%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU806^2 : 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 : n022.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 09:37:22 AM UTC 2026
% Result : Theorem 1.40s 1.26s
% Output : Refutation 1.40s
% Verified :
% SZS Type : Refutation
% Derivation depth : 19
% Number of leaves : 11
% Syntax : Number of formulae : 61 ( 16 unt; 0 typ; 0 def)
% Number of atoms : 605 ( 189 equ; 0 cnn)
% Maximal formula atoms : 10 ( 9 avg)
% Number of connectives : 576 ( 89 ~; 62 |; 50 &; 337 @)
% ( 5 <=>; 33 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 5 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 19 ( 16 usr; 14 con; 0-2 aty)
% Number of variables : 135 ( 0 ^; 94 !; 41 ?; 135 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_2,type,
setadjoin: $i > $i > $i ).
thf(func_def_13,type,
sK3: $i > $i ).
thf(func_def_15,type,
sK5: $i > $i ).
thf(f103,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f102]) ).
thf(f102,plain,
$true != $true,
inference(superposition,[status(thm)],[f100,f64]) ).
thf(f64,plain,
( ( in @ sK12 @ sK12 )
= $true ),
inference(cnf_transformation,[status(thm)],[f45]) ).
thf(f45,plain,
( ( ( in @ sK12 @ sK12 )
= $true )
& ( foundationAx = $true )
& ( setadjoinIL = $true )
& ( in__Cong = $true )
& ( uniqinunit = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK12])],[f20,f44]) ).
thf(f44,plain,
( ? [X0: $i] :
( ( in @ X0 @ X0 )
= $true )
=> ( ( in @ sK12 @ sK12 )
= $true ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f20,plain,
( ? [X0: $i] :
( ( in @ X0 @ X0 )
= $true )
& ( foundationAx = $true )
& ( setadjoinIL = $true )
& ( in__Cong = $true )
& ( uniqinunit = $true ) ),
inference(flattening,[status(thm)],[f19]) ).
thf(f19,plain,
( ? [X0: $i] :
( ( in @ X0 @ X0 )
= $true )
& ( in__Cong = $true )
& ( uniqinunit = $true )
& ( setadjoinIL = $true )
& ( foundationAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f18]) ).
thf(f18,plain,
~ ( ( foundationAx = $true )
=> ( ( setadjoinIL = $true )
=> ( ( uniqinunit = $true )
=> ( ( in__Cong = $true )
=> ! [X0: $i] :
( ( in @ X0 @ X0 )
!= $true ) ) ) ) ),
inference(flattening,[status(thm)],[f11]) ).
thf(f11,plain,
~ ( ( foundationAx = $true )
=> ( ( setadjoinIL = $true )
=> ( ( uniqinunit = $true )
=> ( ( in__Cong = $true )
=> ! [X0: $i] :
( ( in @ X0 @ X0 )
!= $true ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f10]) ).
thf(f10,plain,
~ ( foundationAx
=> ( setadjoinIL
=> ( uniqinunit
=> ( in__Cong
=> ! [X0: $i] :
~ ( in @ X0 @ X0 ) ) ) ) ),
inference(rectify,[status(thm)],[f6]) ).
thf(f6,negated_conjecture,
~ ( foundationAx
=> ( setadjoinIL
=> ( uniqinunit
=> ( in__Cong
=> ! [X0: $i] :
~ ( in @ X0 @ X0 ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f5]) ).
thf(f5,conjecture,
( foundationAx
=> ( setadjoinIL
=> ( uniqinunit
=> ( in__Cong
=> ! [X0: $i] :
~ ( in @ X0 @ X0 ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.TIBw4yTLEl/DTF2THF_31874.p',notinself) ).
thf(f100,plain,
! [X0: $i] :
( ( in @ X0 @ X0 )
!= $true ),
inference(trivial_inequality_removal,[status(thm)],[f97]) ).
thf(f97,plain,
! [X0: $i] :
( ( ( in @ X0 @ X0 )
!= $true )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f96,f81]) ).
thf(f81,plain,
! [X2: $i,X3: $i] :
( ( in @ X2 @ ( setadjoin @ X2 @ X3 ) )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f78]) ).
thf(f78,plain,
! [X2: $i,X3: $i] :
( ( $true != $true )
| ( ( in @ X2 @ ( setadjoin @ X2 @ X3 ) )
= $true ) ),
inference(definition_unfolding,[status(thm)],[f58,f62]) ).
thf(f62,plain,
setadjoinIL = $true,
inference(cnf_transformation,[status(thm)],[f45]) ).
thf(f58,plain,
! [X2: $i,X3: $i] :
( ( ( in @ X2 @ ( setadjoin @ X2 @ X3 ) )
= $true )
| ( setadjoinIL != $true ) ),
inference(cnf_transformation,[status(thm)],[f43]) ).
thf(f43,plain,
( ( ( setadjoinIL = $true )
| ( ( in @ sK10 @ ( setadjoin @ sK10 @ sK11 ) )
!= $true ) )
& ( ! [X2: $i,X3: $i] :
( ( in @ X2 @ ( setadjoin @ X2 @ X3 ) )
= $true )
| ( setadjoinIL != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK10,sK11])],[f41,f42]) ).
thf(f42,plain,
( ? [X0: $i,X1: $i] :
( ( in @ X0 @ ( setadjoin @ X0 @ X1 ) )
!= $true )
=> ( ( in @ sK10 @ ( setadjoin @ sK10 @ sK11 ) )
!= $true ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f41,plain,
( ( ( setadjoinIL = $true )
| ? [X0: $i,X1: $i] :
( ( in @ X0 @ ( setadjoin @ X0 @ X1 ) )
!= $true ) )
& ( ! [X2: $i,X3: $i] :
( ( in @ X2 @ ( setadjoin @ X2 @ X3 ) )
= $true )
| ( setadjoinIL != $true ) ) ),
inference(rectify,[status(thm)],[f40]) ).
thf(f40,plain,
( ( ( setadjoinIL = $true )
| ? [X1: $i,X0: $i] :
( ( in @ X1 @ ( setadjoin @ X1 @ X0 ) )
!= $true ) )
& ( ! [X1: $i,X0: $i] :
( ( in @ X1 @ ( setadjoin @ X1 @ X0 ) )
= $true )
| ( setadjoinIL != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f15]) ).
thf(f15,plain,
( ( setadjoinIL = $true )
<=> ! [X1: $i,X0: $i] :
( ( in @ X1 @ ( setadjoin @ X1 @ X0 ) )
= $true ) ),
inference(fool_elimination,[status(thm)],[f14]) ).
thf(f14,plain,
( setadjoinIL
= ( ! [X0: $i,X1: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X0 ) ) ) ),
inference(rectify,[status(thm)],[f2]) ).
thf(f2,axiom,
( setadjoinIL
= ( ! [X3: $i,X1: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X3 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.TIBw4yTLEl/DTF2THF_31874.p',setadjoinIL) ).
thf(f96,plain,
! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) )
!= $true )
| ( ( in @ X1 @ X0 )
!= $true ) ),
inference(superposition,[status(thm)],[f85,f94]) ).
thf(f94,plain,
! [X0: $i] :
( ( sK5 @ ( setadjoin @ X0 @ emptyset ) )
= X0 ),
inference(trivial_inequality_removal,[status(thm)],[f92]) ).
thf(f92,plain,
! [X0: $i] :
( ( ( sK5 @ ( setadjoin @ X0 @ emptyset ) )
= X0 )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f83,f89]) ).
thf(f89,plain,
! [X0: $i,X1: $i] :
( ( in @ ( sK5 @ ( setadjoin @ X0 @ X1 ) ) @ ( setadjoin @ X0 @ X1 ) )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f88]) ).
thf(f88,plain,
! [X0: $i,X1: $i] :
( ( ( in @ ( sK5 @ ( setadjoin @ X0 @ X1 ) ) @ ( setadjoin @ X0 @ X1 ) )
= $true )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f82,f81]) ).
thf(f82,plain,
! [X7: $i,X4: $i] :
( ( ( in @ X7 @ X4 )
!= $true )
| ( ( in @ ( sK5 @ X4 ) @ X4 )
= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f72]) ).
thf(f72,plain,
! [X7: $i,X4: $i] :
( ( ( in @ X7 @ X4 )
!= $true )
| ( $true != $true )
| ( ( in @ ( sK5 @ X4 ) @ X4 )
= $true ) ),
inference(definition_unfolding,[status(thm)],[f49,f63]) ).
thf(f63,plain,
foundationAx = $true,
inference(cnf_transformation,[status(thm)],[f45]) ).
thf(f49,plain,
! [X7: $i,X4: $i] :
( ( ( in @ ( sK5 @ X4 ) @ X4 )
= $true )
| ( ( in @ X7 @ X4 )
!= $true )
| ( foundationAx != $true ) ),
inference(cnf_transformation,[status(thm)],[f34]) ).
thf(f34,plain,
( ( ( foundationAx = $true )
| ( ! [X1: $i] :
( ( ( ( in @ ( sK3 @ X1 ) @ X1 )
= $true )
& ( ( in @ ( sK3 @ X1 ) @ sK2 )
= $true ) )
| ( ( in @ X1 @ sK2 )
!= $true ) )
& ( $true
= ( in @ sK4 @ sK2 ) ) ) )
& ( ! [X4: $i] :
( ( ! [X6: $i] :
( ( ( in @ X6 @ ( sK5 @ X4 ) )
!= $true )
| ( ( in @ X6 @ X4 )
!= $true ) )
& ( ( in @ ( sK5 @ X4 ) @ X4 )
= $true ) )
| ! [X7: $i] :
( ( in @ X7 @ X4 )
!= $true ) )
| ( foundationAx != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK2,sK3,sK4,sK5])],[f29,f33,f32,f31,f30]) ).
thf(f30,plain,
( ? [X0: $i] :
( ! [X1: $i] :
( ? [X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
& ( ( in @ X2 @ X0 )
= $true ) )
| ( ( in @ X1 @ X0 )
!= $true ) )
& ? [X3: $i] :
( ( in @ X3 @ X0 )
= $true ) )
=> ( ! [X1: $i] :
( ? [X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
& ( ( in @ X2 @ sK2 )
= $true ) )
| ( ( in @ X1 @ sK2 )
!= $true ) )
& ? [X3: $i] :
( ( in @ X3 @ sK2 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f31,plain,
! [X1: $i] :
( ? [X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
& ( ( in @ X2 @ sK2 )
= $true ) )
=> ( ( ( in @ ( sK3 @ X1 ) @ X1 )
= $true )
& ( ( in @ ( sK3 @ X1 ) @ sK2 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f32,plain,
( ? [X3: $i] :
( ( in @ X3 @ sK2 )
= $true )
=> ( $true
= ( in @ sK4 @ sK2 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f33,plain,
! [X4: $i] :
( ? [X5: $i] :
( ! [X6: $i] :
( ( ( in @ X6 @ X5 )
!= $true )
| ( ( in @ X6 @ X4 )
!= $true ) )
& ( ( in @ X5 @ X4 )
= $true ) )
=> ( ! [X6: $i] :
( ( ( in @ X6 @ ( sK5 @ X4 ) )
!= $true )
| ( ( in @ X6 @ X4 )
!= $true ) )
& ( ( in @ ( sK5 @ X4 ) @ X4 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f29,plain,
( ( ( foundationAx = $true )
| ? [X0: $i] :
( ! [X1: $i] :
( ? [X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
& ( ( in @ X2 @ X0 )
= $true ) )
| ( ( in @ X1 @ X0 )
!= $true ) )
& ? [X3: $i] :
( ( in @ X3 @ X0 )
= $true ) ) )
& ( ! [X4: $i] :
( ? [X5: $i] :
( ! [X6: $i] :
( ( ( in @ X6 @ X5 )
!= $true )
| ( ( in @ X6 @ X4 )
!= $true ) )
& ( ( in @ X5 @ X4 )
= $true ) )
| ! [X7: $i] :
( ( in @ X7 @ X4 )
!= $true ) )
| ( foundationAx != $true ) ) ),
inference(rectify,[status(thm)],[f28]) ).
thf(f28,plain,
( ( ( foundationAx = $true )
| ? [X0: $i] :
( ! [X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ X2 )
= $true )
& ( ( in @ X3 @ X0 )
= $true ) )
| ( ( in @ X2 @ X0 )
!= $true ) )
& ? [X1: $i] :
( ( in @ X1 @ X0 )
= $true ) ) )
& ( ! [X0: $i] :
( ? [X2: $i] :
( ! [X3: $i] :
( ( ( in @ X3 @ X2 )
!= $true )
| ( ( in @ X3 @ X0 )
!= $true ) )
& ( ( in @ X2 @ X0 )
= $true ) )
| ! [X1: $i] :
( ( in @ X1 @ X0 )
!= $true ) )
| ( foundationAx != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f21]) ).
thf(f21,plain,
( ( foundationAx = $true )
<=> ! [X0: $i] :
( ? [X2: $i] :
( ! [X3: $i] :
( ( ( in @ X3 @ X2 )
!= $true )
| ( ( in @ X3 @ X0 )
!= $true ) )
& ( ( in @ X2 @ X0 )
= $true ) )
| ! [X1: $i] :
( ( in @ X1 @ X0 )
!= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f13]) ).
thf(f13,plain,
( ! [X0: $i] :
( ? [X1: $i] :
( ( in @ X1 @ X0 )
= $true )
=> ? [X2: $i] :
( ( ( in @ X2 @ X0 )
= $true )
& ~ ? [X3: $i] :
( ( ( in @ X3 @ X0 )
= $true )
& ( ( in @ X3 @ X2 )
= $true ) ) ) )
<=> ( foundationAx = $true ) ),
inference(fool_elimination,[status(thm)],[f12]) ).
thf(f12,plain,
( foundationAx
= ( ! [X0: $i] :
( ? [X1: $i] : ( in @ X1 @ X0 )
=> ? [X2: $i] :
( ( in @ X2 @ X0 )
& ~ ? [X3: $i] :
( ( in @ X3 @ X0 )
& ( in @ X3 @ X2 ) ) ) ) ) ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
( foundationAx
= ( ! [X0: $i] :
( ? [X1: $i] : ( in @ X1 @ X0 )
=> ? [X2: $i] :
( ( in @ X2 @ X0 )
& ~ ? [X1: $i] :
( ( in @ X1 @ X0 )
& ( in @ X1 @ X2 ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.TIBw4yTLEl/DTF2THF_31874.p',foundationAx) ).
thf(f83,plain,
! [X2: $i,X3: $i] :
( ( ( in @ X3 @ ( setadjoin @ X2 @ emptyset ) )
!= $true )
| ( X2 = X3 ) ),
inference(trivial_inequality_removal,[status(thm)],[f67]) ).
thf(f67,plain,
! [X2: $i,X3: $i] :
( ( X2 = X3 )
| ( ( in @ X3 @ ( setadjoin @ X2 @ emptyset ) )
!= $true )
| ( $true != $true ) ),
inference(definition_unfolding,[status(thm)],[f46,f60]) ).
thf(f60,plain,
uniqinunit = $true,
inference(cnf_transformation,[status(thm)],[f45]) ).
thf(f46,plain,
! [X2: $i,X3: $i] :
( ( ( in @ X3 @ ( setadjoin @ X2 @ emptyset ) )
!= $true )
| ( X2 = X3 )
| ( uniqinunit != $true ) ),
inference(cnf_transformation,[status(thm)],[f27]) ).
thf(f27,plain,
( ( ( uniqinunit = $true )
| ( ( ( in @ sK1 @ ( setadjoin @ sK0 @ emptyset ) )
= $true )
& ( sK0 != sK1 ) ) )
& ( ! [X2: $i,X3: $i] :
( ( ( in @ X3 @ ( setadjoin @ X2 @ emptyset ) )
!= $true )
| ( X2 = X3 ) )
| ( uniqinunit != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f25,f26]) ).
thf(f26,plain,
( ? [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) )
= $true )
& ( X0 != X1 ) )
=> ( ( ( in @ sK1 @ ( setadjoin @ sK0 @ emptyset ) )
= $true )
& ( sK0 != sK1 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f25,plain,
( ( ( uniqinunit = $true )
| ? [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) )
= $true )
& ( X0 != X1 ) ) )
& ( ! [X2: $i,X3: $i] :
( ( ( in @ X3 @ ( setadjoin @ X2 @ emptyset ) )
!= $true )
| ( X2 = X3 ) )
| ( uniqinunit != $true ) ) ),
inference(rectify,[status(thm)],[f24]) ).
thf(f24,plain,
( ( ( uniqinunit = $true )
| ? [X1: $i,X0: $i] :
( ( ( in @ X0 @ ( setadjoin @ X1 @ emptyset ) )
= $true )
& ( X0 != X1 ) ) )
& ( ! [X1: $i,X0: $i] :
( ( ( in @ X0 @ ( setadjoin @ X1 @ emptyset ) )
!= $true )
| ( X0 = X1 ) )
| ( uniqinunit != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f22]) ).
thf(f22,plain,
( ( uniqinunit = $true )
<=> ! [X1: $i,X0: $i] :
( ( ( in @ X0 @ ( setadjoin @ X1 @ emptyset ) )
!= $true )
| ( X0 = X1 ) ) ),
inference(ennf_transformation,[status(thm)],[f17]) ).
thf(f17,plain,
( ! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( setadjoin @ X1 @ emptyset ) )
= $true )
=> ( X0 = X1 ) )
<=> ( uniqinunit = $true ) ),
inference(fool_elimination,[status(thm)],[f16]) ).
thf(f16,plain,
! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( setadjoin @ X1 @ emptyset ) )
=> ( X0 = X1 ) )
= uniqinunit ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,axiom,
! [X1: $i,X3: $i] :
( ( ( in @ X1 @ ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X3 ) )
= uniqinunit ),
file('/export/starexec/sandbox2/tmp/tmp.TIBw4yTLEl/DTF2THF_31874.p',uniqinunit) ).
thf(f85,plain,
! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( sK5 @ X1 ) )
!= $true )
| ( ( in @ X0 @ X1 )
!= $true ) ),
inference(condensation,[status(thm)],[f84]) ).
thf(f84,plain,
! [X6: $i,X7: $i,X4: $i] :
( ( ( in @ X6 @ X4 )
!= $true )
| ( ( in @ X6 @ ( sK5 @ X4 ) )
!= $true )
| ( ( in @ X7 @ X4 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f71]) ).
thf(f71,plain,
! [X6: $i,X7: $i,X4: $i] :
( ( ( in @ X7 @ X4 )
!= $true )
| ( ( in @ X6 @ X4 )
!= $true )
| ( $true != $true )
| ( ( in @ X6 @ ( sK5 @ X4 ) )
!= $true ) ),
inference(definition_unfolding,[status(thm)],[f50,f63]) ).
thf(f50,plain,
! [X6: $i,X7: $i,X4: $i] :
( ( ( in @ X6 @ ( sK5 @ X4 ) )
!= $true )
| ( ( in @ X6 @ X4 )
!= $true )
| ( ( in @ X7 @ X4 )
!= $true )
| ( foundationAx != $true ) ),
inference(cnf_transformation,[status(thm)],[f34]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU806^2 : TPTP v9.2.1. Released v3.7.0.
% 0.10/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.12/0.33 % Computer : n022.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 : Tue Feb 24 12:34:36 EST 2026
% 0.12/0.33 % CPUTime :
% 0.12/0.33 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.42 ---- Original DTF file ---
% 0.16/0.42 thf(spec,logic,$$dhol).
% 0.16/0.42 %------------------------------------------------------------------------------
% 0.16/0.42 % File : SEU806^2 : TPTP v9.2.1. Released v3.7.0.
% 0.16/0.42 % Domain : Set Theory
% 0.16/0.42 % Problem : The Foundation Axiom
% 0.16/0.42 % Version : Especial > Reduced > Especial.
% 0.16/0.42 % English : (! A:i.~(in A A))
% 0.16/0.42
% 0.16/0.42 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.16/0.42 % Source : [Bro08]
% 0.16/0.42 % Names : ZFC308l [Bro08]
% 0.16/0.42
% 0.16/0.42 % Status : Theorem
% 0.16/0.42 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.31 v8.1.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.12 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.00 v6.1.0, 0.14 v5.5.0, 0.17 v5.4.0, 0.20 v5.3.0, 0.40 v5.2.0, 0.20 v5.1.0, 0.40 v5.0.0, 0.20 v4.1.0, 0.00 v4.0.1, 0.33 v4.0.0, 0.00 v3.7.0
% 0.16/0.42 % Syntax : Number of formulae : 12 ( 4 unt; 7 typ; 4 def)
% 0.16/0.42 % Number of atoms : 24 ( 7 equ; 0 cnn)
% 0.16/0.42 % Maximal formula atoms : 5 ( 4 avg)
% 0.16/0.42 % Number of connectives : 35 ( 2 ~; 0 |; 2 &; 22 @)
% 0.16/0.42 % ( 1 <=>; 8 =>; 0 <=; 0 <~>)
% 0.16/0.42 % Maximal formula depth : 9 ( 3 avg)
% 0.16/0.42 % Number of types : 2 ( 0 usr)
% 0.16/0.42 % Number of type conns : 4 ( 4 >; 0 *; 0 +; 0 <<)
% 0.16/0.42 % Number of symbols : 8 ( 7 usr; 5 con; 0-2 aty)
% 0.16/0.42 % Number of variables : 13 ( 0 ^; 10 !; 3 ?; 13 :)
% 0.16/0.42 % SPC : TH0_THM_EQU_NAR
% 0.16/0.42
% 0.16/0.42 % Comments : http://mathgate.info/detsetitem.php?id=435
% 0.16/0.42 %------------------------------------------------------------------------------
% 0.16/0.42 thf(in_type,type,
% 0.16/0.42 in: $i > $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(emptyset_type,type,
% 0.16/0.42 emptyset: $i ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoin_type,type,
% 0.16/0.42 setadjoin: $i > $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(foundationAx_type,type,
% 0.16/0.42 foundationAx: $o ).
% 0.16/0.42
% 0.16/0.42 thf(foundationAx,definition,
% 0.16/0.42 ( foundationAx
% 0.16/0.42 = ( ! [A: $i] :
% 0.16/0.42 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.16/0.42 => ? [B: $i] :
% 0.16/0.42 ( ( in @ B @ A )
% 0.16/0.42 & ~ ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinIL_type,type,
% 0.16/0.42 setadjoinIL: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinIL,definition,
% 0.16/0.42 ( setadjoinIL
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(uniqinunit_type,type,
% 0.16/0.42 uniqinunit: $o ).
% 0.16/0.42
% 0.16/0.42 thf(uniqinunit,definition,
% 0.16/0.42 ( uniqinunit
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.42 => ( Xx = Xy ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(in__Cong_type,type,
% 0.16/0.42 in__Cong: $o ).
% 0.16/0.42
% 0.16/0.42 thf(in__Cong,definition,
% 0.16/0.42 ( in__Cong
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( A = B )
% 0.16/0.42 => ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xx = Xy )
% 0.16/0.42 => ( ( in @ Xx @ A )
% 0.16/0.42 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(notinself,conjecture,
% 0.16/0.42 ( foundationAx
% 0.16/0.42 => ( setadjoinIL
% 0.16/0.42 => ( uniqinunit
% 0.16/0.42 => ( in__Cong
% 0.16/0.42 => ! [A: $i] :
% 0.16/0.42 ~ ( in @ A @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 %------------------------------------------------------------------------------
% 0.16/0.42 ------------------------
% 1.40/1.22 ---- Embedded in THF ---
% 1.40/1.23 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.40/1.23 %%% Generated on Tue Feb 24 12:34:37 EST 2026
% 1.40/1.23 %%% using '$$dhol' embedding, version 1.3.0.
% 1.40/1.23 %%% Logic specification used:
% 1.40/1.23 %%% thf(spec, logic, $$dhol).
% 1.40/1.23
% 1.40/1.23 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.TIBw4yTLEl/DTF2DTF31874.p
% See solution above
% 1.40/1.23 ------------------------
% 1.40/1.23 ---- Cleaned THF ---
% 1.40/1.24 thf(in_type,type,
% 1.40/1.24 in: $i > $i > $o ).
% 1.40/1.24
% 1.40/1.24 thf(emptyset_type,type,
% 1.40/1.24 emptyset: $i ).
% 1.40/1.24
% 1.40/1.24 thf(setadjoin_type,type,
% 1.40/1.24 setadjoin: $i > $i > $i ).
% 1.40/1.24
% 1.40/1.24 thf(foundationAx_type,type,
% 1.40/1.24 foundationAx: $o ).
% 1.40/1.24
% 1.40/1.24 thf(foundationAx,definition,
% 1.40/1.24 ( foundationAx
% 1.40/1.24 = ( ! [A: $i] :
% 1.40/1.24 ( ? [Xx: $i] : ( in @ Xx @ A )
% 1.40/1.24 => ? [B: $i] :
% 1.40/1.24 ( ( in @ B @ A )
% 1.40/1.24 & ~ ? [Xx: $i] :
% 1.40/1.24 ( ( in @ Xx @ B )
% 1.40/1.24 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 1.40/1.24
% 1.40/1.24 thf(setadjoinIL_type,type,
% 1.40/1.24 setadjoinIL: $o ).
% 1.40/1.24
% 1.40/1.24 thf(setadjoinIL,definition,
% 1.40/1.24 ( setadjoinIL
% 1.40/1.24 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 1.40/1.24
% 1.40/1.24 thf(uniqinunit_type,type,
% 1.40/1.24 uniqinunit: $o ).
% 1.40/1.24
% 1.40/1.24 thf(uniqinunit,definition,
% 1.40/1.24 ( uniqinunit
% 1.40/1.24 = ( ! [Xx: $i,Xy: $i] :
% 1.40/1.24 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 1.40/1.24 => ( Xx = Xy ) ) ) ) ).
% 1.40/1.24
% 1.40/1.24 thf(in__Cong_type,type,
% 1.40/1.24 in__Cong: $o ).
% 1.40/1.24
% 1.40/1.24 thf(in__Cong,definition,
% 1.40/1.24 ( in__Cong
% 1.40/1.24 = ( ! [A: $i,B: $i] :
% 1.40/1.24 ( ( A = B )
% 1.40/1.24 => ! [Xx: $i,Xy: $i] :
% 1.40/1.24 ( ( Xx = Xy )
% 1.40/1.24 => ( ( in @ Xx @ A )
% 1.40/1.24 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 1.40/1.24
% 1.40/1.24 thf(notinself,conjecture,
% 1.40/1.24 ( foundationAx
% 1.40/1.24 => ( setadjoinIL
% 1.40/1.24 => ( uniqinunit
% 1.40/1.24 => ( in__Cong
% 1.40/1.24 => ! [A: $i] :
% 1.40/1.24 ~ ( in @ A @ A ) ) ) ) ) ).
% 1.40/1.24 ------------------------
% 1.40/1.25 % (31974)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_31874 for (2999ds/183Mi)
% 1.40/1.26 % (31979)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_31874 for (2999ds/275Mi)
% 1.40/1.26 % (31980)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_31874 for (2999ds/18Mi)
% 1.40/1.26 % (31977)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_31874 for (2999ds/2Mi)
% 1.40/1.26 % (31978)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_31874 for (2999ds/2Mi)
% 1.40/1.26 % (31975)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_31874 for (2999ds/4Mi)
% 1.40/1.26 % (31976)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_31874 for (2999ds/27Mi)
% 1.40/1.26 % (31977)Instruction limit reached!
% 1.40/1.26 % (31977)------------------------------
% 1.40/1.26 % (31977)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.40/1.26 % (31977)Termination reason: Unknown
% 1.40/1.26 % (31978)Instruction limit reached!
% 1.40/1.26 % (31978)------------------------------
% 1.40/1.26 % (31978)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.40/1.26 % (31977)Termination phase: Saturation
% 1.40/1.26
% 1.40/1.26 % (31977)Memory used [KB]: 1023
% 1.40/1.26 % (31977)Time elapsed: 0.003 s
% 1.40/1.26 % (31977)Instructions burned: 3 (million)
% 1.40/1.26 % (31977)------------------------------
% 1.40/1.26 % (31977)------------------------------
% 1.40/1.26 % (31978)Termination reason: Unknown
% 1.40/1.26 % (31978)Termination phase: Saturation
% 1.40/1.26
% 1.40/1.26 % (31978)Memory used [KB]: 1023
% 1.40/1.26 % (31978)Time elapsed: 0.003 s
% 1.40/1.26 % (31978)Instructions burned: 3 (million)
% 1.40/1.26 % (31978)------------------------------
% 1.40/1.26 % (31978)------------------------------
% 1.40/1.26 % (31979)Refutation not found, incomplete strategy
% 1.40/1.26 % (31979)------------------------------
% 1.40/1.26 % (31979)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.40/1.26 % (31979)Termination reason: Refutation not found, incomplete strategy
% 1.40/1.26
% 1.40/1.26
% 1.40/1.26 % (31979)Memory used [KB]: 5500
% 1.40/1.26 % (31979)Time elapsed: 0.004 s
% 1.40/1.26 % (31979)Instructions burned: 3 (million)
% 1.40/1.26 % (31979)------------------------------
% 1.40/1.26 % (31979)------------------------------
% 1.40/1.26 % (31975)Instruction limit reached!
% 1.40/1.26 % (31975)------------------------------
% 1.40/1.26 % (31975)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.40/1.26 % (31975)Termination reason: Unknown
% 1.40/1.26 % (31975)Termination phase: Saturation
% 1.40/1.26
% 1.40/1.26 % (31975)Memory used [KB]: 5500
% 1.40/1.26 % (31975)Time elapsed: 0.004 s
% 1.40/1.26 % (31975)Instructions burned: 5 (million)
% 1.40/1.26 % (31975)------------------------------
% 1.40/1.26 % (31975)------------------------------
% 1.40/1.26 % (31980)First to succeed.
% 1.40/1.26 % (31980)Refutation found. Thanks to Tanya!
% 1.40/1.26 % SZS status Theorem for DTF2THF_31874
% 1.40/1.26 % SZS output start Proof for DTF2THF_31874
% 1.40/1.26 thf(func_def_0, type, in: $i > $i > $o).
% 1.40/1.26 thf(func_def_2, type, setadjoin: $i > $i > $i).
% 1.40/1.26 thf(func_def_13, type, sK3: $i > $i).
% 1.40/1.26 thf(func_def_15, type, sK5: $i > $i).
% 1.40/1.26 thf(f103,plain,(
% 1.40/1.26 $false),
% 1.40/1.26 inference(trivial_inequality_removal,[status(thm)],[f102])).
% 1.40/1.26 thf(f102,plain,(
% 1.40/1.26 ($true != $true)),
% 1.40/1.26 inference(superposition,[status(thm)],[f100,f64])).
% 1.40/1.26 thf(f64,plain,(
% 1.40/1.26 ((in @ sK12 @ sK12) = $true)),
% 1.40/1.26 inference(cnf_transformation,[status(thm)],[f45])).
% 1.40/1.26 thf(f45,plain,(
% 1.40/1.26 ((in @ sK12 @ sK12) = $true) & (foundationAx = $true) & (setadjoinIL = $true) & (in__Cong = $true) & (uniqinunit = $true)),
% 1.40/1.26 inference(skolemisation,[status(esa),new_symbols(skolem,[sK12])],[f20,f44])).
% 1.40/1.26 thf(f44,plain,(
% 1.40/1.26 ? [X0 : $i] : ((in @ X0 @ X0) = $true) => ((in @ sK12 @ sK12) = $true)),
% 1.40/1.26 introduced(definition,[],[choice_axiom])).
% 1.40/1.26 thf(f20,plain,(
% 1.40/1.26 ? [X0 : $i] : ((in @ X0 @ X0) = $true) & (foundationAx = $true) & (setadjoinIL = $true) & (in__Cong = $true) & (uniqinunit = $true)),
% 1.40/1.26 inference(flattening,[status(thm)],[f19])).
% 1.40/1.26 thf(f19,plain,(
% 1.40/1.26 (((? [X0 : $i] : ((in @ X0 @ X0) = $true) & (in__Cong = $true)) & (uniqinunit = $true)) & (setadjoinIL = $true)) & (foundationAx = $true)),
% 1.40/1.26 inference(ennf_transformation,[status(thm)],[f18])).
% 1.40/1.26 thf(f18,plain,(
% 1.40/1.26 ~((foundationAx = $true) => ((setadjoinIL = $true) => ((uniqinunit = $true) => ((in__Cong = $true) => ! [X0 : $i] : ((in @ X0 @ X0) != $true)))))),
% 1.40/1.26 inference(flattening,[status(thm)],[f11])).
% 1.40/1.26 thf(f11,plain,(
% 1.40/1.26 ~((foundationAx = $true) => ((setadjoinIL = $true) => ((uniqinunit = $true) => ((in__Cong = $true) => ! [X0 : $i] : ~((in @ X0 @ X0) = $true)))))),
% 1.40/1.26 inference(fool_elimination,[status(thm)],[f10])).
% 1.40/1.26 thf(f10,plain,(
% 1.40/1.26 ~(foundationAx => (setadjoinIL => (uniqinunit => (in__Cong => ! [X0 : $i] : ~(in @ X0 @ X0)))))),
% 1.40/1.26 inference(rectify,[status(thm)],[f6])).
% 1.40/1.26 thf(f6,negated_conjecture,(
% 1.40/1.26 ~(foundationAx => (setadjoinIL => (uniqinunit => (in__Cong => ! [X0 : $i] : ~(in @ X0 @ X0)))))),
% 1.40/1.26 inference(negated_conjecture,[status(cth)],[f5])).
% 1.40/1.26 thf(f5,conjecture,(
% 1.40/1.26 foundationAx => (setadjoinIL => (uniqinunit => (in__Cong => ! [X0 : $i] : ~(in @ X0 @ X0))))),
% 1.40/1.26 file('/export/starexec/sandbox2/tmp/tmp.TIBw4yTLEl/DTF2THF_31874.p',notinself)).
% 1.40/1.26 thf(f100,plain,(
% 1.40/1.26 ( ! [X0 : $i] : (((in @ X0 @ X0) != $true)) )),
% 1.40/1.26 inference(trivial_inequality_removal,[status(thm)],[f97])).
% 1.40/1.26 thf(f97,plain,(
% 1.40/1.26 ( ! [X0 : $i] : (((in @ X0 @ X0) != $true) | ($true != $true)) )),
% 1.40/1.26 inference(superposition,[status(thm)],[f96,f81])).
% 1.40/1.26 thf(f81,plain,(
% 1.40/1.26 ( ! [X2 : $i,X3 : $i] : (((in @ X2 @ (setadjoin @ X2 @ X3)) = $true)) )),
% 1.40/1.26 inference(trivial_inequality_removal,[status(thm)],[f78])).
% 1.40/1.26 thf(f78,plain,(
% 1.40/1.26 ( ! [X2 : $i,X3 : $i] : (($true != $true) | ((in @ X2 @ (setadjoin @ X2 @ X3)) = $true)) )),
% 1.40/1.26 inference(definition_unfolding,[status(thm)],[f58,f62])).
% 1.40/1.26 thf(f62,plain,(
% 1.40/1.26 (setadjoinIL = $true)),
% 1.40/1.26 inference(cnf_transformation,[status(thm)],[f45])).
% 1.40/1.26 thf(f58,plain,(
% 1.40/1.26 ( ! [X2 : $i,X3 : $i] : (((in @ X2 @ (setadjoin @ X2 @ X3)) = $true) | (setadjoinIL != $true)) )),
% 1.40/1.26 inference(cnf_transformation,[status(thm)],[f43])).
% 1.40/1.26 thf(f43,plain,(
% 1.40/1.26 ((setadjoinIL = $true) | ((in @ sK10 @ (setadjoin @ sK10 @ sK11)) != $true)) & (! [X2 : $i,X3 : $i] : ((in @ X2 @ (setadjoin @ X2 @ X3)) = $true) | (setadjoinIL != $true))),
% 1.40/1.26 inference(skolemisation,[status(esa),new_symbols(skolem,[sK10,sK11])],[f41,f42])).
% 1.40/1.26 thf(f42,plain,(
% 1.40/1.26 ? [X0 : $i,X1 : $i] : ((in @ X0 @ (setadjoin @ X0 @ X1)) != $true) => ((in @ sK10 @ (setadjoin @ sK10 @ sK11)) != $true)),
% 1.40/1.26 introduced(definition,[],[choice_axiom])).
% 1.40/1.26 thf(f41,plain,(
% 1.40/1.26 ((setadjoinIL = $true) | ? [X0 : $i,X1 : $i] : ((in @ X0 @ (setadjoin @ X0 @ X1)) != $true)) & (! [X2 : $i,X3 : $i] : ((in @ X2 @ (setadjoin @ X2 @ X3)) = $true) | (setadjoinIL != $true))),
% 1.40/1.26 inference(rectify,[status(thm)],[f40])).
% 1.40/1.26 thf(f40,plain,(
% 1.40/1.26 ((setadjoinIL = $true) | ? [X1 : $i,X0 : $i] : ((in @ X1 @ (setadjoin @ X1 @ X0)) != $true)) & (! [X1 : $i,X0 : $i] : ((in @ X1 @ (setadjoin @ X1 @ X0)) = $true) | (setadjoinIL != $true))),
% 1.40/1.26 inference(nnf_transformation,[status(thm)],[f15])).
% 1.40/1.26 thf(f15,plain,(
% 1.40/1.26 (setadjoinIL = $true) <=> ! [X1 : $i,X0 : $i] : ((in @ X1 @ (setadjoin @ X1 @ X0)) = $true)),
% 1.40/1.26 inference(fool_elimination,[status(thm)],[f14])).
% 1.40/1.26 thf(f14,plain,(
% 1.40/1.26 (setadjoinIL = ! [X0 : $i,X1 : $i] : (in @ X1 @ (setadjoin @ X1 @ X0)))),
% 1.40/1.26 inference(rectify,[status(thm)],[f2])).
% 1.40/1.26 thf(f2,axiom,(
% 1.40/1.26 (setadjoinIL = ! [X3 : $i,X1 : $i] : (in @ X1 @ (setadjoin @ X1 @ X3)))),
% 1.40/1.26 file('/export/starexec/sandbox2/tmp/tmp.TIBw4yTLEl/DTF2THF_31874.p',setadjoinIL)).
% 1.40/1.26 thf(f96,plain,(
% 1.40/1.26 ( ! [X0 : $i,X1 : $i] : (((in @ X1 @ (setadjoin @ X0 @ emptyset)) != $true) | ((in @ X1 @ X0) != $true)) )),
% 1.40/1.26 inference(superposition,[status(thm)],[f85,f94])).
% 1.40/1.26 thf(f94,plain,(
% 1.40/1.26 ( ! [X0 : $i] : (((sK5 @ (setadjoin @ X0 @ emptyset)) = X0)) )),
% 1.40/1.26 inference(trivial_inequality_removal,[status(thm)],[f92])).
% 1.40/1.26 thf(f92,plain,(
% 1.40/1.26 ( ! [X0 : $i] : (((sK5 @ (setadjoin @ X0 @ emptyset)) = X0) | ($true != $true)) )),
% 1.40/1.26 inference(superposition,[status(thm)],[f83,f89])).
% 1.40/1.26 thf(f89,plain,(
% 1.40/1.26 ( ! [X0 : $i,X1 : $i] : (((in @ (sK5 @ (setadjoin @ X0 @ X1)) @ (setadjoin @ X0 @ X1)) = $true)) )),
% 1.40/1.26 inference(trivial_inequality_removal,[status(thm)],[f88])).
% 1.40/1.26 thf(f88,plain,(
% 1.40/1.26 ( ! [X0 : $i,X1 : $i] : (((in @ (sK5 @ (setadjoin @ X0 @ X1)) @ (setadjoin @ X0 @ X1)) = $true) | ($true != $true)) )),
% 1.40/1.26 inference(superposition,[status(thm)],[f82,f81])).
% 1.40/1.26 thf(f82,plain,(
% 1.40/1.26 ( ! [X7 : $i,X4 : $i] : (((in @ X7 @ X4) != $true) | ((in @ (sK5 @ X4) @ X4) = $true)) )),
% 1.40/1.26 inference(trivial_inequality_removal,[status(thm)],[f72])).
% 1.40/1.26 thf(f72,plain,(
% 1.40/1.26 ( ! [X7 : $i,X4 : $i] : (((in @ X7 @ X4) != $true) | ($true != $true) | ((in @ (sK5 @ X4) @ X4) = $true)) )),
% 1.40/1.26 inference(definition_unfolding,[status(thm)],[f49,f63])).
% 1.40/1.26 thf(f63,plain,(
% 1.40/1.26 (foundationAx = $true)),
% 1.40/1.26 inference(cnf_transformation,[status(thm)],[f45])).
% 1.40/1.26 thf(f49,plain,(
% 1.40/1.26 ( ! [X7 : $i,X4 : $i] : (((in @ (sK5 @ X4) @ X4) = $true) | ((in @ X7 @ X4) != $true) | (foundationAx != $true)) )),
% 1.40/1.26 inference(cnf_transformation,[status(thm)],[f34])).
% 1.40/1.26 thf(f34,plain,(
% 1.40/1.26 ((foundationAx = $true) | (! [X1 : $i] : ((((in @ (sK3 @ X1) @ X1) = $true) & ((in @ (sK3 @ X1) @ sK2) = $true)) | ((in @ X1 @ sK2) != $true)) & ($true = (in @ sK4 @ sK2)))) & (! [X4 : $i] : ((! [X6 : $i] : (((in @ X6 @ (sK5 @ X4)) != $true) | ((in @ X6 @ X4) != $true)) & ((in @ (sK5 @ X4) @ X4) = $true)) | ! [X7 : $i] : ((in @ X7 @ X4) != $true)) | (foundationAx != $true))),
% 1.40/1.26 inference(skolemisation,[status(esa),new_symbols(skolem,[sK2,sK3,sK4,sK5])],[f29,f33,f32,f31,f30])).
% 1.40/1.26 thf(f30,plain,(
% 1.40/1.26 ? [X0 : $i] : (! [X1 : $i] : (? [X2 : $i] : (((in @ X2 @ X1) = $true) & ((in @ X2 @ X0) = $true)) | ((in @ X1 @ X0) != $true)) & ? [X3 : $i] : ((in @ X3 @ X0) = $true)) => (! [X1 : $i] : (? [X2 : $i] : (((in @ X2 @ X1) = $true) & ((in @ X2 @ sK2) = $true)) | ((in @ X1 @ sK2) != $true)) & ? [X3 : $i] : ((in @ X3 @ sK2) = $true))),
% 1.40/1.26 introduced(definition,[],[choice_axiom])).
% 1.40/1.26 thf(f31,plain,(
% 1.40/1.26 ! [X1 : $i] : (? [X2 : $i] : (((in @ X2 @ X1) = $true) & ((in @ X2 @ sK2) = $true)) => (((in @ (sK3 @ X1) @ X1) = $true) & ((in @ (sK3 @ X1) @ sK2) = $true)))),
% 1.40/1.26 introduced(definition,[],[choice_axiom])).
% 1.40/1.26 thf(f32,plain,(
% 1.40/1.26 ? [X3 : $i] : ((in @ X3 @ sK2) = $true) => ($true = (in @ sK4 @ sK2))),
% 1.40/1.26 introduced(definition,[],[choice_axiom])).
% 1.40/1.26 thf(f33,plain,(
% 1.40/1.26 ! [X4 : $i] : (? [X5 : $i] : (! [X6 : $i] : (((in @ X6 @ X5) != $true) | ((in @ X6 @ X4) != $true)) & ((in @ X5 @ X4) = $true)) => (! [X6 : $i] : (((in @ X6 @ (sK5 @ X4)) != $true) | ((in @ X6 @ X4) != $true)) & ((in @ (sK5 @ X4) @ X4) = $true)))),
% 1.40/1.26 introduced(definition,[],[choice_axiom])).
% 1.40/1.26 thf(f29,plain,(
% 1.40/1.26 ((foundationAx = $true) | ? [X0 : $i] : (! [X1 : $i] : (? [X2 : $i] : (((in @ X2 @ X1) = $true) & ((in @ X2 @ X0) = $true)) | ((in @ X1 @ X0) != $true)) & ? [X3 : $i] : ((in @ X3 @ X0) = $true))) & (! [X4 : $i] : (? [X5 : $i] : (! [X6 : $i] : (((in @ X6 @ X5) != $true) | ((in @ X6 @ X4) != $true)) & ((in @ X5 @ X4) = $true)) | ! [X7 : $i] : ((in @ X7 @ X4) != $true)) | (foundationAx != $true))),
% 1.40/1.26 inference(rectify,[status(thm)],[f28])).
% 1.40/1.26 thf(f28,plain,(
% 1.40/1.26 ((foundationAx = $true) | ? [X0 : $i] : (! [X2 : $i] : (? [X3 : $i] : (((in @ X3 @ X2) = $true) & ((in @ X3 @ X0) = $true)) | ((in @ X2 @ X0) != $true)) & ? [X1 : $i] : ((in @ X1 @ X0) = $true))) & (! [X0 : $i] : (? [X2 : $i] : (! [X3 : $i] : (((in @ X3 @ X2) != $true) | ((in @ X3 @ X0) != $true)) & ((in @ X2 @ X0) = $true)) | ! [X1 : $i] : ((in @ X1 @ X0) != $true)) | (foundationAx != $true))),
% 1.40/1.26 inference(nnf_transformation,[status(thm)],[f21])).
% 1.40/1.26 thf(f21,plain,(
% 1.40/1.26 (foundationAx = $true) <=> ! [X0 : $i] : (? [X2 : $i] : (! [X3 : $i] : (((in @ X3 @ X2) != $true) | ((in @ X3 @ X0) != $true)) & ((in @ X2 @ X0) = $true)) | ! [X1 : $i] : ((in @ X1 @ X0) != $true))),
% 1.40/1.26 inference(ennf_transformation,[status(thm)],[f13])).
% 1.40/1.26 thf(f13,plain,(
% 1.40/1.26 ! [X0 : $i] : (? [X1 : $i] : ((in @ X1 @ X0) = $true) => ? [X2 : $i] : (((in @ X2 @ X0) = $true) & ~? [X3 : $i] : (((in @ X3 @ X0) = $true) & ((in @ X3 @ X2) = $true)))) <=> (foundationAx = $true)),
% 1.40/1.26 inference(fool_elimination,[status(thm)],[f12])).
% 1.40/1.26 thf(f12,plain,(
% 1.40/1.26 (foundationAx = ! [X0 : $i] : (? [X1 : $i] : (in @ X1 @ X0) => ? [X2 : $i] : ((in @ X2 @ X0) & ~? [X3 : $i] : ((in @ X3 @ X0) & (in @ X3 @ X2)))))),
% 1.40/1.26 inference(rectify,[status(thm)],[f1])).
% 1.40/1.26 thf(f1,axiom,(
% 1.40/1.26 (foundationAx = ! [X0 : $i] : (? [X1 : $i] : (in @ X1 @ X0) => ? [X2 : $i] : ((in @ X2 @ X0) & ~? [X1 : $i] : ((in @ X1 @ X0) & (in @ X1 @ X2)))))),
% 1.40/1.26 file('/export/starexec/sandbox2/tmp/tmp.TIBw4yTLEl/DTF2THF_31874.p',foundationAx)).
% 1.40/1.26 thf(f83,plain,(
% 1.40/1.26 ( ! [X2 : $i,X3 : $i] : (((in @ X3 @ (setadjoin @ X2 @ emptyset)) != $true) | (X2 = X3)) )),
% 1.40/1.26 inference(trivial_inequality_removal,[status(thm)],[f67])).
% 1.40/1.26 thf(f67,plain,(
% 1.40/1.26 ( ! [X2 : $i,X3 : $i] : ((X2 = X3) | ((in @ X3 @ (setadjoin @ X2 @ emptyset)) != $true) | ($true != $true)) )),
% 1.40/1.26 inference(definition_unfolding,[status(thm)],[f46,f60])).
% 1.40/1.26 thf(f60,plain,(
% 1.40/1.26 (uniqinunit = $true)),
% 1.40/1.26 inference(cnf_transformation,[status(thm)],[f45])).
% 1.40/1.26 thf(f46,plain,(
% 1.40/1.26 ( ! [X2 : $i,X3 : $i] : (((in @ X3 @ (setadjoin @ X2 @ emptyset)) != $true) | (X2 = X3) | (uniqinunit != $true)) )),
% 1.40/1.26 inference(cnf_transformation,[status(thm)],[f27])).
% 1.40/1.26 thf(f27,plain,(
% 1.40/1.26 ((uniqinunit = $true) | (((in @ sK1 @ (setadjoin @ sK0 @ emptyset)) = $true) & (sK0 != sK1))) & (! [X2 : $i,X3 : $i] : (((in @ X3 @ (setadjoin @ X2 @ emptyset)) != $true) | (X2 = X3)) | (uniqinunit != $true))),
% 1.40/1.26 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f25,f26])).
% 1.40/1.26 thf(f26,plain,(
% 1.40/1.26 ? [X0 : $i,X1 : $i] : (((in @ X1 @ (setadjoin @ X0 @ emptyset)) = $true) & (X0 != X1)) => (((in @ sK1 @ (setadjoin @ sK0 @ emptyset)) = $true) & (sK0 != sK1))),
% 1.40/1.26 introduced(definition,[],[choice_axiom])).
% 1.40/1.26 thf(f25,plain,(
% 1.40/1.26 ((uniqinunit = $true) | ? [X0 : $i,X1 : $i] : (((in @ X1 @ (setadjoin @ X0 @ emptyset)) = $true) & (X0 != X1))) & (! [X2 : $i,X3 : $i] : (((in @ X3 @ (setadjoin @ X2 @ emptyset)) != $true) | (X2 = X3)) | (uniqinunit != $true))),
% 1.40/1.26 inference(rectify,[status(thm)],[f24])).
% 1.40/1.26 thf(f24,plain,(
% 1.40/1.26 ((uniqinunit = $true) | ? [X1 : $i,X0 : $i] : (((in @ X0 @ (setadjoin @ X1 @ emptyset)) = $true) & (X0 != X1))) & (! [X1 : $i,X0 : $i] : (((in @ X0 @ (setadjoin @ X1 @ emptyset)) != $true) | (X0 = X1)) | (uniqinunit != $true))),
% 1.40/1.26 inference(nnf_transformation,[status(thm)],[f22])).
% 1.40/1.26 thf(f22,plain,(
% 1.40/1.26 (uniqinunit = $true) <=> ! [X1 : $i,X0 : $i] : (((in @ X0 @ (setadjoin @ X1 @ emptyset)) != $true) | (X0 = X1))),
% 1.40/1.26 inference(ennf_transformation,[status(thm)],[f17])).
% 1.40/1.26 thf(f17,plain,(
% 1.40/1.26 ! [X0 : $i,X1 : $i] : (((in @ X0 @ (setadjoin @ X1 @ emptyset)) = $true) => (X0 = X1)) <=> (uniqinunit = $true)),
% 1.40/1.26 inference(fool_elimination,[status(thm)],[f16])).
% 1.40/1.26 thf(f16,plain,(
% 1.40/1.26 (! [X0 : $i,X1 : $i] : ((in @ X0 @ (setadjoin @ X1 @ emptyset)) => (X0 = X1)) = uniqinunit)),
% 1.40/1.26 inference(rectify,[status(thm)],[f3])).
% 1.40/1.26 thf(f3,axiom,(
% 1.40/1.26 (! [X1 : $i,X3 : $i] : ((in @ X1 @ (setadjoin @ X3 @ emptyset)) => (X1 = X3)) = uniqinunit)),
% 1.40/1.26 file('/export/starexec/sandbox2/tmp/tmp.TIBw4yTLEl/DTF2THF_31874.p',uniqinunit)).
% 1.40/1.26 thf(f85,plain,(
% 1.40/1.26 ( ! [X0 : $i,X1 : $i] : (((in @ X0 @ (sK5 @ X1)) != $true) | ((in @ X0 @ X1) != $true)) )),
% 1.40/1.26 inference(condensation,[status(thm)],[f84])).
% 1.40/1.26 thf(f84,plain,(
% 1.40/1.26 ( ! [X6 : $i,X7 : $i,X4 : $i] : (((in @ X6 @ X4) != $true) | ((in @ X6 @ (sK5 @ X4)) != $true) | ((in @ X7 @ X4) != $true)) )),
% 1.40/1.26 inference(trivial_inequality_removal,[status(thm)],[f71])).
% 1.40/1.26 thf(f71,plain,(
% 1.40/1.26 ( ! [X6 : $i,X7 : $i,X4 : $i] : (((in @ X7 @ X4) != $true) | ((in @ X6 @ X4) != $true) | ($true != $true) | ((in @ X6 @ (sK5 @ X4)) != $true)) )),
% 1.40/1.26 inference(definition_unfolding,[status(thm)],[f50,f63])).
% 1.40/1.26 thf(f50,plain,(
% 1.40/1.26 ( ! [X6 : $i,X7 : $i,X4 : $i] : (((in @ X6 @ (sK5 @ X4)) != $true) | ((in @ X6 @ X4) != $true) | ((in @ X7 @ X4) != $true) | (foundationAx != $true)) )),
% 1.40/1.26 inference(cnf_transformation,[status(thm)],[f34])).
% 1.40/1.26 % SZS output end Proof for DTF2THF_31874
% 1.40/1.26 % (31980)------------------------------
% 1.40/1.26 % (31980)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.40/1.26 % (31980)Termination reason: Refutation
% 1.40/1.26
% 1.40/1.26 % (31980)Memory used [KB]: 5500
% 1.40/1.26 % (31980)Time elapsed: 0.007 s
% 1.40/1.26 % (31980)Instructions burned: 6 (million)
% 1.40/1.26 % (31980)------------------------------
% 1.40/1.26 % (31980)------------------------------
% 1.40/1.26 % (31973)Success in time 0.013 s
%------------------------------------------------------------------------------