%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU622^2 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n027.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 : Wed Feb 25 09:35:41 AM UTC 2026
% Result : Theorem 1.47s 1.28s
% Output : Refutation 1.47s
% Verified :
% SZS Type : Refutation
% Derivation depth : 14
% Number of leaves : 6
% Syntax : Number of formulae : 39 ( 11 unt; 0 typ; 0 def)
% Number of atoms : 398 ( 106 equ; 0 cnn)
% Maximal formula atoms : 6 ( 10 avg)
% Number of connectives : 380 ( 49 ~; 32 |; 27 &; 247 @)
% ( 4 <=>; 21 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 5 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 16 ( 13 usr; 11 con; 0-2 aty)
% Number of variables : 66 ( 0 ^; 48 !; 18 ?; 66 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_2,type,
setadjoin: $i > $i > $i ).
thf(func_def_3,type,
powerset: $i > $i ).
thf(func_def_4,type,
subset: $i > $i > $o ).
thf(f48,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f47,f35]) ).
thf(f35,plain,
( $true
= ( in @ sK4 @ sK5 ) ),
inference(cnf_transformation,[status(thm)],[f25]) ).
thf(f25,plain,
( ( $true
= ( in @ sK4 @ sK5 ) )
& ( $true
!= ( in @ ( setadjoin @ sK4 @ emptyset ) @ ( powerset @ sK5 ) ) )
& ( powersetI1 = $true )
& ( singletonsubset = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK4,sK5])],[f13,f24]) ).
thf(f24,plain,
( ? [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ X1 ) )
& ( $true
!= ( in @ ( setadjoin @ X0 @ emptyset ) @ ( powerset @ X1 ) ) ) )
=> ( ( $true
= ( in @ sK4 @ sK5 ) )
& ( $true
!= ( in @ ( setadjoin @ sK4 @ emptyset ) @ ( powerset @ sK5 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f13,plain,
( ? [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ X1 ) )
& ( $true
!= ( in @ ( setadjoin @ X0 @ emptyset ) @ ( powerset @ X1 ) ) ) )
& ( powersetI1 = $true )
& ( singletonsubset = $true ) ),
inference(flattening,[status(thm)],[f12]) ).
thf(f12,plain,
( ? [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ X1 ) )
& ( $true
!= ( in @ ( setadjoin @ X0 @ emptyset ) @ ( powerset @ X1 ) ) ) )
& ( singletonsubset = $true )
& ( powersetI1 = $true ) ),
inference(ennf_transformation,[status(thm)],[f7]) ).
thf(f7,plain,
~ ( ( powersetI1 = $true )
=> ( ( singletonsubset = $true )
=> ! [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ X1 ) )
=> ( $true
= ( in @ ( setadjoin @ X0 @ emptyset ) @ ( powerset @ X1 ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f6]) ).
thf(f6,plain,
~ ( powersetI1
=> ( singletonsubset
=> ! [X0: $i,X1: $i] :
( ( in @ X0 @ X1 )
=> ( in @ ( setadjoin @ X0 @ emptyset ) @ ( powerset @ X1 ) ) ) ) ),
inference(rectify,[status(thm)],[f4]) ).
thf(f4,negated_conjecture,
~ ( powersetI1
=> ( singletonsubset
=> ! [X2: $i,X0: $i] :
( ( in @ X2 @ X0 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X0 ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f3]) ).
thf(f3,conjecture,
( powersetI1
=> ( singletonsubset
=> ! [X2: $i,X0: $i] :
( ( in @ X2 @ X0 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X0 ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.Nqk7FWTlF7/DTF2THF_7579.p',singletoninpowerset) ).
thf(f47,plain,
( $true
!= ( in @ sK4 @ sK5 ) ),
inference(trivial_inequality_removal,[status(thm)],[f46]) ).
thf(f46,plain,
( ( $true
!= ( in @ sK4 @ sK5 ) )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f45,f42]) ).
thf(f42,plain,
! [X2: $i,X3: $i] :
( ( $true
= ( subset @ ( setadjoin @ X2 @ emptyset ) @ X3 ) )
| ( $true
!= ( in @ X2 @ X3 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f41]) ).
thf(f41,plain,
! [X2: $i,X3: $i] :
( ( $true
= ( subset @ ( setadjoin @ X2 @ emptyset ) @ X3 ) )
| ( $true != $true )
| ( $true
!= ( in @ X2 @ X3 ) ) ),
inference(definition_unfolding,[status(thm)],[f29,f32]) ).
thf(f32,plain,
singletonsubset = $true,
inference(cnf_transformation,[status(thm)],[f25]) ).
thf(f29,plain,
! [X2: $i,X3: $i] :
( ( $true
!= ( in @ X2 @ X3 ) )
| ( $true
= ( subset @ ( setadjoin @ X2 @ emptyset ) @ X3 ) )
| ( singletonsubset != $true ) ),
inference(cnf_transformation,[status(thm)],[f23]) ).
thf(f23,plain,
( ( ( singletonsubset = $true )
| ( ( $true
= ( in @ sK2 @ sK3 ) )
& ( $true
!= ( subset @ ( setadjoin @ sK2 @ emptyset ) @ sK3 ) ) ) )
& ( ! [X2: $i,X3: $i] :
( ( $true
!= ( in @ X2 @ X3 ) )
| ( $true
= ( subset @ ( setadjoin @ X2 @ emptyset ) @ X3 ) ) )
| ( singletonsubset != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK2,sK3])],[f21,f22]) ).
thf(f22,plain,
( ? [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ X1 ) )
& ( $true
!= ( subset @ ( setadjoin @ X0 @ emptyset ) @ X1 ) ) )
=> ( ( $true
= ( in @ sK2 @ sK3 ) )
& ( $true
!= ( subset @ ( setadjoin @ sK2 @ emptyset ) @ sK3 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f21,plain,
( ( ( singletonsubset = $true )
| ? [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ X1 ) )
& ( $true
!= ( subset @ ( setadjoin @ X0 @ emptyset ) @ X1 ) ) ) )
& ( ! [X2: $i,X3: $i] :
( ( $true
!= ( in @ X2 @ X3 ) )
| ( $true
= ( subset @ ( setadjoin @ X2 @ emptyset ) @ X3 ) ) )
| ( singletonsubset != $true ) ) ),
inference(rectify,[status(thm)],[f20]) ).
thf(f20,plain,
( ( ( singletonsubset = $true )
| ? [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ X1 ) )
& ( $true
!= ( subset @ ( setadjoin @ X0 @ emptyset ) @ X1 ) ) ) )
& ( ! [X0: $i,X1: $i] :
( ( $true
!= ( in @ X0 @ X1 ) )
| ( $true
= ( subset @ ( setadjoin @ X0 @ emptyset ) @ X1 ) ) )
| ( singletonsubset != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f15]) ).
thf(f15,plain,
( ( singletonsubset = $true )
<=> ! [X0: $i,X1: $i] :
( ( $true
!= ( in @ X0 @ X1 ) )
| ( $true
= ( subset @ ( setadjoin @ X0 @ emptyset ) @ X1 ) ) ) ),
inference(ennf_transformation,[status(thm)],[f9]) ).
thf(f9,plain,
( ( singletonsubset = $true )
<=> ! [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ X1 ) )
=> ( $true
= ( subset @ ( setadjoin @ X0 @ emptyset ) @ X1 ) ) ) ),
inference(fool_elimination,[status(thm)],[f8]) ).
thf(f8,plain,
! [X0: $i,X1: $i] :
( ( ( in @ X0 @ X1 )
=> ( subset @ ( setadjoin @ X0 @ emptyset ) @ X1 ) )
= singletonsubset ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
! [X2: $i,X0: $i] :
( ( ( in @ X2 @ X0 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X0 ) )
= singletonsubset ),
file('/export/starexec/sandbox2/tmp/tmp.Nqk7FWTlF7/DTF2THF_7579.p',singletonsubset) ).
thf(f45,plain,
( $true
!= ( subset @ ( setadjoin @ sK4 @ emptyset ) @ sK5 ) ),
inference(trivial_inequality_removal,[status(thm)],[f44]) ).
thf(f44,plain,
( ( $true
!= ( subset @ ( setadjoin @ sK4 @ emptyset ) @ sK5 ) )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f34,f43]) ).
thf(f43,plain,
! [X2: $i,X3: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X3 ) ) )
| ( ( subset @ X2 @ X3 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f38]) ).
thf(f38,plain,
! [X2: $i,X3: $i] :
( ( ( subset @ X2 @ X3 )
!= $true )
| ( $true != $true )
| ( $true
= ( in @ X2 @ ( powerset @ X3 ) ) ) ),
inference(definition_unfolding,[status(thm)],[f26,f33]) ).
thf(f33,plain,
powersetI1 = $true,
inference(cnf_transformation,[status(thm)],[f25]) ).
thf(f26,plain,
! [X2: $i,X3: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X3 ) ) )
| ( ( subset @ X2 @ X3 )
!= $true )
| ( powersetI1 != $true ) ),
inference(cnf_transformation,[status(thm)],[f19]) ).
thf(f19,plain,
( ( ( powersetI1 = $true )
| ( ( $true
!= ( in @ sK0 @ ( powerset @ sK1 ) ) )
& ( $true
= ( subset @ sK0 @ sK1 ) ) ) )
& ( ! [X2: $i,X3: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X3 ) ) )
| ( ( subset @ X2 @ X3 )
!= $true ) )
| ( powersetI1 != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f17,f18]) ).
thf(f18,plain,
( ? [X0: $i,X1: $i] :
( ( $true
!= ( in @ X0 @ ( powerset @ X1 ) ) )
& ( $true
= ( subset @ X0 @ X1 ) ) )
=> ( ( $true
!= ( in @ sK0 @ ( powerset @ sK1 ) ) )
& ( $true
= ( subset @ sK0 @ sK1 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f17,plain,
( ( ( powersetI1 = $true )
| ? [X0: $i,X1: $i] :
( ( $true
!= ( in @ X0 @ ( powerset @ X1 ) ) )
& ( $true
= ( subset @ X0 @ X1 ) ) ) )
& ( ! [X2: $i,X3: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X3 ) ) )
| ( ( subset @ X2 @ X3 )
!= $true ) )
| ( powersetI1 != $true ) ) ),
inference(rectify,[status(thm)],[f16]) ).
thf(f16,plain,
( ( ( powersetI1 = $true )
| ? [X0: $i,X1: $i] :
( ( $true
!= ( in @ X0 @ ( powerset @ X1 ) ) )
& ( $true
= ( subset @ X0 @ X1 ) ) ) )
& ( ! [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) )
| ( $true
!= ( subset @ X0 @ X1 ) ) )
| ( powersetI1 != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f14]) ).
thf(f14,plain,
( ( powersetI1 = $true )
<=> ! [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) )
| ( $true
!= ( subset @ X0 @ X1 ) ) ) ),
inference(ennf_transformation,[status(thm)],[f11]) ).
thf(f11,plain,
( ( powersetI1 = $true )
<=> ! [X0: $i,X1: $i] :
( ( $true
= ( subset @ X0 @ X1 ) )
=> ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f10]) ).
thf(f10,plain,
! [X0: $i,X1: $i] :
( ( ( subset @ X0 @ X1 )
=> ( in @ X0 @ ( powerset @ X1 ) ) )
= powersetI1 ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
! [X1: $i,X0: $i] :
( ( ( subset @ X1 @ X0 )
=> ( in @ X1 @ ( powerset @ X0 ) ) )
= powersetI1 ),
file('/export/starexec/sandbox2/tmp/tmp.Nqk7FWTlF7/DTF2THF_7579.p',powersetI1) ).
thf(f34,plain,
( $true
!= ( in @ ( setadjoin @ sK4 @ emptyset ) @ ( powerset @ sK5 ) ) ),
inference(cnf_transformation,[status(thm)],[f25]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU622^2 : TPTP v9.2.1. Released v3.7.0.
% 0.11/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.15/0.33 % Computer : n027.cluster.edu
% 0.15/0.33 % Model : x86_64 x86_64
% 0.15/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.33 % Memory : 8042.1875MB
% 0.15/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.33 % CPULimit : 300
% 0.15/0.33 % WCLimit : 300
% 0.15/0.33 % DateTime : Tue Feb 24 12:28:52 EST 2026
% 0.15/0.33 % CPUTime :
% 0.15/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.29/0.46 ---- Original DTF file ---
% 0.29/0.46 thf(spec,logic,$$dhol).
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 % File : SEU622^2 : TPTP v9.2.1. Released v3.7.0.
% 0.29/0.46 % Domain : Set Theory
% 0.29/0.46 % Problem : Ordered Pairs - Cartesian Products
% 0.29/0.46 % Version : Especial > Reduced > Especial.
% 0.29/0.46 % English : (! A:i.! x:i.in x A -> in (setadjoin x emptyset) (powerset A))
% 0.29/0.46
% 0.29/0.46 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.29/0.46 % Source : [Bro08]
% 0.29/0.46 % Names : ZFC124l [Bro08]
% 0.29/0.46
% 0.29/0.46 % Status : Theorem
% 0.29/0.46 % Rating : 0.00 v8.2.0, 0.08 v8.1.0, 0.00 v7.1.0, 0.12 v7.0.0, 0.00 v6.0.0, 0.14 v5.5.0, 0.17 v5.4.0, 0.20 v5.3.0, 0.40 v5.2.0, 0.20 v4.1.0, 0.00 v3.7.0
% 0.29/0.46 % Syntax : Number of formulae : 10 ( 2 unt; 7 typ; 2 def)
% 0.29/0.46 % Number of atoms : 12 ( 2 equ; 0 cnn)
% 0.29/0.46 % Maximal formula atoms : 4 ( 4 avg)
% 0.29/0.46 % Number of connectives : 23 ( 0 ~; 0 |; 0 &; 18 @)
% 0.29/0.46 % ( 0 <=>; 5 =>; 0 <=; 0 <~>)
% 0.29/0.46 % Maximal formula depth : 10 ( 4 avg)
% 0.29/0.46 % Number of types : 2 ( 0 usr)
% 0.29/0.46 % Number of type conns : 7 ( 7 >; 0 *; 0 +; 0 <<)
% 0.29/0.46 % Number of symbols : 8 ( 7 usr; 3 con; 0-2 aty)
% 0.29/0.46 % Number of variables : 6 ( 0 ^; 6 !; 0 ?; 6 :)
% 0.29/0.46 % SPC : TH0_THM_EQU_NAR
% 0.29/0.46
% 0.29/0.46 % Comments : http://mathgate.info/detsetitem.php?id=179
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 thf(in_type,type,
% 0.29/0.46 in: $i > $i > $o ).
% 0.29/0.46
% 0.29/0.46 thf(emptyset_type,type,
% 0.29/0.46 emptyset: $i ).
% 0.29/0.46
% 0.29/0.46 thf(setadjoin_type,type,
% 0.29/0.46 setadjoin: $i > $i > $i ).
% 0.29/0.46
% 0.29/0.46 thf(powerset_type,type,
% 0.29/0.46 powerset: $i > $i ).
% 0.29/0.46
% 0.29/0.46 thf(subset_type,type,
% 0.29/0.46 subset: $i > $i > $o ).
% 0.29/0.46
% 0.29/0.46 thf(powersetI1_type,type,
% 0.29/0.46 powersetI1: $o ).
% 0.29/0.46
% 0.29/0.46 thf(powersetI1,definition,
% 0.29/0.46 ( powersetI1
% 0.29/0.46 = ( ! [A: $i,B: $i] :
% 0.29/0.46 ( ( subset @ B @ A )
% 0.29/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(singletonsubset_type,type,
% 0.29/0.46 singletonsubset: $o ).
% 0.29/0.46
% 0.29/0.46 thf(singletonsubset,definition,
% 0.29/0.46 ( singletonsubset
% 0.29/0.46 = ( ! [A: $i,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(singletoninpowerset,conjecture,
% 0.29/0.46 ( powersetI1
% 0.29/0.46 => ( singletonsubset
% 0.29/0.46 => ! [A: $i,Xx: $i] :
% 0.29/0.46 ( ( in @ Xx @ A )
% 0.29/0.46 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 ------------------------
% 1.47/1.25 ---- Embedded in THF ---
% 1.47/1.25 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.47/1.25 %%% Generated on Tue Feb 24 12:28:53 EST 2026
% 1.47/1.25 %%% using '$$dhol' embedding, version 1.3.0.
% 1.47/1.25 %%% Logic specification used:
% 1.47/1.25 %%% thf(spec, logic, $$dhol).
% 1.47/1.25
% 1.47/1.25 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.Nqk7FWTlF7/DTF2DTF7579.p
% See solution above
% 1.47/1.25 ------------------------
% 1.47/1.26 ---- Cleaned THF ---
% 1.47/1.26 thf(in_type,type,
% 1.47/1.26 in: $i > $i > $o ).
% 1.47/1.26
% 1.47/1.26 thf(emptyset_type,type,
% 1.47/1.26 emptyset: $i ).
% 1.47/1.26
% 1.47/1.26 thf(setadjoin_type,type,
% 1.47/1.26 setadjoin: $i > $i > $i ).
% 1.47/1.26
% 1.47/1.26 thf(powerset_type,type,
% 1.47/1.26 powerset: $i > $i ).
% 1.47/1.26
% 1.47/1.26 thf(subset_type,type,
% 1.47/1.26 subset: $i > $i > $o ).
% 1.47/1.26
% 1.47/1.26 thf(powersetI1_type,type,
% 1.47/1.26 powersetI1: $o ).
% 1.47/1.26
% 1.47/1.26 thf(powersetI1,definition,
% 1.47/1.26 ( powersetI1
% 1.47/1.26 = ( ! [A: $i,B: $i] :
% 1.47/1.26 ( ( subset @ B @ A )
% 1.47/1.26 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 1.47/1.26
% 1.47/1.26 thf(singletonsubset_type,type,
% 1.47/1.26 singletonsubset: $o ).
% 1.47/1.26
% 1.47/1.26 thf(singletonsubset,definition,
% 1.47/1.26 ( singletonsubset
% 1.47/1.26 = ( ! [A: $i,Xx: $i] :
% 1.47/1.26 ( ( in @ Xx @ A )
% 1.47/1.26 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 1.47/1.26
% 1.47/1.26 thf(singletoninpowerset,conjecture,
% 1.47/1.26 ( powersetI1
% 1.47/1.26 => ( singletonsubset
% 1.47/1.26 => ! [A: $i,Xx: $i] :
% 1.47/1.26 ( ( in @ Xx @ A )
% 1.47/1.26 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 1.47/1.26 ------------------------
% 1.47/1.27 % (7747)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_7579 for (2999ds/275Mi)
% 1.47/1.27 % (7743)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_7579 for (2999ds/4Mi)
% 1.47/1.27 % (7747)First to succeed.
% 1.47/1.28 % (7743)Also succeeded, but the first one will report.
% 1.47/1.28 % (7745)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_7579 for (2999ds/2Mi)
% 1.47/1.28 % (7747)Refutation found. Thanks to Tanya!
% 1.47/1.28 % SZS status Theorem for DTF2THF_7579
% 1.47/1.28 % SZS output start Proof for DTF2THF_7579
% 1.47/1.28 thf(func_def_0, type, in: $i > $i > $o).
% 1.47/1.28 thf(func_def_2, type, setadjoin: $i > $i > $i).
% 1.47/1.28 thf(func_def_3, type, powerset: $i > $i).
% 1.47/1.28 thf(func_def_4, type, subset: $i > $i > $o).
% 1.47/1.28 thf(f48,plain,(
% 1.47/1.28 $false),
% 1.47/1.28 inference(subsumption_resolution,[status(thm)],[f47,f35])).
% 1.47/1.28 thf(f35,plain,(
% 1.47/1.28 ($true = (in @ sK4 @ sK5))),
% 1.47/1.28 inference(cnf_transformation,[status(thm)],[f25])).
% 1.47/1.28 thf(f25,plain,(
% 1.47/1.28 (($true = (in @ sK4 @ sK5)) & ($true != (in @ (setadjoin @ sK4 @ emptyset) @ (powerset @ sK5)))) & (powersetI1 = $true) & (singletonsubset = $true)),
% 1.47/1.28 inference(skolemisation,[status(esa),new_symbols(skolem,[sK4,sK5])],[f13,f24])).
% 1.47/1.28 thf(f24,plain,(
% 1.47/1.28 ? [X0 : $i,X1 : $i] : (($true = (in @ X0 @ X1)) & ($true != (in @ (setadjoin @ X0 @ emptyset) @ (powerset @ X1)))) => (($true = (in @ sK4 @ sK5)) & ($true != (in @ (setadjoin @ sK4 @ emptyset) @ (powerset @ sK5))))),
% 1.47/1.28 introduced(definition,[],[choice_axiom])).
% 1.47/1.28 thf(f13,plain,(
% 1.47/1.28 ? [X0 : $i,X1 : $i] : (($true = (in @ X0 @ X1)) & ($true != (in @ (setadjoin @ X0 @ emptyset) @ (powerset @ X1)))) & (powersetI1 = $true) & (singletonsubset = $true)),
% 1.47/1.28 inference(flattening,[status(thm)],[f12])).
% 1.47/1.28 thf(f12,plain,(
% 1.47/1.28 (? [X0 : $i,X1 : $i] : (($true = (in @ X0 @ X1)) & ($true != (in @ (setadjoin @ X0 @ emptyset) @ (powerset @ X1)))) & (singletonsubset = $true)) & (powersetI1 = $true)),
% 1.47/1.28 inference(ennf_transformation,[status(thm)],[f7])).
% 1.47/1.28 thf(f7,plain,(
% 1.47/1.28 ~((powersetI1 = $true) => ((singletonsubset = $true) => ! [X0 : $i,X1 : $i] : (($true = (in @ X0 @ X1)) => ($true = (in @ (setadjoin @ X0 @ emptyset) @ (powerset @ X1))))))),
% 1.47/1.28 inference(fool_elimination,[status(thm)],[f6])).
% 1.47/1.28 thf(f6,plain,(
% 1.47/1.28 ~(powersetI1 => (singletonsubset => ! [X0 : $i,X1 : $i] : ((in @ X0 @ X1) => (in @ (setadjoin @ X0 @ emptyset) @ (powerset @ X1)))))),
% 1.47/1.28 inference(rectify,[status(thm)],[f4])).
% 1.47/1.28 thf(f4,negated_conjecture,(
% 1.47/1.28 ~(powersetI1 => (singletonsubset => ! [X2 : $i,X0 : $i] : ((in @ X2 @ X0) => (in @ (setadjoin @ X2 @ emptyset) @ (powerset @ X0)))))),
% 1.47/1.28 inference(negated_conjecture,[status(cth)],[f3])).
% 1.47/1.28 thf(f3,conjecture,(
% 1.47/1.28 powersetI1 => (singletonsubset => ! [X2 : $i,X0 : $i] : ((in @ X2 @ X0) => (in @ (setadjoin @ X2 @ emptyset) @ (powerset @ X0))))),
% 1.47/1.28 file('/export/starexec/sandbox2/tmp/tmp.Nqk7FWTlF7/DTF2THF_7579.p',singletoninpowerset)).
% 1.47/1.28 thf(f47,plain,(
% 1.47/1.28 ($true != (in @ sK4 @ sK5))),
% 1.47/1.28 inference(trivial_inequality_removal,[status(thm)],[f46])).
% 1.47/1.28 thf(f46,plain,(
% 1.47/1.28 ($true != (in @ sK4 @ sK5)) | ($true != $true)),
% 1.47/1.28 inference(superposition,[status(thm)],[f45,f42])).
% 1.47/1.28 thf(f42,plain,(
% 1.47/1.28 ( ! [X2 : $i,X3 : $i] : (($true = (subset @ (setadjoin @ X2 @ emptyset) @ X3)) | ($true != (in @ X2 @ X3))) )),
% 1.47/1.28 inference(trivial_inequality_removal,[status(thm)],[f41])).
% 1.47/1.28 thf(f41,plain,(
% 1.47/1.28 ( ! [X2 : $i,X3 : $i] : (($true = (subset @ (setadjoin @ X2 @ emptyset) @ X3)) | ($true != $true) | ($true != (in @ X2 @ X3))) )),
% 1.47/1.28 inference(definition_unfolding,[status(thm)],[f29,f32])).
% 1.47/1.28 thf(f32,plain,(
% 1.47/1.28 (singletonsubset = $true)),
% 1.47/1.28 inference(cnf_transformation,[status(thm)],[f25])).
% 1.47/1.28 thf(f29,plain,(
% 1.47/1.28 ( ! [X2 : $i,X3 : $i] : (($true != (in @ X2 @ X3)) | ($true = (subset @ (setadjoin @ X2 @ emptyset) @ X3)) | (singletonsubset != $true)) )),
% 1.47/1.28 inference(cnf_transformation,[status(thm)],[f23])).
% 1.47/1.28 thf(f23,plain,(
% 1.47/1.28 ((singletonsubset = $true) | (($true = (in @ sK2 @ sK3)) & ($true != (subset @ (setadjoin @ sK2 @ emptyset) @ sK3)))) & (! [X2 : $i,X3 : $i] : (($true != (in @ X2 @ X3)) | ($true = (subset @ (setadjoin @ X2 @ emptyset) @ X3))) | (singletonsubset != $true))),
% 1.47/1.28 inference(skolemisation,[status(esa),new_symbols(skolem,[sK2,sK3])],[f21,f22])).
% 1.47/1.28 thf(f22,plain,(
% 1.47/1.28 ? [X0 : $i,X1 : $i] : (($true = (in @ X0 @ X1)) & ($true != (subset @ (setadjoin @ X0 @ emptyset) @ X1))) => (($true = (in @ sK2 @ sK3)) & ($true != (subset @ (setadjoin @ sK2 @ emptyset) @ sK3)))),
% 1.47/1.28 introduced(definition,[],[choice_axiom])).
% 1.47/1.28 thf(f21,plain,(
% 1.47/1.28 ((singletonsubset = $true) | ? [X0 : $i,X1 : $i] : (($true = (in @ X0 @ X1)) & ($true != (subset @ (setadjoin @ X0 @ emptyset) @ X1)))) & (! [X2 : $i,X3 : $i] : (($true != (in @ X2 @ X3)) | ($true = (subset @ (setadjoin @ X2 @ emptyset) @ X3))) | (singletonsubset != $true))),
% 1.47/1.28 inference(rectify,[status(thm)],[f20])).
% 1.47/1.28 thf(f20,plain,(
% 1.47/1.28 ((singletonsubset = $true) | ? [X0 : $i,X1 : $i] : (($true = (in @ X0 @ X1)) & ($true != (subset @ (setadjoin @ X0 @ emptyset) @ X1)))) & (! [X0 : $i,X1 : $i] : (($true != (in @ X0 @ X1)) | ($true = (subset @ (setadjoin @ X0 @ emptyset) @ X1))) | (singletonsubset != $true))),
% 1.47/1.28 inference(nnf_transformation,[status(thm)],[f15])).
% 1.47/1.28 thf(f15,plain,(
% 1.47/1.28 (singletonsubset = $true) <=> ! [X0 : $i,X1 : $i] : (($true != (in @ X0 @ X1)) | ($true = (subset @ (setadjoin @ X0 @ emptyset) @ X1)))),
% 1.47/1.28 inference(ennf_transformation,[status(thm)],[f9])).
% 1.47/1.28 thf(f9,plain,(
% 1.47/1.28 (singletonsubset = $true) <=> ! [X0 : $i,X1 : $i] : (($true = (in @ X0 @ X1)) => ($true = (subset @ (setadjoin @ X0 @ emptyset) @ X1)))),
% 1.47/1.28 inference(fool_elimination,[status(thm)],[f8])).
% 1.47/1.28 thf(f8,plain,(
% 1.47/1.28 (! [X0 : $i,X1 : $i] : ((in @ X0 @ X1) => (subset @ (setadjoin @ X0 @ emptyset) @ X1)) = singletonsubset)),
% 1.47/1.28 inference(rectify,[status(thm)],[f2])).
% 1.47/1.28 thf(f2,axiom,(
% 1.47/1.28 (! [X2 : $i,X0 : $i] : ((in @ X2 @ X0) => (subset @ (setadjoin @ X2 @ emptyset) @ X0)) = singletonsubset)),
% 1.47/1.28 file('/export/starexec/sandbox2/tmp/tmp.Nqk7FWTlF7/DTF2THF_7579.p',singletonsubset)).
% 1.47/1.28 thf(f45,plain,(
% 1.47/1.28 ($true != (subset @ (setadjoin @ sK4 @ emptyset) @ sK5))),
% 1.47/1.28 inference(trivial_inequality_removal,[status(thm)],[f44])).
% 1.47/1.28 thf(f44,plain,(
% 1.47/1.28 ($true != (subset @ (setadjoin @ sK4 @ emptyset) @ sK5)) | ($true != $true)),
% 1.47/1.28 inference(superposition,[status(thm)],[f34,f43])).
% 1.47/1.28 thf(f43,plain,(
% 1.47/1.28 ( ! [X2 : $i,X3 : $i] : (($true = (in @ X2 @ (powerset @ X3))) | ((subset @ X2 @ X3) != $true)) )),
% 1.47/1.28 inference(trivial_inequality_removal,[status(thm)],[f38])).
% 1.47/1.28 thf(f38,plain,(
% 1.47/1.28 ( ! [X2 : $i,X3 : $i] : (((subset @ X2 @ X3) != $true) | ($true != $true) | ($true = (in @ X2 @ (powerset @ X3)))) )),
% 1.47/1.28 inference(definition_unfolding,[status(thm)],[f26,f33])).
% 1.47/1.28 thf(f33,plain,(
% 1.47/1.28 (powersetI1 = $true)),
% 1.47/1.28 inference(cnf_transformation,[status(thm)],[f25])).
% 1.47/1.28 thf(f26,plain,(
% 1.47/1.28 ( ! [X2 : $i,X3 : $i] : (($true = (in @ X2 @ (powerset @ X3))) | ((subset @ X2 @ X3) != $true) | (powersetI1 != $true)) )),
% 1.47/1.28 inference(cnf_transformation,[status(thm)],[f19])).
% 1.47/1.28 thf(f19,plain,(
% 1.47/1.28 ((powersetI1 = $true) | (($true != (in @ sK0 @ (powerset @ sK1))) & ($true = (subset @ sK0 @ sK1)))) & (! [X2 : $i,X3 : $i] : (($true = (in @ X2 @ (powerset @ X3))) | ((subset @ X2 @ X3) != $true)) | (powersetI1 != $true))),
% 1.47/1.28 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f17,f18])).
% 1.47/1.28 thf(f18,plain,(
% 1.47/1.28 ? [X0 : $i,X1 : $i] : (($true != (in @ X0 @ (powerset @ X1))) & ($true = (subset @ X0 @ X1))) => (($true != (in @ sK0 @ (powerset @ sK1))) & ($true = (subset @ sK0 @ sK1)))),
% 1.47/1.28 introduced(definition,[],[choice_axiom])).
% 1.47/1.28 thf(f17,plain,(
% 1.47/1.28 ((powersetI1 = $true) | ? [X0 : $i,X1 : $i] : (($true != (in @ X0 @ (powerset @ X1))) & ($true = (subset @ X0 @ X1)))) & (! [X2 : $i,X3 : $i] : (($true = (in @ X2 @ (powerset @ X3))) | ((subset @ X2 @ X3) != $true)) | (powersetI1 != $true))),
% 1.47/1.28 inference(rectify,[status(thm)],[f16])).
% 1.47/1.28 thf(f16,plain,(
% 1.47/1.28 ((powersetI1 = $true) | ? [X0 : $i,X1 : $i] : (($true != (in @ X0 @ (powerset @ X1))) & ($true = (subset @ X0 @ X1)))) & (! [X0 : $i,X1 : $i] : (($true = (in @ X0 @ (powerset @ X1))) | ($true != (subset @ X0 @ X1))) | (powersetI1 != $true))),
% 1.47/1.28 inference(nnf_transformation,[status(thm)],[f14])).
% 1.47/1.28 thf(f14,plain,(
% 1.47/1.28 (powersetI1 = $true) <=> ! [X0 : $i,X1 : $i] : (($true = (in @ X0 @ (powerset @ X1))) | ($true != (subset @ X0 @ X1)))),
% 1.47/1.28 inference(ennf_transformation,[status(thm)],[f11])).
% 1.47/1.28 thf(f11,plain,(
% 1.47/1.28 (powersetI1 = $true) <=> ! [X0 : $i,X1 : $i] : (($true = (subset @ X0 @ X1)) => ($true = (in @ X0 @ (powerset @ X1))))),
% 1.47/1.28 inference(fool_elimination,[status(thm)],[f10])).
% 1.47/1.28 thf(f10,plain,(
% 1.47/1.28 (! [X0 : $i,X1 : $i] : ((subset @ X0 @ X1) => (in @ X0 @ (powerset @ X1))) = powersetI1)),
% 1.47/1.28 inference(rectify,[status(thm)],[f1])).
% 1.47/1.28 thf(f1,axiom,(
% 1.47/1.28 (! [X1 : $i,X0 : $i] : ((subset @ X1 @ X0) => (in @ X1 @ (powerset @ X0))) = powersetI1)),
% 1.47/1.28 file('/export/starexec/sandbox2/tmp/tmp.Nqk7FWTlF7/DTF2THF_7579.p',powersetI1)).
% 1.47/1.28 thf(f34,plain,(
% 1.47/1.28 ($true != (in @ (setadjoin @ sK4 @ emptyset) @ (powerset @ sK5)))),
% 1.47/1.28 inference(cnf_transformation,[status(thm)],[f25])).
% 1.47/1.28 % SZS output end Proof for DTF2THF_7579
% 1.47/1.28 % (7747)------------------------------
% 1.47/1.28 % (7747)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.47/1.28 % (7747)Termination reason: Refutation
% 1.47/1.28
% 1.47/1.28 % (7747)Memory used [KB]: 5500
% 1.47/1.28 % (7747)Time elapsed: 0.004 s
% 1.47/1.28 % (7747)Instructions burned: 2 (million)
% 1.47/1.28 % (7747)------------------------------
% 1.47/1.28 % (7747)------------------------------
% 1.47/1.28 % (7741)Success in time 0.014 s
%------------------------------------------------------------------------------