member(X,universal_class)
| member(image(Xf,X),universal_class)
| ~ function(Xf) ) )),
inference(bind,[status(th)],[]) ).
tff(104,plain,
( ! [Xf: $i,X: $i] :
( ~ function(Xf)
| ~ member(X,universal_class)
| member(image(Xf,X),universal_class) )
<=> ! [Xf: $i,X: $i] :
( ~ member(X,universal_class)
| member(image(Xf,X),universal_class)
| ~ function(Xf) ) ),
inference(quant_intro,[status(thm)],[103]) ).
tff(105,axiom,
! [Xf: $i,X: $i] :
( ~ function(Xf)
| ~ member(X,universal_class)
| member(image(Xf,X),universal_class) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax',replacement) ).
tff(106,plain,
! [Xf: $i,X: $i] :
( ~ member(X,universal_class)
| member(image(Xf,X),universal_class)
| ~ function(Xf) ),
inference(modus_ponens,[status(thm)],[105,104]) ).
tff(107,plain,
! [Xf: $i,X: $i] :
( ~ member(X,universal_class)
| member(image(Xf,X),universal_class)
| ~ function(Xf) ),
inference(modus_ponens,[status(thm)],[106,102]) ).
tff(108,plain,
! [Xf: $i,X: $i] :
( ~ member(X,universal_class)
| member(image(Xf,X),universal_class)
| ~ function(Xf) ),
inference(skolemize,[status(sab)],[107]) ).
tff(109,plain,
! [Xf: $i,X: $i] :
( ~ member(X,universal_class)
| member(image(Xf,X),universal_class)
| ~ function(Xf) ),
inference(modus_ponens,[status(thm)],[108,101]) ).
tff(110,plain,
( ( ~ ! [Xf: $i,X: $i] :
( ~ member(X,universal_class)
| member(image(Xf,X),universal_class)
| ~ function(Xf) )
| ~ function(choice)
| ~ member(singleton(singleton(x)),universal_class)
| member(image(choice,singleton(singleton(x))),universal_class) )
<=> ( ~ ! [Xf: $i,X: $i] :
( ~ member(X,universal_class)
| member(image(Xf,X),universal_class)
| ~ function(Xf) )
| ~ function(choice)
| ~ member(singleton(singleton(x)),universal_class)
| member(image(choice,singleton(singleton(x))),universal_class) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(111,plain,
( ( ~ member(singleton(singleton(x)),universal_class)
| member(image(choice,singleton(singleton(x))),universal_class)
| ~ function(choice) )
<=> ( ~ function(choice)
| ~ member(singleton(singleton(x)),universal_class)
| member(image(choice,singleton(singleton(x))),universal_class) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(112,plain,
( ( ~ ! [Xf: $i,X: $i] :
( ~ member(X,universal_class)
| member(image(Xf,X),universal_class)
| ~ function(Xf) )
| ~ member(singleton(singleton(x)),universal_class)
| member(image(choice,singleton(singleton(x))),universal_class)
| ~ function(choice) )
<=> ( ~ ! [Xf: $i,X: $i] :
( ~ member(X,universal_class)
| member(image(Xf,X),universal_class)
| ~ function(Xf) )
| ~ function(choice)
| ~ member(singleton(singleton(x)),universal_class)
| member(image(choice,singleton(singleton(x))),universal_class) ) ),
inference(monotonicity,[status(thm)],[111]) ).
tff(113,plain,
( ( ~ ! [Xf: $i,X: $i] :
( ~ member(X,universal_class)
| member(image(Xf,X),universal_class)
| ~ function(Xf) )
| ~ member(singleton(singleton(x)),universal_class)
| member(image(choice,singleton(singleton(x))),universal_class)
| ~ function(choice) )
<=> ( ~ ! [Xf: $i,X: $i] :
( ~ member(X,universal_class)
| member(image(Xf,X),universal_class)
| ~ function(Xf) )
| ~ function(choice)
| ~ member(singleton(singleton(x)),universal_class)
| member(image(choice,singleton(singleton(x))),universal_class) ) ),
inference(transitivity,[status(thm)],[112,110]) ).
tff(114,plain,
( ~ ! [Xf: $i,X: $i] :
( ~ member(X,universal_class)
| member(image(Xf,X),universal_class)
| ~ function(Xf) )
| ~ member(singleton(singleton(x)),universal_class)
| member(image(choice,singleton(singleton(x))),universal_class)
| ~ function(choice) ),
inference(quant_inst,[status(thm)],[]) ).
tff(115,plain,
( ~ ! [Xf: $i,X: $i] :
( ~ member(X,universal_class)
| member(image(Xf,X),universal_class)
| ~ function(Xf) )
| ~ function(choice)
| ~ member(singleton(singleton(x)),universal_class)
| member(image(choice,singleton(singleton(x))),universal_class) ),
inference(modus_ponens,[status(thm)],[114,113]) ).
tff(116,plain,
( ~ member(singleton(singleton(x)),universal_class)
| member(image(choice,singleton(singleton(x))),universal_class) ),
inference(unit_resolution,[status(thm)],[115,109,99]) ).
tff(117,plain,
~ member(singleton(singleton(x)),universal_class),
inference(unit_resolution,[status(thm)],[116,96]) ).
tff(118,plain,
~ member(unordered_pair(singleton(x),singleton(x)),universal_class),
inference(modus_ponens,[status(thm)],[117,12]) ).
tff(119,plain,
( ~ ! [Y: $i,X: $i] : member(unordered_pair(X,Y),universal_class)
| member(unordered_pair(singleton(x),singleton(x)),universal_class) ),
inference(quant_inst,[status(thm)],[]) ).
tff(120,plain,
member(unordered_pair(singleton(x),singleton(x)),universal_class),
inference(unit_resolution,[status(thm)],[119,36]) ).
tff(121,plain,
$false,
inference(unit_resolution,[status(thm)],[120,118]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : SET082-6 : TPTP v9.0.0. Bugfixed v2.1.0.
% 0.03/0.12 % Command : run_E %s %d THM
% 0.12/0.33 % Computer : n007.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Fri Jun 20 04:15:05 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.20/0.42 % SZS status Unsatisfiable
% 0.20/0.42 % SZS output start Proof
% See solution above
% 0.20/0.44 % E exiting
%------------------------------------------------------------------------------