%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET099-7 : TPTP v9.0.0. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n002.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 05:25:13 AM UTC 2025
% Result : Unsatisfiable 0.20s 0.47s
% Output : Proof 0.20s
% Verified :
% SZS Type : Refutation
% Derivation depth : 12
% Number of leaves : 43
% Syntax : Number of formulae : 99 ( 29 unt; 0 typ; 0 def)
% Number of atoms : 381 ( 87 equ)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 438 ( 178 ~; 216 |; 0 &)
% ( 44 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 4 avg)
% Maximal term depth : 6 ( 2 avg)
% Number of FOOLs : 22 ( 22 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 8 ( 6 usr; 1 prp; 0-3 aty)
% Number of functors : 7 ( 7 usr; 3 con; 0-2 aty)
% Number of variables : 177 ( 159 !; 0 ?; 177 :)
% Comments :
%------------------------------------------------------------------------------
tff(subclass_type,type,
subclass: ( $i * $i ) > $o ).
tff(universal_class_type,type,
universal_class: $i ).
tff(x_type,type,
x: $i ).
tff(member_type,type,
member: ( $i * $i ) > $o ).
tff(not_subclass_element_type,type,
not_subclass_element: ( $i * $i ) > $i ).
tff(null_class_type,type,
null_class: $i ).
tff(singleton_type,type,
singleton: $i > $i ).
tff(complement_type,type,
complement: $i > $i ).
tff(intersection_type,type,
intersection: ( $i * $i ) > $i ).
tff(1,plain,
( ( not_subclass_element(intersection(complement(singleton(not_subclass_element(x,null_class))),x),null_class) = not_subclass_element(x,null_class) )
<=> ( not_subclass_element(intersection(complement(singleton(not_subclass_element(x,null_class))),x),null_class) = not_subclass_element(x,null_class) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(2,axiom,
not_subclass_element(intersection(complement(singleton(not_subclass_element(x,null_class))),x),null_class) = not_subclass_element(x,null_class),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_corollary_2_to_number_of_elements_in_class_1) ).
tff(3,plain,
not_subclass_element(intersection(complement(singleton(not_subclass_element(x,null_class))),x),null_class) = not_subclass_element(x,null_class),
inference(modus_ponens,[status(thm)],[2,1]) ).
tff(4,plain,
not_subclass_element(x,null_class) = not_subclass_element(intersection(complement(singleton(not_subclass_element(x,null_class))),x),null_class),
inference(symmetry,[status(thm)],[3]) ).
tff(5,plain,
( member(not_subclass_element(x,null_class),intersection(complement(singleton(not_subclass_element(x,null_class))),x))
<=> member(not_subclass_element(intersection(complement(singleton(not_subclass_element(x,null_class))),x),null_class),intersection(complement(singleton(not_subclass_element(x,null_class))),x)) ),
inference(monotonicity,[status(thm)],[4]) ).
tff(6,plain,
( member(not_subclass_element(intersection(complement(singleton(not_subclass_element(x,null_class))),x),null_class),intersection(complement(singleton(not_subclass_element(x,null_class))),x))
<=> member(not_subclass_element(x,null_class),intersection(complement(singleton(not_subclass_element(x,null_class))),x)) ),
inference(symmetry,[status(thm)],[5]) ).
tff(7,plain,
( ( x != null_class )
<=> ( x != null_class ) ),
inference(rewrite,[status(thm)],[]) ).
tff(8,axiom,
x != null_class,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_corollary_2_to_number_of_elements_in_class_3) ).
tff(9,plain,
x != null_class,
inference(modus_ponens,[status(thm)],[8,7]) ).
tff(10,plain,
( ( singleton(not_subclass_element(x,null_class)) != x )
<=> ( singleton(not_subclass_element(x,null_class)) != x ) ),
inference(rewrite,[status(thm)],[]) ).
tff(11,axiom,
singleton(not_subclass_element(x,null_class)) != x,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',prove_corollary_2_to_number_of_elements_in_class_2) ).
tff(12,plain,
singleton(not_subclass_element(x,null_class)) != x,
inference(modus_ponens,[status(thm)],[11,10]) ).
tff(13,plain,
^ [X: $i] :
refl(( ( ( X = null_class )
| ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) )
<=> ( ( X = null_class )
| ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) ) )),
inference(bind,[status(th)],[]) ).
tff(14,plain,
( ! [X: $i] :
( ( X = null_class )
| ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) )
<=> ! [X: $i] :
( ( X = null_class )
| ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) ) ),
inference(quant_intro,[status(thm)],[13]) ).
tff(15,plain,
( ! [X: $i] :
( ( X = null_class )
| ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) )
<=> ! [X: $i] :
( ( X = null_class )
| ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(16,plain,
^ [X: $i] :
trans(monotonicity(rewrite(( ( member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X))
| ( singleton(not_subclass_element(X,null_class)) = X ) )
<=> ( ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) ) )),
( ( member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X))
| ( singleton(not_subclass_element(X,null_class)) = X )
| ( X = null_class ) )
<=> ( ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X))
| ( X = null_class ) ) )),
rewrite(( ( ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X))
| ( X = null_class ) )
<=> ( ( X = null_class )
| ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) ) )),
( ( member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X))
| ( singleton(not_subclass_element(X,null_class)) = X )
| ( X = null_class ) )
<=> ( ( X = null_class )
| ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) ) )),
inference(bind,[status(th)],[]) ).
tff(17,plain,
( ! [X: $i] :
( member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X))
| ( singleton(not_subclass_element(X,null_class)) = X )
| ( X = null_class ) )
<=> ! [X: $i] :
( ( X = null_class )
| ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) ) ),
inference(quant_intro,[status(thm)],[16]) ).
tff(18,axiom,
! [X: $i] :
( member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X))
| ( singleton(not_subclass_element(X,null_class)) = X )
| ( X = null_class ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',number_of_elements_in_class) ).
tff(19,plain,
! [X: $i] :
( ( X = null_class )
| ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) ),
inference(modus_ponens,[status(thm)],[18,17]) ).
tff(20,plain,
! [X: $i] :
( ( X = null_class )
| ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) ),
inference(modus_ponens,[status(thm)],[19,15]) ).
tff(21,plain,
! [X: $i] :
( ( X = null_class )
| ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) ),
inference(skolemize,[status(sab)],[20]) ).
tff(22,plain,
! [X: $i] :
( ( X = null_class )
| ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) ),
inference(modus_ponens,[status(thm)],[21,14]) ).
tff(23,plain,
( ( ~ ! [X: $i] :
( ( X = null_class )
| ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) )
| ( x = null_class )
| ( singleton(not_subclass_element(x,null_class)) = x )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(x,null_class))),x),null_class),intersection(complement(singleton(not_subclass_element(x,null_class))),x)) )
<=> ( ~ ! [X: $i] :
( ( X = null_class )
| ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) )
| ( x = null_class )
| ( singleton(not_subclass_element(x,null_class)) = x )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(x,null_class))),x),null_class),intersection(complement(singleton(not_subclass_element(x,null_class))),x)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(24,plain,
( ~ ! [X: $i] :
( ( X = null_class )
| ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) )
| ( x = null_class )
| ( singleton(not_subclass_element(x,null_class)) = x )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(x,null_class))),x),null_class),intersection(complement(singleton(not_subclass_element(x,null_class))),x)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(25,plain,
( ~ ! [X: $i] :
( ( X = null_class )
| ( singleton(not_subclass_element(X,null_class)) = X )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(X,null_class))),X),null_class),intersection(complement(singleton(not_subclass_element(X,null_class))),X)) )
| ( x = null_class )
| ( singleton(not_subclass_element(x,null_class)) = x )
| member(not_subclass_element(intersection(complement(singleton(not_subclass_element(x,null_class))),x),null_class),intersection(complement(singleton(not_subclass_element(x,null_class))),x)) ),
inference(modus_ponens,[status(thm)],[24,23]) ).
tff(26,plain,
member(not_subclass_element(intersection(complement(singleton(not_subclass_element(x,null_class))),x),null_class),intersection(complement(singleton(not_subclass_element(x,null_class))),x)),
inference(unit_resolution,[status(thm)],[25,22,12,9]) ).
tff(27,plain,
member(not_subclass_element(x,null_class),intersection(complement(singleton(not_subclass_element(x,null_class))),x)),
inference(modus_ponens,[status(thm)],[26,6]) ).
tff(28,plain,
^ [Z: $i,Y: $i,X: $i] :
refl(( ( ~ member(Z,intersection(X,Y))
| member(Z,X) )
<=> ( ~ member(Z,intersection(X,Y))
| member(Z,X) ) )),
inference(bind,[status(th)],[]) ).
tff(29,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) ) ),
inference(quant_intro,[status(thm)],[28]) ).
tff(30,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(31,axiom,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax',intersection1) ).
tff(32,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) ),
inference(modus_ponens,[status(thm)],[31,30]) ).
tff(33,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) ),
inference(skolemize,[status(sab)],[32]) ).
tff(34,plain,
! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) ),
inference(modus_ponens,[status(thm)],[33,29]) ).
tff(35,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) )
| ~ member(not_subclass_element(x,null_class),intersection(complement(singleton(not_subclass_element(x,null_class))),x))
| member(not_subclass_element(x,null_class),complement(singleton(not_subclass_element(x,null_class)))) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) )
| ~ member(not_subclass_element(x,null_class),intersection(complement(singleton(not_subclass_element(x,null_class))),x))
| member(not_subclass_element(x,null_class),complement(singleton(not_subclass_element(x,null_class)))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(36,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) )
| ~ member(not_subclass_element(x,null_class),intersection(complement(singleton(not_subclass_element(x,null_class))),x))
| member(not_subclass_element(x,null_class),complement(singleton(not_subclass_element(x,null_class)))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(37,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ~ member(Z,intersection(X,Y))
| member(Z,X) )
| ~ member(not_subclass_element(x,null_class),intersection(complement(singleton(not_subclass_element(x,null_class))),x))
| member(not_subclass_element(x,null_class),complement(singleton(not_subclass_element(x,null_class)))) ),
inference(modus_ponens,[status(thm)],[36,35]) ).
tff(38,plain,
( ~ member(not_subclass_element(x,null_class),intersection(complement(singleton(not_subclass_element(x,null_class))),x))
| member(not_subclass_element(x,null_class),complement(singleton(not_subclass_element(x,null_class)))) ),
inference(unit_resolution,[status(thm)],[37,34]) ).
tff(39,plain,
member(not_subclass_element(x,null_class),complement(singleton(not_subclass_element(x,null_class)))),
inference(unit_resolution,[status(thm)],[38,27]) ).
tff(40,plain,
^ [Z: $i,X: $i] :
refl(( ( ~ member(Z,X)
| ~ member(Z,complement(X)) )
<=> ( ~ member(Z,X)
| ~ member(Z,complement(X)) ) )),
inference(bind,[status(th)],[]) ).
tff(41,plain,
( ! [Z: $i,X: $i] :
( ~ member(Z,X)
| ~ member(Z,complement(X)) )
<=> ! [Z: $i,X: $i] :
( ~ member(Z,X)
| ~ member(Z,complement(X)) ) ),
inference(quant_intro,[status(thm)],[40]) ).
tff(42,plain,
( ! [Z: $i,X: $i] :
( ~ member(Z,X)
| ~ member(Z,complement(X)) )
<=> ! [Z: $i,X: $i] :
( ~ member(Z,X)
| ~ member(Z,complement(X)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(43,plain,
^ [Z: $i,X: $i] :
rewrite(( ( ~ member(Z,complement(X))
| ~ member(Z,X) )
<=> ( ~ member(Z,X)
| ~ member(Z,complement(X)) ) )),
inference(bind,[status(th)],[]) ).
tff(44,plain,
( ! [Z: $i,X: $i] :
( ~ member(Z,complement(X))
| ~ member(Z,X) )
<=> ! [Z: $i,X: $i] :
( ~ member(Z,X)
| ~ member(Z,complement(X)) ) ),
inference(quant_intro,[status(thm)],[43]) ).
tff(45,axiom,
! [Z: $i,X: $i] :
( ~ member(Z,complement(X))
| ~ member(Z,X) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax',complement1) ).
tff(46,plain,
! [Z: $i,X: $i] :
( ~ member(Z,X)
| ~ member(Z,complement(X)) ),
inference(modus_ponens,[status(thm)],[45,44]) ).
tff(47,plain,
! [Z: $i,X: $i] :
( ~ member(Z,X)
| ~ member(Z,complement(X)) ),
inference(modus_ponens,[status(thm)],[46,42]) ).
tff(48,plain,
! [Z: $i,X: $i] :
( ~ member(Z,X)
| ~ member(Z,complement(X)) ),
inference(skolemize,[status(sab)],[47]) ).
tff(49,plain,
! [Z: $i,X: $i] :
( ~ member(Z,X)
| ~ member(Z,complement(X)) ),
inference(modus_ponens,[status(thm)],[48,41]) ).
tff(50,plain,
( ( ~ ! [Z: $i,X: $i] :
( ~ member(Z,X)
| ~ member(Z,complement(X)) )
| ~ member(not_subclass_element(x,null_class),singleton(not_subclass_element(x,null_class)))
| ~ member(not_subclass_element(x,null_class),complement(singleton(not_subclass_element(x,null_class)))) )
<=> ( ~ ! [Z: $i,X: $i] :
( ~ member(Z,X)
| ~ member(Z,complement(X)) )
| ~ member(not_subclass_element(x,null_class),singleton(not_subclass_element(x,null_class)))
| ~ member(not_subclass_element(x,null_class),complement(singleton(not_subclass_element(x,null_class)))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(51,plain,
( ~ ! [Z: $i,X: $i] :
( ~ member(Z,X)
| ~ member(Z,complement(X)) )
| ~ member(not_subclass_element(x,null_class),singleton(not_subclass_element(x,null_class)))
| ~ member(not_subclass_element(x,null_class),complement(singleton(not_subclass_element(x,null_class)))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(52,plain,
( ~ ! [Z: $i,X: $i] :
( ~ member(Z,X)
| ~ member(Z,complement(X)) )
| ~ member(not_subclass_element(x,null_class),singleton(not_subclass_element(x,null_class)))
| ~ member(not_subclass_element(x,null_class),complement(singleton(not_subclass_element(x,null_class)))) ),
inference(modus_ponens,[status(thm)],[51,50]) ).
tff(53,plain,
~ member(not_subclass_element(x,null_class),singleton(not_subclass_element(x,null_class))),
inference(unit_resolution,[status(thm)],[52,49,39]) ).
tff(54,plain,
^ [X: $i] :
refl(( ( ~ member(X,universal_class)
| member(X,singleton(X)) )
<=> ( ~ member(X,universal_class)
| member(X,singleton(X)) ) )),
inference(bind,[status(th)],[]) ).
tff(55,plain,
( ! [X: $i] :
( ~ member(X,universal_class)
| member(X,singleton(X)) )
<=> ! [X: $i] :
( ~ member(X,universal_class)
| member(X,singleton(X)) ) ),
inference(quant_intro,[status(thm)],[54]) ).
tff(56,plain,
( ! [X: $i] :
( ~ member(X,universal_class)
| member(X,singleton(X)) )
<=> ! [X: $i] :
( ~ member(X,universal_class)
| member(X,singleton(X)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(57,axiom,
! [X: $i] :
( ~ member(X,universal_class)
| member(X,singleton(X)) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',set_in_its_singleton) ).
tff(58,plain,
! [X: $i] :
( ~ member(X,universal_class)
| member(X,singleton(X)) ),
inference(modus_ponens,[status(thm)],[57,56]) ).
tff(59,plain,
! [X: $i] :
( ~ member(X,universal_class)
| member(X,singleton(X)) ),
inference(skolemize,[status(sab)],[58]) ).
tff(60,plain,
! [X: $i] :
( ~ member(X,universal_class)
| member(X,singleton(X)) ),
inference(modus_ponens,[status(thm)],[59,55]) ).
tff(61,plain,
( ( ~ ! [X: $i] :
( ~ member(X,universal_class)
| member(X,singleton(X)) )
| ~ member(not_subclass_element(x,null_class),universal_class)
| member(not_subclass_element(x,null_class),singleton(not_subclass_element(x,null_class))) )
<=> ( ~ ! [X: $i] :
( ~ member(X,universal_class)
| member(X,singleton(X)) )
| ~ member(not_subclass_element(x,null_class),universal_class)
| member(not_subclass_element(x,null_class),singleton(not_subclass_element(x,null_class))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(62,plain,
( ~ ! [X: $i] :
( ~ member(X,universal_class)
| member(X,singleton(X)) )
| ~ member(not_subclass_element(x,null_class),universal_class)
| member(not_subclass_element(x,null_class),singleton(not_subclass_element(x,null_class))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(63,plain,
( ~ ! [X: $i] :
( ~ member(X,universal_class)
| member(X,singleton(X)) )
| ~ member(not_subclass_element(x,null_class),universal_class)
| member(not_subclass_element(x,null_class),singleton(not_subclass_element(x,null_class))) ),
inference(modus_ponens,[status(thm)],[62,61]) ).
tff(64,plain,
( ~ member(not_subclass_element(x,null_class),universal_class)
| member(not_subclass_element(x,null_class),singleton(not_subclass_element(x,null_class))) ),
inference(unit_resolution,[status(thm)],[63,60]) ).
tff(65,plain,
~ member(not_subclass_element(x,null_class),universal_class),
inference(unit_resolution,[status(thm)],[64,53]) ).
tff(66,plain,
^ [Z: $i] :
refl(( ( ( Z = null_class )
| member(not_subclass_element(Z,null_class),Z) )
<=> ( ( Z = null_class )
| member(not_subclass_element(Z,null_class),Z) ) )),
inference(bind,[status(th)],[]) ).
tff(67,plain,
( ! [Z: $i] :
( ( Z = null_class )
| member(not_subclass_element(Z,null_class),Z) )
<=> ! [Z: $i] :
( ( Z = null_class )
| member(not_subclass_element(Z,null_class),Z) ) ),
inference(quant_intro,[status(thm)],[66]) ).
tff(68,plain,
( ! [Z: $i] :
( ( Z = null_class )
| member(not_subclass_element(Z,null_class),Z) )
<=> ! [Z: $i] :
( ( Z = null_class )
| member(not_subclass_element(Z,null_class),Z) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(69,axiom,
! [Z: $i] :
( ( Z = null_class )
| member(not_subclass_element(Z,null_class),Z) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',null_class_is_unique) ).
tff(70,plain,
! [Z: $i] :
( ( Z = null_class )
| member(not_subclass_element(Z,null_class),Z) ),
inference(modus_ponens,[status(thm)],[69,68]) ).
tff(71,plain,
! [Z: $i] :
( ( Z = null_class )
| member(not_subclass_element(Z,null_class),Z) ),
inference(skolemize,[status(sab)],[70]) ).
tff(72,plain,
! [Z: $i] :
( ( Z = null_class )
| member(not_subclass_element(Z,null_class),Z) ),
inference(modus_ponens,[status(thm)],[71,67]) ).
tff(73,plain,
( ( ~ ! [Z: $i] :
( ( Z = null_class )
| member(not_subclass_element(Z,null_class),Z) )
| ( x = null_class )
| member(not_subclass_element(x,null_class),x) )
<=> ( ~ ! [Z: $i] :
( ( Z = null_class )
| member(not_subclass_element(Z,null_class),Z) )
| ( x = null_class )
| member(not_subclass_element(x,null_class),x) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(74,plain,
( ~ ! [Z: $i] :
( ( Z = null_class )
| member(not_subclass_element(Z,null_class),Z) )
| ( x = null_class )
| member(not_subclass_element(x,null_class),x) ),
inference(quant_inst,[status(thm)],[]) ).
tff(75,plain,
( ~ ! [Z: $i] :
( ( Z = null_class )
| member(not_subclass_element(Z,null_class),Z) )
| ( x = null_class )
| member(not_subclass_element(x,null_class),x) ),
inference(modus_ponens,[status(thm)],[74,73]) ).
tff(76,plain,
member(not_subclass_element(x,null_class),x),
inference(unit_resolution,[status(thm)],[75,72,9]) ).
tff(77,plain,
^ [Y: $i,U: $i,X: $i] :
refl(( ( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) )
<=> ( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) ) )),
inference(bind,[status(th)],[]) ).
tff(78,plain,
( ! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) )
<=> ! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) ) ),
inference(quant_intro,[status(thm)],[77]) ).
tff(79,plain,
( ! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) )
<=> ! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(80,plain,
^ [Y: $i,U: $i,X: $i] :
trans(monotonicity(rewrite(( ( ~ subclass(X,Y)
| ~ member(U,X) )
<=> ( ~ member(U,X)
| ~ subclass(X,Y) ) )),
( ( ~ subclass(X,Y)
| ~ member(U,X)
| member(U,Y) )
<=> ( ~ member(U,X)
| ~ subclass(X,Y)
| member(U,Y) ) )),
rewrite(( ( ~ member(U,X)
| ~ subclass(X,Y)
| member(U,Y) )
<=> ( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) ) )),
( ( ~ subclass(X,Y)
| ~ member(U,X)
| member(U,Y) )
<=> ( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) ) )),
inference(bind,[status(th)],[]) ).
tff(81,plain,
( ! [Y: $i,U: $i,X: $i] :
( ~ subclass(X,Y)
| ~ member(U,X)
| member(U,Y) )
<=> ! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) ) ),
inference(quant_intro,[status(thm)],[80]) ).
tff(82,axiom,
! [Y: $i,U: $i,X: $i] :
( ~ subclass(X,Y)
| ~ member(U,X)
| member(U,Y) ),
file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax',subclass_members) ).
tff(83,plain,
! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) ),
inference(modus_ponens,[status(thm)],[82,81]) ).
tff(84,plain,
! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) ),
inference(modus_ponens,[status(thm)],[83,79]) ).
tff(85,plain,
! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) ),
inference(skolemize,[status(sab)],[84]) ).
tff(86,plain,
! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) ),
inference(modus_ponens,[status(thm)],[85,78]) ).
tff(87,plain,
( ( ~ ! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) )
| member(not_subclass_element(x,null_class),universal_class)
| ~ member(not_subclass_element(x,null_class),x)
| ~ subclass(x,universal_class) )
<=> ( ~ ! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) )
| member(not_subclass_element(x,null_class),universal_class)
| ~ member(not_subclass_element(x,null_class),x)
| ~ subclass(x,universal_class) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(88,plain,
( ~ ! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) )
| member(not_subclass_element(x,null_class),universal_class)
| ~ member(not_subclass_element(x,null_class),x)
| ~ subclass(x,universal_class) ),
inference(quant_inst,[status(thm)],[]) ).
tff(89,plain,
( ~ ! [Y: $i,U: $i,X: $i] :
( member(U,Y)
| ~ member(U,X)
| ~ subclass(X,Y) )
| member(not_subclass_element(x,null_class),universal_class)
| ~ member(not_subclass_element(x,null_class),x)
| ~ subclass(x,universal_class) ),
inference(modus_ponens,[status(thm)],[88,87]) ).
tff(90,plain,
~ subclass(x,universal_class),
inference(unit_resolution,[status(thm)],[89,86,76,65]) ).
tff(91,plain,
^ [X: $i] :
refl(( subclass(X,universal_class)
<=> subclass(X,universal_class) )),
inference(bind,[status(th)],[]) ).
tff(92,plain,
( ! [X: $i] : subclass(X,universal_class)
<=> ! [X: $i] : subclass(X,universal_class) ),
inference(quant_intro,[status(thm)],[91]) ).
tff(93,plain,
( ! [X: $i] : subclass(X,universal_class)
<=> ! [X: $i] : subclass(X,universal_class) ),
inference(rewrite,[status(thm)],[]) ).
tff(94,axiom,
! [X: $i] : subclass(X,universal_class),
file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax',class_elements_are_sets) ).
tff(95,plain,
! [X: $i] : subclass(X,universal_class),
inference(modus_ponens,[status(thm)],[94,93]) ).
tff(96,plain,
! [X: $i] : subclass(X,universal_class),
inference(skolemize,[status(sab)],[95]) ).
tff(97,plain,
! [X: $i] : subclass(X,universal_class),
inference(modus_ponens,[status(thm)],[96,92]) ).
tff(98,plain,
( ~ ! [X: $i] : subclass(X,universal_class)
| subclass(x,universal_class) ),
inference(quant_inst,[status(thm)],[]) ).
tff(99,plain,
$false,
inference(unit_resolution,[status(thm)],[98,97,90]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.12 % Problem : SET099-7 : TPTP v9.0.0. Bugfixed v2.1.0.
% 0.06/0.12 % Command : run_E %s %d THM
% 0.12/0.34 % Computer : n002.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Fri Jun 20 04:18:31 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.20/0.47 % SZS status Unsatisfiable
% 0.20/0.47 % SZS output start Proof
% See solution above
% 0.20/0.48 % E exiting
%------------------------------------------------------------------------------