%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET090-7 : TPTP v9.0.0. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n005.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 05:25:11 AM UTC 2025
% Result : Unsatisfiable 0.20s 0.40s
% Output : Proof 0.20s
% Verified :
% SZS Type : Refutation
% Derivation depth : 7
% Number of leaves : 25
% Syntax : Number of formulae : 56 ( 19 unt; 0 typ; 0 def)
% Number of atoms : 273 ( 157 equ)
% Maximal formula atoms : 15 ( 4 avg)
% Number of connectives : 347 ( 147 ~; 172 |; 0 &)
% ( 28 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 5 avg)
% Maximal term depth : 4 ( 2 avg)
% Number of FOOLs : 17 ( 17 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 7 ( 5 usr; 1 prp; 0-3 aty)
% Number of functors : 5 ( 5 usr; 2 con; 0-2 aty)
% Number of variables : 99 ( 90 !; 0 ?; 99 :)
% Comments :
%------------------------------------------------------------------------------
tff(singleton_type,type,
singleton: $i > $i ).
tff(member_of_type,type,
member_of: $i > $i ).
tff(u_type,type,
u: $i ).
tff(unordered_pair_type,type,
unordered_pair: ( $i * $i ) > $i ).
tff(member_type,type,
member: ( $i * $i ) > $o ).
tff(universal_class_type,type,
universal_class: $i ).
tff(1,plain,
( member(u,universal_class)
<=> member(u,universal_class) ),
inference(rewrite,[status(thm)],[]) ).
tff(2,axiom,
member(u,universal_class),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_member_of_singleton_is_unique_1) ).
tff(3,plain,
member(u,universal_class),
inference(modus_ponens,[status(thm)],[2,1]) ).
tff(4,plain,
^ [Y: $i] :
refl(( ( ~ member(Y,universal_class)
| ( singleton(member_of(singleton(Y))) = singleton(Y) ) )
<=> ( ~ member(Y,universal_class)
| ( singleton(member_of(singleton(Y))) = singleton(Y) ) ) )),
inference(bind,[status(th)],[]) ).
tff(5,plain,
( ! [Y: $i] :
( ~ member(Y,universal_class)
| ( singleton(member_of(singleton(Y))) = singleton(Y) ) )
<=> ! [Y: $i] :
( ~ member(Y,universal_class)
| ( singleton(member_of(singleton(Y))) = singleton(Y) ) ) ),
inference(quant_intro,[status(thm)],[4]) ).
tff(6,plain,
( ! [Y: $i] :
( ~ member(Y,universal_class)
| ( singleton(member_of(singleton(Y))) = singleton(Y) ) )
<=> ! [Y: $i] :
( ~ member(Y,universal_class)
| ( singleton(member_of(singleton(Y))) = singleton(Y) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(7,axiom,
! [Y: $i] :
( ~ member(Y,universal_class)
| ( singleton(member_of(singleton(Y))) = singleton(Y) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',member_exists2) ).
tff(8,plain,
! [Y: $i] :
( ~ member(Y,universal_class)
| ( singleton(member_of(singleton(Y))) = singleton(Y) ) ),
inference(modus_ponens,[status(thm)],[7,6]) ).
tff(9,plain,
! [Y: $i] :
( ~ member(Y,universal_class)
| ( singleton(member_of(singleton(Y))) = singleton(Y) ) ),
inference(skolemize,[status(sab)],[8]) ).
tff(10,plain,
! [Y: $i] :
( ~ member(Y,universal_class)
| ( singleton(member_of(singleton(Y))) = singleton(Y) ) ),
inference(modus_ponens,[status(thm)],[9,5]) ).
tff(11,plain,
( ( ~ ! [Y: $i] :
( ~ member(Y,universal_class)
| ( singleton(member_of(singleton(Y))) = singleton(Y) ) )
| ~ member(u,universal_class)
| ( singleton(member_of(singleton(u))) = singleton(u) ) )
<=> ( ~ ! [Y: $i] :
( ~ member(Y,universal_class)
| ( singleton(member_of(singleton(Y))) = singleton(Y) ) )
| ~ member(u,universal_class)
| ( singleton(member_of(singleton(u))) = singleton(u) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(12,plain,
( ~ ! [Y: $i] :
( ~ member(Y,universal_class)
| ( singleton(member_of(singleton(Y))) = singleton(Y) ) )
| ~ member(u,universal_class)
| ( singleton(member_of(singleton(u))) = singleton(u) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(13,plain,
( ~ ! [Y: $i] :
( ~ member(Y,universal_class)
| ( singleton(member_of(singleton(Y))) = singleton(Y) ) )
| ~ member(u,universal_class)
| ( singleton(member_of(singleton(u))) = singleton(u) ) ),
inference(modus_ponens,[status(thm)],[12,11]) ).
tff(14,plain,
singleton(member_of(singleton(u))) = singleton(u),
inference(unit_resolution,[status(thm)],[13,10,3]) ).
tff(15,plain,
singleton(u) = singleton(member_of(singleton(u))),
inference(symmetry,[status(thm)],[14]) ).
tff(16,plain,
^ [X: $i] :
refl(( ( unordered_pair(X,X) = singleton(X) )
<=> ( unordered_pair(X,X) = singleton(X) ) )),
inference(bind,[status(th)],[]) ).
tff(17,plain,
( ! [X: $i] : ( unordered_pair(X,X) = singleton(X) )
<=> ! [X: $i] : ( unordered_pair(X,X) = singleton(X) ) ),
inference(quant_intro,[status(thm)],[16]) ).
tff(18,plain,
( ! [X: $i] : ( unordered_pair(X,X) = singleton(X) )
<=> ! [X: $i] : ( unordered_pair(X,X) = singleton(X) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(19,axiom,
! [X: $i] : ( unordered_pair(X,X) = singleton(X) ),
file('/export/starexec/sandbox/benchmark/Axioms/SET004-0.ax',singleton_set) ).
tff(20,plain,
! [X: $i] : ( unordered_pair(X,X) = singleton(X) ),
inference(modus_ponens,[status(thm)],[19,18]) ).
tff(21,plain,
! [X: $i] : ( unordered_pair(X,X) = singleton(X) ),
inference(skolemize,[status(sab)],[20]) ).
tff(22,plain,
! [X: $i] : ( unordered_pair(X,X) = singleton(X) ),
inference(modus_ponens,[status(thm)],[21,17]) ).
tff(23,plain,
( ~ ! [X: $i] : ( unordered_pair(X,X) = singleton(X) )
| ( unordered_pair(u,u) = singleton(u) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(24,plain,
unordered_pair(u,u) = singleton(u),
inference(unit_resolution,[status(thm)],[23,22]) ).
tff(25,plain,
unordered_pair(u,u) = singleton(member_of(singleton(u))),
inference(transitivity,[status(thm)],[24,15]) ).
tff(26,plain,
^ [Y: $i] :
refl(( ( ~ member(Y,universal_class)
| member(member_of(singleton(Y)),universal_class) )
<=> ( ~ member(Y,universal_class)
| member(member_of(singleton(Y)),universal_class) ) )),
inference(bind,[status(th)],[]) ).
tff(27,plain,
( ! [Y: $i] :
( ~ member(Y,universal_class)
| member(member_of(singleton(Y)),universal_class) )
<=> ! [Y: $i] :
( ~ member(Y,universal_class)
| member(member_of(singleton(Y)),universal_class) ) ),
inference(quant_intro,[status(thm)],[26]) ).
tff(28,plain,
( ! [Y: $i] :
( ~ member(Y,universal_class)
| member(member_of(singleton(Y)),universal_class) )
<=> ! [Y: $i] :
( ~ member(Y,universal_class)
| member(member_of(singleton(Y)),universal_class) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(29,axiom,
! [Y: $i] :
( ~ member(Y,universal_class)
| member(member_of(singleton(Y)),universal_class) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',member_exists1) ).
tff(30,plain,
! [Y: $i] :
( ~ member(Y,universal_class)
| member(member_of(singleton(Y)),universal_class) ),
inference(modus_ponens,[status(thm)],[29,28]) ).
tff(31,plain,
! [Y: $i] :
( ~ member(Y,universal_class)
| member(member_of(singleton(Y)),universal_class) ),
inference(skolemize,[status(sab)],[30]) ).
tff(32,plain,
! [Y: $i] :
( ~ member(Y,universal_class)
| member(member_of(singleton(Y)),universal_class) ),
inference(modus_ponens,[status(thm)],[31,27]) ).
tff(33,plain,
( ( ~ ! [Y: $i] :
( ~ member(Y,universal_class)
| member(member_of(singleton(Y)),universal_class) )
| ~ member(u,universal_class)
| member(member_of(singleton(u)),universal_class) )
<=> ( ~ ! [Y: $i] :
( ~ member(Y,universal_class)
| member(member_of(singleton(Y)),universal_class) )
| ~ member(u,universal_class)
| member(member_of(singleton(u)),universal_class) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(34,plain,
( ~ ! [Y: $i] :
( ~ member(Y,universal_class)
| member(member_of(singleton(Y)),universal_class) )
| ~ member(u,universal_class)
| member(member_of(singleton(u)),universal_class) ),
inference(quant_inst,[status(thm)],[]) ).
tff(35,plain,
( ~ ! [Y: $i] :
( ~ member(Y,universal_class)
| member(member_of(singleton(Y)),universal_class) )
| ~ member(u,universal_class)
| member(member_of(singleton(u)),universal_class) ),
inference(modus_ponens,[status(thm)],[34,33]) ).
tff(36,plain,
member(member_of(singleton(u)),universal_class),
inference(unit_resolution,[status(thm)],[35,32,3]) ).
tff(37,plain,
( ( member_of(singleton(u)) != u )
<=> ( member_of(singleton(u)) != u ) ),
inference(rewrite,[status(thm)],[]) ).
tff(38,axiom,
member_of(singleton(u)) != u,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',prove_member_of_singleton_is_unique_2) ).
tff(39,plain,
member_of(singleton(u)) != u,
inference(modus_ponens,[status(thm)],[38,37]) ).
tff(40,plain,
^ [Z: $i,Y: $i,X: $i] :
refl(( ( ( X = Y )
| ~ member(X,universal_class)
| ( X = Z )
| ( unordered_pair(Y,Z) != singleton(X) ) )
<=> ( ( X = Y )
| ~ member(X,universal_class)
| ( X = Z )
| ( unordered_pair(Y,Z) != singleton(X) ) ) )),
inference(bind,[status(th)],[]) ).
tff(41,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ member(X,universal_class)
| ( X = Z )
| ( unordered_pair(Y,Z) != singleton(X) ) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ member(X,universal_class)
| ( X = Z )
| ( unordered_pair(Y,Z) != singleton(X) ) ) ),
inference(quant_intro,[status(thm)],[40]) ).
tff(42,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ member(X,universal_class)
| ( X = Z )
| ( unordered_pair(Y,Z) != singleton(X) ) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ member(X,universal_class)
| ( X = Z )
| ( unordered_pair(Y,Z) != singleton(X) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(43,plain,
^ [Z: $i,Y: $i,X: $i] :
trans(monotonicity(trans(monotonicity(rewrite(( ( ( unordered_pair(Y,Z) != singleton(X) )
| ~ member(X,universal_class) )
<=> ( ~ member(X,universal_class)
| ( unordered_pair(Y,Z) != singleton(X) ) ) )),
( ( ( unordered_pair(Y,Z) != singleton(X) )
| ~ member(X,universal_class)
| ( X = Y ) )
<=> ( ~ member(X,universal_class)
| ( unordered_pair(Y,Z) != singleton(X) )
| ( X = Y ) ) )),
rewrite(( ( ~ member(X,universal_class)
| ( unordered_pair(Y,Z) != singleton(X) )
| ( X = Y ) )
<=> ( ( X = Y )
| ~ member(X,universal_class)
| ( unordered_pair(Y,Z) != singleton(X) ) ) )),
( ( ( unordered_pair(Y,Z) != singleton(X) )
| ~ member(X,universal_class)
| ( X = Y ) )
<=> ( ( X = Y )
| ~ member(X,universal_class)
| ( unordered_pair(Y,Z) != singleton(X) ) ) )),
( ( ( unordered_pair(Y,Z) != singleton(X) )
| ~ member(X,universal_class)
| ( X = Y )
| ( X = Z ) )
<=> ( ( X = Y )
| ~ member(X,universal_class)
| ( unordered_pair(Y,Z) != singleton(X) )
| ( X = Z ) ) )),
rewrite(( ( ( X = Y )
| ~ member(X,universal_class)
| ( unordered_pair(Y,Z) != singleton(X) )
| ( X = Z ) )
<=> ( ( X = Y )
| ~ member(X,universal_class)
| ( X = Z )
| ( unordered_pair(Y,Z) != singleton(X) ) ) )),
( ( ( unordered_pair(Y,Z) != singleton(X) )
| ~ member(X,universal_class)
| ( X = Y )
| ( X = Z ) )
<=> ( ( X = Y )
| ~ member(X,universal_class)
| ( X = Z )
| ( unordered_pair(Y,Z) != singleton(X) ) ) )),
inference(bind,[status(th)],[]) ).
tff(44,plain,
( ! [Z: $i,Y: $i,X: $i] :
( ( unordered_pair(Y,Z) != singleton(X) )
| ~ member(X,universal_class)
| ( X = Y )
| ( X = Z ) )
<=> ! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ member(X,universal_class)
| ( X = Z )
| ( unordered_pair(Y,Z) != singleton(X) ) ) ),
inference(quant_intro,[status(thm)],[43]) ).
tff(45,axiom,
! [Z: $i,Y: $i,X: $i] :
( ( unordered_pair(Y,Z) != singleton(X) )
| ~ member(X,universal_class)
| ( X = Y )
| ( X = Z ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',singleton_in_unordered_pair3) ).
tff(46,plain,
! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ member(X,universal_class)
| ( X = Z )
| ( unordered_pair(Y,Z) != singleton(X) ) ),
inference(modus_ponens,[status(thm)],[45,44]) ).
tff(47,plain,
! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ member(X,universal_class)
| ( X = Z )
| ( unordered_pair(Y,Z) != singleton(X) ) ),
inference(modus_ponens,[status(thm)],[46,42]) ).
tff(48,plain,
! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ member(X,universal_class)
| ( X = Z )
| ( unordered_pair(Y,Z) != singleton(X) ) ),
inference(skolemize,[status(sab)],[47]) ).
tff(49,plain,
! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ member(X,universal_class)
| ( X = Z )
| ( unordered_pair(Y,Z) != singleton(X) ) ),
inference(modus_ponens,[status(thm)],[48,41]) ).
tff(50,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ member(X,universal_class)
| ( X = Z )
| ( unordered_pair(Y,Z) != singleton(X) ) )
| ( member_of(singleton(u)) = u )
| ~ member(member_of(singleton(u)),universal_class)
| ( unordered_pair(u,u) != singleton(member_of(singleton(u))) ) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ member(X,universal_class)
| ( X = Z )
| ( unordered_pair(Y,Z) != singleton(X) ) )
| ( member_of(singleton(u)) = u )
| ~ member(member_of(singleton(u)),universal_class)
| ( unordered_pair(u,u) != singleton(member_of(singleton(u))) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(51,plain,
( ( ( member_of(singleton(u)) = u )
| ~ member(member_of(singleton(u)),universal_class)
| ( member_of(singleton(u)) = u )
| ( unordered_pair(u,u) != singleton(member_of(singleton(u))) ) )
<=> ( ( member_of(singleton(u)) = u )
| ~ member(member_of(singleton(u)),universal_class)
| ( unordered_pair(u,u) != singleton(member_of(singleton(u))) ) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(52,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ member(X,universal_class)
| ( X = Z )
| ( unordered_pair(Y,Z) != singleton(X) ) )
| ( member_of(singleton(u)) = u )
| ~ member(member_of(singleton(u)),universal_class)
| ( member_of(singleton(u)) = u )
| ( unordered_pair(u,u) != singleton(member_of(singleton(u))) ) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ member(X,universal_class)
| ( X = Z )
| ( unordered_pair(Y,Z) != singleton(X) ) )
| ( member_of(singleton(u)) = u )
| ~ member(member_of(singleton(u)),universal_class)
| ( unordered_pair(u,u) != singleton(member_of(singleton(u))) ) ) ),
inference(monotonicity,[status(thm)],[51]) ).
tff(53,plain,
( ( ~ ! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ member(X,universal_class)
| ( X = Z )
| ( unordered_pair(Y,Z) != singleton(X) ) )
| ( member_of(singleton(u)) = u )
| ~ member(member_of(singleton(u)),universal_class)
| ( member_of(singleton(u)) = u )
| ( unordered_pair(u,u) != singleton(member_of(singleton(u))) ) )
<=> ( ~ ! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ member(X,universal_class)
| ( X = Z )
| ( unordered_pair(Y,Z) != singleton(X) ) )
| ( member_of(singleton(u)) = u )
| ~ member(member_of(singleton(u)),universal_class)
| ( unordered_pair(u,u) != singleton(member_of(singleton(u))) ) ) ),
inference(transitivity,[status(thm)],[52,50]) ).
tff(54,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ member(X,universal_class)
| ( X = Z )
| ( unordered_pair(Y,Z) != singleton(X) ) )
| ( member_of(singleton(u)) = u )
| ~ member(member_of(singleton(u)),universal_class)
| ( member_of(singleton(u)) = u )
| ( unordered_pair(u,u) != singleton(member_of(singleton(u))) ) ),
inference(quant_inst,[status(thm)],[]) ).
tff(55,plain,
( ~ ! [Z: $i,Y: $i,X: $i] :
( ( X = Y )
| ~ member(X,universal_class)
| ( X = Z )
| ( unordered_pair(Y,Z) != singleton(X) ) )
| ( member_of(singleton(u)) = u )
| ~ member(member_of(singleton(u)),universal_class)
| ( unordered_pair(u,u) != singleton(member_of(singleton(u))) ) ),
inference(modus_ponens,[status(thm)],[54,53]) ).
tff(56,plain,
$false,
inference(unit_resolution,[status(thm)],[55,49,39,36,25]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.12 % Problem : SET090-7 : TPTP v9.0.0. Bugfixed v2.1.0.
% 0.06/0.12 % Command : run_E %s %d THM
% 0.11/0.33 % Computer : n005.cluster.edu
% 0.11/0.33 % Model : x86_64 x86_64
% 0.11/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.33 % Memory : 8042.1875MB
% 0.11/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.33 % CPULimit : 300
% 0.11/0.33 % WCLimit : 300
% 0.11/0.33 % DateTime : Fri Jun 20 04:15:05 EDT 2025
% 0.11/0.33 % CPUTime :
% 0.20/0.40 % SZS status Unsatisfiable
% 0.20/0.40 % SZS output start Proof
% See solution above
% 0.20/0.42 % E exiting
%------------------------------------------------------------------------------