%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SET055^12 : TPTP v9.2.1. Released v8.1.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n031.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 09:31:05 AM UTC 2026
% Result : Theorem 2.37s 1.51s
% Output : Refutation 2.37s
% Verified :
% SZS Type : Refutation
% Derivation depth : 6
% Number of leaves : 2
% Syntax : Number of formulae : 11 ( 6 unt; 0 typ; 0 def)
% Number of atoms : 41 ( 5 equ; 0 cnn)
% Maximal formula atoms : 3 ( 3 avg)
% Number of connectives : 45 ( 5 ~; 0 |; 0 &; 40 @)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 7 ( 4 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 44 ( 41 usr; 3 con; 0-3 aty)
% Number of variables : 10 ( 10 ^; 0 !; 0 ?; 10 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
mworld: $tType ).
thf(func_def_0,type,
mworld: $tType ).
thf(func_def_1,type,
mrel: mworld > mworld > $o ).
thf(func_def_2,type,
mactual: mworld ).
thf(func_def_3,type,
mlocal: ( mworld > $o ) > $o ).
thf(func_def_5,type,
mnot: ( mworld > $o ) > mworld > $o ).
thf(func_def_6,type,
mand: ( mworld > $o ) > ( mworld > $o ) > mworld > $o ).
thf(func_def_7,type,
mor: ( mworld > $o ) > ( mworld > $o ) > mworld > $o ).
thf(func_def_8,type,
mimplies: ( mworld > $o ) > ( mworld > $o ) > mworld > $o ).
thf(func_def_9,type,
mequiv: ( mworld > $o ) > ( mworld > $o ) > mworld > $o ).
thf(func_def_10,type,
mbox: ( mworld > $o ) > mworld > $o ).
thf(func_def_11,type,
mdia: ( mworld > $o ) > mworld > $o ).
thf(func_def_12,type,
eiw_di: $i > mworld > $o ).
thf(func_def_13,type,
mforall_di: ( $i > mworld > $o ) > mworld > $o ).
thf(func_def_14,type,
mexists_di: ( $i > mworld > $o ) > mworld > $o ).
thf(func_def_20,type,
qmltpeq: $i > $i > mworld > $o ).
thf(func_def_21,type,
inductive: $i > mworld > $o ).
thf(func_def_22,type,
disjoint: $i > $i > mworld > $o ).
thf(func_def_23,type,
function: $i > mworld > $o ).
thf(func_def_24,type,
member: $i > $i > mworld > $o ).
thf(func_def_25,type,
subclass: $i > $i > mworld > $o ).
thf(func_def_26,type,
singleton: $i > $i ).
thf(func_def_27,type,
image: $i > $i > $i ).
thf(func_def_28,type,
inverse: $i > $i ).
thf(func_def_29,type,
rotate: $i > $i ).
thf(func_def_30,type,
unordered_pair: $i > $i > $i ).
thf(func_def_31,type,
successor: $i > $i ).
thf(func_def_32,type,
apply: $i > $i > $i ).
thf(func_def_33,type,
power_class: $i > $i ).
thf(func_def_34,type,
range_of: $i > $i ).
thf(func_def_35,type,
union: $i > $i > $i ).
thf(func_def_36,type,
restrict: $i > $i > $i > $i ).
thf(func_def_37,type,
second: $i > $i ).
thf(func_def_38,type,
ordered_pair: $i > $i > $i ).
thf(func_def_39,type,
domain_of: $i > $i ).
thf(func_def_40,type,
sum_class: $i > $i ).
thf(func_def_41,type,
compose: $i > $i > $i ).
thf(func_def_42,type,
intersection: $i > $i > $i ).
thf(func_def_43,type,
cross_product: $i > $i > $i ).
thf(func_def_44,type,
complement: $i > $i ).
thf(func_def_45,type,
flip: $i > $i ).
thf(func_def_46,type,
first: $i > $i ).
thf(func_def_62,type,
ph1:
!>[X0: $tType] : X0 ).
thf(f293,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f291,f292]) ).
thf(f292,plain,
( $true
= ( mlocal
@ ( mforall_di
@ ^ [Y0: $i] : ( qmltpeq @ Y0 @ Y0 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f245]) ).
thf(f245,plain,
( $true
= ( mlocal
@ ( mforall_di
@ ^ [Y0: $i] : ( qmltpeq @ Y0 @ Y0 ) ) ) ),
inference(fool_elimination,[status(thm)],[f244]) ).
thf(f244,plain,
( mlocal
@ ( mforall_di
@ ^ [X0: $i] : ( qmltpeq @ X0 @ X0 ) ) ),
inference(rectify,[status(thm)],[f13]) ).
thf(f13,axiom,
( mlocal
@ ( mforall_di
@ ^ [X5: $i] : ( qmltpeq @ X5 @ X5 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.xzz2ub9STI/DTF2THF_16121.p',reflexivity) ).
thf(f291,plain,
( $true
!= ( mlocal
@ ( mforall_di
@ ^ [Y0: $i] : ( qmltpeq @ Y0 @ Y0 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f290]) ).
thf(f290,plain,
( $true
!= ( mlocal
@ ( mforall_di
@ ^ [Y0: $i] : ( qmltpeq @ Y0 @ Y0 ) ) ) ),
inference(flattening,[status(thm)],[f211]) ).
thf(f211,plain,
( $true
!= ( mlocal
@ ( mforall_di
@ ^ [Y0: $i] : ( qmltpeq @ Y0 @ Y0 ) ) ) ),
inference(fool_elimination,[status(thm)],[f210]) ).
thf(f210,plain,
~ ( mlocal
@ ( mforall_di
@ ^ [X0: $i] : ( qmltpeq @ X0 @ X0 ) ) ),
inference(rectify,[status(thm)],[f97]) ).
thf(f97,negated_conjecture,
~ ( mlocal
@ ( mforall_di
@ ^ [X5: $i] : ( qmltpeq @ X5 @ X5 ) ) ),
inference(negated_conjecture,[status(cth)],[f96]) ).
thf(f96,conjecture,
( mlocal
@ ( mforall_di
@ ^ [X5: $i] : ( qmltpeq @ X5 @ X5 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.xzz2ub9STI/DTF2THF_16121.p',reflexivity_0) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SET055^12 : TPTP v9.2.1. Released v8.1.0.
% 0.00/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n031.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 11:51:48 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.27/0.46 ---- Original DTF file ---
% 0.27/0.46 thf(spec,logic,$$dhol).
% 0.27/0.46 %------------------------------------------------------------------------------
% 0.27/0.46 % File : SET055^12 : TPTP v9.2.1. Released v8.1.0.
% 0.27/0.46 % Domain : Set Theory
% 0.27/0.46 % Problem : TPTP problem SET055+1.p with axiomatized equality
% 0.27/0.46 % Version : [BP13] axioms.
% 0.27/0.46 % English :
% 0.27/0.46
% 0.27/0.46 % Refs : [RO12] Raths & Otten (2012), The QMLTP Problem Library for Fi
% 0.27/0.46 % : [BP13] Benzmueller & Paulson (2013), Quantified Multimodal Lo
% 0.27/0.46 % : [Ste22] Steen (2022), An Extensible Logic Embedding Tool for L
% 0.27/0.46 % Source : [TPTP]
% 0.27/0.46 % Names : SET055+1 [QMLTP]
% 0.27/0.46
% 0.27/0.46 % Status : Theorem
% 0.27/0.46 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.00 v8.2.0, 0.08 v8.1.0
% 0.27/0.46 % Syntax : Number of formulae : 142 ( 12 unt; 46 typ; 10 def)
% 0.27/0.46 % Number of atoms : 598 ( 10 equ; 0 cnn)
% 0.27/0.46 % Maximal formula atoms : 11 ( 6 avg)
% 0.27/0.46 % Number of connectives : 1140 ( 1 ~; 1 |; 3 &;1131 @)
% 0.27/0.46 % ( 1 <=>; 3 =>; 0 <=; 0 <~>)
% 0.27/0.46 % Maximal formula depth : 19 ( 10 avg)
% 0.27/0.47 % Number of types : 3 ( 1 usr)
% 0.27/0.47 % Number of type conns : 106 ( 106 >; 0 *; 0 +; 0 <<)
% 0.27/0.47 % Number of symbols : 46 ( 45 usr; 6 con; 0-3 aty)
% 0.27/0.47 % Number of variables : 226 ( 218 ^; 5 !; 3 ?; 226 :)
% 0.27/0.47 % SPC : TH0_THM_EQU_NAR
% 0.27/0.47
% 0.27/0.47 % Comments : This output was generated by embedproblem, version 1.7.1 (library
% 0.27/0.47 % version 1.3). Generated on Thu Apr 28 13:18:18 EDT 2022 using
% 0.27/0.47 % 'modal' embedding, version 1.5.2. Logic specification used:
% 0.27/0.47 % $modal == [$constants == $rigid,$quantification == $varying,
% 0.27/0.47 % $modalities == $modal_system_S5U].
% 0.27/0.47 %------------------------------------------------------------------------------
% 0.27/0.47 thf(mworld,type,
% 0.27/0.47 mworld: $tType ).
% 0.27/0.47
% 0.27/0.47 thf(mrel_type,type,
% 0.27/0.47 mrel: mworld > mworld > $o ).
% 0.27/0.47
% 0.27/0.47 thf(mactual_type,type,
% 0.27/0.47 mactual: mworld ).
% 0.27/0.47
% 0.27/0.47 thf(mlocal_type,type,
% 0.27/0.47 mlocal: ( mworld > $o ) > $o ).
% 0.27/0.47
% 0.27/0.47 thf(mlocal_def,definition,
% 0.27/0.47 ( mlocal
% 0.27/0.47 = ( ^ [Phi: mworld > $o] : ( Phi @ mactual ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(mnot_type,type,
% 0.27/0.47 mnot: ( mworld > $o ) > mworld > $o ).
% 0.27/0.47
% 0.27/0.47 thf(mand_type,type,
% 0.27/0.47 mand: ( mworld > $o ) > ( mworld > $o ) > mworld > $o ).
% 0.27/0.47
% 0.27/0.47 thf(mor_type,type,
% 0.27/0.47 mor: ( mworld > $o ) > ( mworld > $o ) > mworld > $o ).
% 0.27/0.47
% 0.27/0.47 thf(mimplies_type,type,
% 0.27/0.47 mimplies: ( mworld > $o ) > ( mworld > $o ) > mworld > $o ).
% 0.27/0.47
% 0.27/0.47 thf(mequiv_type,type,
% 0.27/0.47 mequiv: ( mworld > $o ) > ( mworld > $o ) > mworld > $o ).
% 0.27/0.47
% 0.27/0.47 thf(mnot_def,definition,
% 0.27/0.47 ( mnot
% 0.27/0.47 = ( ^ [A: mworld > $o,W: mworld] :
% 0.27/0.47 ~ ( A @ W ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(mand_def,definition,
% 0.27/0.47 ( mand
% 0.27/0.47 = ( ^ [A: mworld > $o,B: mworld > $o,W: mworld] :
% 0.27/0.47 ( ( A @ W )
% 0.27/0.47 & ( B @ W ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(mor_def,definition,
% 0.27/0.47 ( mor
% 0.27/0.47 = ( ^ [A: mworld > $o,B: mworld > $o,W: mworld] :
% 0.27/0.47 ( ( A @ W )
% 0.27/0.47 | ( B @ W ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(mimplies_def,definition,
% 0.27/0.47 ( mimplies
% 0.27/0.47 = ( ^ [A: mworld > $o,B: mworld > $o,W: mworld] :
% 0.27/0.47 ( ( A @ W )
% 0.27/0.47 => ( B @ W ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(mequiv_def,definition,
% 0.27/0.47 ( mequiv
% 0.27/0.47 = ( ^ [A: mworld > $o,B: mworld > $o,W: mworld] :
% 0.27/0.47 ( ( A @ W )
% 0.27/0.47 <=> ( B @ W ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(mbox_type,type,
% 0.27/0.47 mbox: ( mworld > $o ) > mworld > $o ).
% 0.27/0.47
% 0.27/0.47 thf(mbox_def,definition,
% 0.27/0.47 ( mbox
% 0.27/0.47 = ( ^ [Phi: mworld > $o,W: mworld] :
% 0.27/0.47 ! [V: mworld] :
% 0.27/0.47 ( ( mrel @ W @ V )
% 0.27/0.47 => ( Phi @ V ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(mdia_type,type,
% 0.27/0.47 mdia: ( mworld > $o ) > mworld > $o ).
% 0.27/0.47
% 0.27/0.47 thf(mdia_def,definition,
% 0.27/0.47 ( mdia
% 0.27/0.47 = ( ^ [Phi: mworld > $o,W: mworld] :
% 0.27/0.47 ? [V: mworld] :
% 0.27/0.47 ( ( mrel @ W @ V )
% 0.27/0.47 & ( Phi @ V ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(mrel_universal,axiom,
% 0.27/0.47 ! [W: mworld,V: mworld] : ( mrel @ W @ V ) ).
% 0.27/0.47
% 0.27/0.47 thf(eiw_di_type,type,
% 0.27/0.47 eiw_di: $i > mworld > $o ).
% 0.27/0.47
% 0.27/0.47 thf(eiw_di_nonempty,axiom,
% 0.27/0.47 ! [W: mworld] :
% 0.27/0.47 ? [X: $i] : ( eiw_di @ X @ W ) ).
% 0.27/0.47
% 0.27/0.47 thf(mforall_di_type,type,
% 0.27/0.47 mforall_di: ( $i > mworld > $o ) > mworld > $o ).
% 0.27/0.47
% 0.27/0.47 thf(mforall_di_def,definition,
% 0.27/0.47 ( mforall_di
% 0.27/0.47 = ( ^ [A: $i > mworld > $o,W: mworld] :
% 0.27/0.47 ! [X: $i] :
% 0.27/0.47 ( ( eiw_di @ X @ W )
% 0.27/0.47 => ( A @ X @ W ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(mexists_di_type,type,
% 0.27/0.47 mexists_di: ( $i > mworld > $o ) > mworld > $o ).
% 0.27/0.47
% 0.27/0.47 thf(mexists_di_def,definition,
% 0.27/0.47 ( mexists_di
% 0.27/0.47 = ( ^ [A: $i > mworld > $o,W: mworld] :
% 0.27/0.47 ? [X: $i] :
% 0.27/0.47 ( ( eiw_di @ X @ W )
% 0.27/0.47 & ( A @ X @ W ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 %%% This output was generated by tptputils, version 1.2.
% 0.27/0.47 %%% Generated on Wed Apr 27 15:37:57 CEST 2022 using command 'downgrade(tff)'.
% 0.27/0.47 thf(universal_class_decl,type,
% 0.27/0.47 universal_class: $i ).
% 0.27/0.47
% 0.27/0.47 thf(identity_relation_decl,type,
% 0.27/0.47 identity_relation: $i ).
% 0.27/0.47
% 0.27/0.47 thf(successor_relation_decl,type,
% 0.27/0.47 successor_relation: $i ).
% 0.27/0.47
% 0.27/0.47 thf(element_relation_decl,type,
% 0.27/0.47 element_relation: $i ).
% 0.27/0.47
% 0.27/0.47 thf(null_class_decl,type,
% 0.27/0.47 null_class: $i ).
% 0.27/0.47
% 0.27/0.47 thf(qmltpeq_decl,type,
% 0.27/0.47 qmltpeq: $i > $i > mworld > $o ).
% 0.27/0.47
% 0.27/0.47 thf(inductive_decl,type,
% 0.27/0.47 inductive: $i > mworld > $o ).
% 0.27/0.47
% 0.27/0.47 thf(disjoint_decl,type,
% 0.27/0.47 disjoint: $i > $i > mworld > $o ).
% 0.27/0.47
% 0.27/0.47 thf(function_decl,type,
% 0.27/0.47 function: $i > mworld > $o ).
% 0.27/0.47
% 0.27/0.47 thf(member_decl,type,
% 0.27/0.47 member: $i > $i > mworld > $o ).
% 0.27/0.47
% 0.27/0.47 thf(subclass_decl,type,
% 0.27/0.47 subclass: $i > $i > mworld > $o ).
% 0.27/0.47
% 0.27/0.47 thf(singleton_decl,type,
% 0.27/0.47 singleton: $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(image_decl,type,
% 0.27/0.47 image: $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(inverse_decl,type,
% 0.27/0.47 inverse: $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(rotate_decl,type,
% 0.27/0.47 rotate: $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(unordered_pair_decl,type,
% 0.27/0.47 unordered_pair: $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(successor_decl,type,
% 0.27/0.47 successor: $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(apply_decl,type,
% 0.27/0.47 apply: $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(power_class_decl,type,
% 0.27/0.47 power_class: $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(range_of_decl,type,
% 0.27/0.47 range_of: $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(union_decl,type,
% 0.27/0.47 union: $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(restrict_decl,type,
% 0.27/0.47 restrict: $i > $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(second_decl,type,
% 0.27/0.47 second: $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(ordered_pair_decl,type,
% 0.27/0.47 ordered_pair: $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(domain_of_decl,type,
% 0.27/0.47 domain_of: $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(sum_class_decl,type,
% 0.27/0.47 sum_class: $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(compose_decl,type,
% 0.27/0.47 compose: $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(intersection_decl,type,
% 0.27/0.47 intersection: $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(cross_product_decl,type,
% 0.27/0.47 cross_product: $i > $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(complement_decl,type,
% 0.27/0.47 complement: $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(flip_decl,type,
% 0.27/0.47 flip: $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(first_decl,type,
% 0.27/0.47 first: $i > $i ).
% 0.27/0.47
% 0.27/0.47 thf(reflexivity,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [X: $i] : ( qmltpeq @ X @ X ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(symmetry,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [X: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [Y: $i] : ( mimplies @ ( qmltpeq @ X @ Y ) @ ( qmltpeq @ Y @ X ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(transitivity,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [X: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [Y: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [Z: $i] : ( mimplies @ ( mand @ ( qmltpeq @ X @ Y ) @ ( qmltpeq @ Y @ Z ) ) @ ( qmltpeq @ X @ Z ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(apply_substitution_1,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( apply @ A @ C ) @ ( apply @ B @ C ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(apply_substitution_2,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( apply @ C @ A ) @ ( apply @ C @ B ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(complement_substitution_1,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( complement @ A ) @ ( complement @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(compose_substitution_1,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( compose @ A @ C ) @ ( compose @ B @ C ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(compose_substitution_2,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( compose @ C @ A ) @ ( compose @ C @ B ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cross_product_substitution_1,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( cross_product @ A @ C ) @ ( cross_product @ B @ C ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cross_product_substitution_2,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( cross_product @ C @ A ) @ ( cross_product @ C @ B ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(domain_of_substitution_1,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( domain_of @ A ) @ ( domain_of @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(first_substitution_1,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( first @ A ) @ ( first @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(flip_substitution_1,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( flip @ A ) @ ( flip @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(image_substitution_1,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( image @ A @ C ) @ ( image @ B @ C ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(image_substitution_2,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( image @ C @ A ) @ ( image @ C @ B ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(intersection_substitution_1,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( intersection @ A @ C ) @ ( intersection @ B @ C ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(intersection_substitution_2,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( intersection @ C @ A ) @ ( intersection @ C @ B ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(inverse_substitution_1,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( inverse @ A ) @ ( inverse @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ordered_pair_substitution_1,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( ordered_pair @ A @ C ) @ ( ordered_pair @ B @ C ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ordered_pair_substitution_2,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( ordered_pair @ C @ A ) @ ( ordered_pair @ C @ B ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(power_class_substitution_1,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( power_class @ A ) @ ( power_class @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(range_of_substitution_1,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( range_of @ A ) @ ( range_of @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(restrict_substitution_1,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [C: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [D: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( restrict @ A @ C @ D ) @ ( restrict @ B @ C @ D ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(restrict_substitution_2,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [C: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [D: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( restrict @ C @ A @ D ) @ ( restrict @ C @ B @ D ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(restrict_substitution_3,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [C: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [D: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( restrict @ C @ D @ A ) @ ( restrict @ C @ D @ B ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(rotate_substitution_1,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( rotate @ A ) @ ( rotate @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(second_substitution_1,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( second @ A ) @ ( second @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(singleton_substitution_1,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( singleton @ A ) @ ( singleton @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(successor_substitution_1,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( successor @ A ) @ ( successor @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(sum_class_substitution_1,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( sum_class @ A ) @ ( sum_class @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(union_substitution_1,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( union @ A @ C ) @ ( union @ B @ C ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(union_substitution_2,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( union @ C @ A ) @ ( union @ C @ B ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(unordered_pair_substitution_1,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( unordered_pair @ A @ C ) @ ( unordered_pair @ B @ C ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(unordered_pair_substitution_2,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( unordered_pair @ C @ A ) @ ( unordered_pair @ C @ B ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(disjoint_substitution_1,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( disjoint @ A @ C ) ) @ ( disjoint @ B @ C ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(disjoint_substitution_2,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( disjoint @ C @ A ) ) @ ( disjoint @ C @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(function_substitution_1,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( function @ A ) ) @ ( function @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(inductive_substitution_1,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( inductive @ A ) ) @ ( inductive @ B ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(member_substitution_1,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( member @ A @ C ) ) @ ( member @ B @ C ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(member_substitution_2,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( member @ C @ A ) ) @ ( member @ C @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(subclass_substitution_1,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( subclass @ A @ C ) ) @ ( subclass @ B @ C ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(subclass_substitution_2,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [A: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [B: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( subclass @ C @ A ) ) @ ( subclass @ C @ B ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(subclass_defn,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [X: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [Y: $i] :
% 0.27/0.47 ( mequiv @ ( subclass @ X @ Y )
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [U: $i] : ( mimplies @ ( member @ U @ X ) @ ( member @ U @ Y ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(class_elements_are_sets,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [X: $i] : ( subclass @ X @ universal_class ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(extensionality,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [X: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [Y: $i] : ( mequiv @ ( qmltpeq @ X @ Y ) @ ( mand @ ( subclass @ X @ Y ) @ ( subclass @ Y @ X ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(unordered_pair_defn,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [U: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [X: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [Y: $i] : ( mequiv @ ( member @ U @ ( unordered_pair @ X @ Y ) ) @ ( mand @ ( member @ U @ universal_class ) @ ( mor @ ( qmltpeq @ U @ X ) @ ( qmltpeq @ U @ Y ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(unordered_pair_0,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [X: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [Y: $i] : ( member @ ( unordered_pair @ X @ Y ) @ universal_class ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(singleton_set_defn,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [X: $i] : ( qmltpeq @ ( singleton @ X ) @ ( unordered_pair @ X @ X ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(ordered_pair_defn,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [X: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [Y: $i] : ( qmltpeq @ ( ordered_pair @ X @ Y ) @ ( unordered_pair @ ( singleton @ X ) @ ( unordered_pair @ X @ ( singleton @ Y ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cross_product_defn,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [U: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [V: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [X: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [Y: $i] : ( mequiv @ ( member @ ( ordered_pair @ U @ V ) @ ( cross_product @ X @ Y ) ) @ ( mand @ ( member @ U @ X ) @ ( member @ V @ Y ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(cross_product_0,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [X: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [Y: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [Z: $i] : ( mimplies @ ( member @ Z @ ( cross_product @ X @ Y ) ) @ ( qmltpeq @ Z @ ( ordered_pair @ ( first @ Z ) @ ( second @ Z ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(element_relation_defn,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [X: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [Y: $i] : ( mequiv @ ( member @ ( ordered_pair @ X @ Y ) @ element_relation ) @ ( mand @ ( member @ Y @ universal_class ) @ ( member @ X @ Y ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(element_relation_0,axiom,
% 0.27/0.47 mlocal @ ( subclass @ element_relation @ ( cross_product @ universal_class @ universal_class ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(intersection_0,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [X: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [Y: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [Z: $i] : ( mequiv @ ( member @ Z @ ( intersection @ X @ Y ) ) @ ( mand @ ( member @ Z @ X ) @ ( member @ Z @ Y ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(complement_0,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [X: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [Z: $i] : ( mequiv @ ( member @ Z @ ( complement @ X ) ) @ ( mand @ ( member @ Z @ universal_class ) @ ( mnot @ ( member @ Z @ X ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(restrict_defn,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [X: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [XR: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [Y: $i] : ( qmltpeq @ ( restrict @ XR @ X @ Y ) @ ( intersection @ XR @ ( cross_product @ X @ Y ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(null_class_defn,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [X: $i] : ( mnot @ ( member @ X @ null_class ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(domain_of_0,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [X: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [Z: $i] : ( mequiv @ ( member @ Z @ ( domain_of @ X ) ) @ ( mand @ ( member @ Z @ universal_class ) @ ( mnot @ ( qmltpeq @ ( restrict @ X @ ( singleton @ Z ) @ universal_class ) @ null_class ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(rotate_defn,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [X: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [U: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [V: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [W: $i] : ( mequiv @ ( member @ ( ordered_pair @ ( ordered_pair @ U @ V ) @ W ) @ ( rotate @ X ) ) @ ( mand @ ( member @ ( ordered_pair @ ( ordered_pair @ U @ V ) @ W ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) ) @ ( member @ ( ordered_pair @ ( ordered_pair @ V @ W ) @ U ) @ X ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(rotate_0,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [X: $i] : ( subclass @ ( rotate @ X ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(flip_defn,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [U: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [V: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [W: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [X: $i] : ( mequiv @ ( member @ ( ordered_pair @ ( ordered_pair @ U @ V ) @ W ) @ ( flip @ X ) ) @ ( mand @ ( member @ ( ordered_pair @ ( ordered_pair @ U @ V ) @ W ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) ) @ ( member @ ( ordered_pair @ ( ordered_pair @ V @ U ) @ W ) @ X ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(flip_0,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [X: $i] : ( subclass @ ( flip @ X ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(union_defn,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [X: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [Y: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [Z: $i] : ( mequiv @ ( member @ Z @ ( union @ X @ Y ) ) @ ( mor @ ( member @ Z @ X ) @ ( member @ Z @ Y ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(successor_defn,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [X: $i] : ( qmltpeq @ ( successor @ X ) @ ( union @ X @ ( singleton @ X ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(successor_relation_defn1,axiom,
% 0.27/0.47 mlocal @ ( subclass @ successor_relation @ ( cross_product @ universal_class @ universal_class ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(successor_relation_defn2,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [X: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [Y: $i] : ( mequiv @ ( member @ ( ordered_pair @ X @ Y ) @ successor_relation ) @ ( mand @ ( member @ X @ universal_class ) @ ( mand @ ( member @ Y @ universal_class ) @ ( qmltpeq @ ( successor @ X ) @ Y ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(inverse_defn,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [Y: $i] : ( qmltpeq @ ( inverse @ Y ) @ ( domain_of @ ( flip @ ( cross_product @ Y @ universal_class ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(range_of_defn,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [Z: $i] : ( qmltpeq @ ( range_of @ Z ) @ ( domain_of @ ( inverse @ Z ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(image_defn,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [X: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [XR: $i] : ( qmltpeq @ ( image @ XR @ X ) @ ( range_of @ ( restrict @ XR @ X @ universal_class ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(inductive_defn,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [X: $i] : ( mequiv @ ( inductive @ X ) @ ( mand @ ( member @ null_class @ X ) @ ( subclass @ ( image @ successor_relation @ X ) @ X ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(infinity,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mexists_di
% 0.27/0.47 @ ^ [X: $i] :
% 0.27/0.47 ( mand @ ( member @ X @ universal_class )
% 0.27/0.47 @ ( mand @ ( inductive @ X )
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [Y: $i] : ( mimplies @ ( inductive @ Y ) @ ( subclass @ X @ Y ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(sum_class_defn,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [U: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [X: $i] :
% 0.27/0.47 ( mequiv @ ( member @ U @ ( sum_class @ X ) )
% 0.27/0.47 @ ( mexists_di
% 0.27/0.47 @ ^ [Y: $i] : ( mand @ ( member @ U @ Y ) @ ( member @ Y @ X ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(sum_class_0,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [X: $i] : ( mimplies @ ( member @ X @ universal_class ) @ ( member @ ( sum_class @ X ) @ universal_class ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(power_class_defn,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [U: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [X: $i] : ( mequiv @ ( member @ U @ ( power_class @ X ) ) @ ( mand @ ( member @ U @ universal_class ) @ ( subclass @ U @ X ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(power_class_0,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [U: $i] : ( mimplies @ ( member @ U @ universal_class ) @ ( member @ ( power_class @ U ) @ universal_class ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(compose_defn1,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [XR: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [YR: $i] : ( subclass @ ( compose @ YR @ XR ) @ ( cross_product @ universal_class @ universal_class ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(compose_defn2,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [XR: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [YR: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [U: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [V: $i] : ( mequiv @ ( member @ ( ordered_pair @ U @ V ) @ ( compose @ YR @ XR ) ) @ ( mand @ ( member @ U @ universal_class ) @ ( member @ V @ ( image @ YR @ ( image @ YR @ ( singleton @ U ) ) ) ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(function_defn,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [XF: $i] : ( mequiv @ ( function @ XF ) @ ( mand @ ( subclass @ XF @ ( cross_product @ universal_class @ universal_class ) ) @ ( subclass @ ( compose @ XF @ ( inverse @ XF ) ) @ identity_relation ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(replacement,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [X: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [XF: $i] : ( mimplies @ ( mand @ ( member @ X @ universal_class ) @ ( function @ XF ) ) @ ( member @ ( image @ XF @ X ) @ universal_class ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(disjoint_defn,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [X: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [Y: $i] :
% 0.27/0.47 ( mequiv @ ( disjoint @ X @ Y )
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [U: $i] : ( mnot @ ( mand @ ( member @ U @ X ) @ ( member @ U @ Y ) ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(regularity,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [X: $i] :
% 0.27/0.47 ( mimplies @ ( mnot @ ( qmltpeq @ X @ null_class ) )
% 0.27/0.47 @ ( mexists_di
% 0.27/0.47 @ ^ [U: $i] : ( mand @ ( member @ U @ universal_class ) @ ( mand @ ( member @ U @ X ) @ ( disjoint @ U @ X ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(apply_defn,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [XF: $i] :
% 0.27/0.47 ( mforall_di
% 0.27/0.47 @ ^ [Y: $i] : ( qmltpeq @ ( apply @ XF @ Y ) @ ( sum_class @ ( image @ XF @ ( singleton @ Y ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(choice,axiom,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mexists_di
% 0.27/0.47 @ ^ [XF: $i] :
% 0.27/0.47 ( mand @ ( function @ XF )
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [Y: $i] : ( mimplies @ ( member @ Y @ universal_class ) @ ( mor @ ( qmltpeq @ Y @ null_class ) @ ( member @ ( apply @ XF @ Y ) @ Y ) ) ) ) ) ) ) ).
% 0.27/0.47
% 0.27/0.47 thf(reflexivity_0,conjecture,
% 0.27/0.47 ( mlocal
% 0.27/0.47 @ ( mforall_di
% 0.27/0.47 @ ^ [X: $i] : ( qmltpeq @ X @ X ) ) ) ).
% 0.27/0.47
% 0.27/0.47 %------------------------------------------------------------------------------
% 0.27/0.47 ------------------------
% 2.29/1.45 ---- Embedded in THF ---
% 2.29/1.45 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.29/1.45 %%% Generated on Tue Feb 24 11:51:49 EST 2026
% 2.29/1.45 %%% using '$$dhol' embedding, version 1.3.0.
% 2.29/1.45 %%% Logic specification used:
% 2.29/1.45 %%% thf(spec, logic, $$dhol).
% 2.29/1.45
% 2.29/1.45 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.xzz2ub9STI/DTF2DTF16121.p
% See solution above
% 2.29/1.45 ------------------------
% 2.29/1.47 ---- Cleaned THF ---
% 2.29/1.47 thf(mworld,type,
% 2.29/1.47 mworld: $tType ).
% 2.29/1.47
% 2.29/1.47 thf(mrel_type,type,
% 2.29/1.47 mrel: mworld > mworld > $o ).
% 2.29/1.47
% 2.29/1.47 thf(mactual_type,type,
% 2.29/1.47 mactual: mworld ).
% 2.29/1.47
% 2.29/1.47 thf(mlocal_type,type,
% 2.29/1.47 mlocal: ( mworld > $o ) > $o ).
% 2.29/1.47
% 2.29/1.47 thf(mlocal_def,definition,
% 2.29/1.47 ( mlocal
% 2.29/1.47 = ( ^ [Phi: mworld > $o] : ( Phi @ mactual ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(mnot_type,type,
% 2.29/1.47 mnot: ( mworld > $o ) > mworld > $o ).
% 2.29/1.47
% 2.29/1.47 thf(mand_type,type,
% 2.29/1.47 mand: ( mworld > $o ) > ( mworld > $o ) > mworld > $o ).
% 2.29/1.47
% 2.29/1.47 thf(mor_type,type,
% 2.29/1.47 mor: ( mworld > $o ) > ( mworld > $o ) > mworld > $o ).
% 2.29/1.47
% 2.29/1.47 thf(mimplies_type,type,
% 2.29/1.47 mimplies: ( mworld > $o ) > ( mworld > $o ) > mworld > $o ).
% 2.29/1.47
% 2.29/1.47 thf(mequiv_type,type,
% 2.29/1.47 mequiv: ( mworld > $o ) > ( mworld > $o ) > mworld > $o ).
% 2.29/1.47
% 2.29/1.47 thf(mnot_def,definition,
% 2.29/1.47 ( mnot
% 2.29/1.47 = ( ^ [A: mworld > $o,W: mworld] :
% 2.29/1.47 ~ ( A @ W ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(mand_def,definition,
% 2.29/1.47 ( mand
% 2.29/1.47 = ( ^ [A: mworld > $o,B: mworld > $o,W: mworld] :
% 2.29/1.47 ( ( A @ W )
% 2.29/1.47 & ( B @ W ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(mor_def,definition,
% 2.29/1.47 ( mor
% 2.29/1.47 = ( ^ [A: mworld > $o,B: mworld > $o,W: mworld] :
% 2.29/1.47 ( ( A @ W )
% 2.29/1.47 | ( B @ W ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(mimplies_def,definition,
% 2.29/1.47 ( mimplies
% 2.29/1.47 = ( ^ [A: mworld > $o,B: mworld > $o,W: mworld] :
% 2.29/1.47 ( ( A @ W )
% 2.29/1.47 => ( B @ W ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(mequiv_def,definition,
% 2.29/1.47 ( mequiv
% 2.29/1.47 = ( ^ [A: mworld > $o,B: mworld > $o,W: mworld] :
% 2.29/1.47 ( ( A @ W )
% 2.29/1.47 <=> ( B @ W ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(mbox_type,type,
% 2.29/1.47 mbox: ( mworld > $o ) > mworld > $o ).
% 2.29/1.47
% 2.29/1.47 thf(mbox_def,definition,
% 2.29/1.47 ( mbox
% 2.29/1.47 = ( ^ [Phi: mworld > $o,W: mworld] :
% 2.29/1.47 ! [V: mworld] :
% 2.29/1.47 ( ( mrel @ W @ V )
% 2.29/1.47 => ( Phi @ V ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(mdia_type,type,
% 2.29/1.47 mdia: ( mworld > $o ) > mworld > $o ).
% 2.29/1.47
% 2.29/1.47 thf(mdia_def,definition,
% 2.29/1.47 ( mdia
% 2.29/1.47 = ( ^ [Phi: mworld > $o,W: mworld] :
% 2.29/1.47 ? [V: mworld] :
% 2.29/1.47 ( ( mrel @ W @ V )
% 2.29/1.47 & ( Phi @ V ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(mrel_universal,axiom,
% 2.29/1.47 ! [W: mworld,V: mworld] : ( mrel @ W @ V ) ).
% 2.29/1.47
% 2.29/1.47 thf(eiw_di_type,type,
% 2.29/1.47 eiw_di: $i > mworld > $o ).
% 2.29/1.47
% 2.29/1.47 thf(eiw_di_nonempty,axiom,
% 2.29/1.47 ! [W: mworld] :
% 2.29/1.47 ? [X: $i] : ( eiw_di @ X @ W ) ).
% 2.29/1.47
% 2.29/1.47 thf(mforall_di_type,type,
% 2.29/1.47 mforall_di: ( $i > mworld > $o ) > mworld > $o ).
% 2.29/1.47
% 2.29/1.47 thf(mforall_di_def,definition,
% 2.29/1.47 ( mforall_di
% 2.29/1.47 = ( ^ [A: $i > mworld > $o,W: mworld] :
% 2.29/1.47 ! [X: $i] :
% 2.29/1.47 ( ( eiw_di @ X @ W )
% 2.29/1.47 => ( A @ X @ W ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(mexists_di_type,type,
% 2.29/1.47 mexists_di: ( $i > mworld > $o ) > mworld > $o ).
% 2.29/1.47
% 2.29/1.47 thf(mexists_di_def,definition,
% 2.29/1.47 ( mexists_di
% 2.29/1.47 = ( ^ [A: $i > mworld > $o,W: mworld] :
% 2.29/1.47 ? [X: $i] :
% 2.29/1.47 ( ( eiw_di @ X @ W )
% 2.29/1.47 & ( A @ X @ W ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(universal_class_decl,type,
% 2.29/1.47 universal_class: $i ).
% 2.29/1.47
% 2.29/1.47 thf(identity_relation_decl,type,
% 2.29/1.47 identity_relation: $i ).
% 2.29/1.47
% 2.29/1.47 thf(successor_relation_decl,type,
% 2.29/1.47 successor_relation: $i ).
% 2.29/1.47
% 2.29/1.47 thf(element_relation_decl,type,
% 2.29/1.47 element_relation: $i ).
% 2.29/1.47
% 2.29/1.47 thf(null_class_decl,type,
% 2.29/1.47 null_class: $i ).
% 2.29/1.47
% 2.29/1.47 thf(qmltpeq_decl,type,
% 2.29/1.47 qmltpeq: $i > $i > mworld > $o ).
% 2.29/1.47
% 2.29/1.47 thf(inductive_decl,type,
% 2.29/1.47 inductive: $i > mworld > $o ).
% 2.29/1.47
% 2.29/1.47 thf(disjoint_decl,type,
% 2.29/1.47 disjoint: $i > $i > mworld > $o ).
% 2.29/1.47
% 2.29/1.47 thf(function_decl,type,
% 2.29/1.47 function: $i > mworld > $o ).
% 2.29/1.47
% 2.29/1.47 thf(member_decl,type,
% 2.29/1.47 member: $i > $i > mworld > $o ).
% 2.29/1.47
% 2.29/1.47 thf(subclass_decl,type,
% 2.29/1.47 subclass: $i > $i > mworld > $o ).
% 2.29/1.47
% 2.29/1.47 thf(singleton_decl,type,
% 2.29/1.47 singleton: $i > $i ).
% 2.29/1.47
% 2.29/1.47 thf(image_decl,type,
% 2.29/1.47 image: $i > $i > $i ).
% 2.29/1.47
% 2.29/1.47 thf(inverse_decl,type,
% 2.29/1.47 inverse: $i > $i ).
% 2.29/1.47
% 2.29/1.47 thf(rotate_decl,type,
% 2.29/1.47 rotate: $i > $i ).
% 2.29/1.47
% 2.29/1.47 thf(unordered_pair_decl,type,
% 2.29/1.47 unordered_pair: $i > $i > $i ).
% 2.29/1.47
% 2.29/1.47 thf(successor_decl,type,
% 2.29/1.47 successor: $i > $i ).
% 2.29/1.47
% 2.29/1.47 thf(apply_decl,type,
% 2.29/1.47 apply: $i > $i > $i ).
% 2.29/1.47
% 2.29/1.47 thf(power_class_decl,type,
% 2.29/1.47 power_class: $i > $i ).
% 2.29/1.47
% 2.29/1.47 thf(range_of_decl,type,
% 2.29/1.47 range_of: $i > $i ).
% 2.29/1.47
% 2.29/1.47 thf(union_decl,type,
% 2.29/1.47 union: $i > $i > $i ).
% 2.29/1.47
% 2.29/1.47 thf(restrict_decl,type,
% 2.29/1.47 restrict: $i > $i > $i > $i ).
% 2.29/1.47
% 2.29/1.47 thf(second_decl,type,
% 2.29/1.47 second: $i > $i ).
% 2.29/1.47
% 2.29/1.47 thf(ordered_pair_decl,type,
% 2.29/1.47 ordered_pair: $i > $i > $i ).
% 2.29/1.47
% 2.29/1.47 thf(domain_of_decl,type,
% 2.29/1.47 domain_of: $i > $i ).
% 2.29/1.47
% 2.29/1.47 thf(sum_class_decl,type,
% 2.29/1.47 sum_class: $i > $i ).
% 2.29/1.47
% 2.29/1.47 thf(compose_decl,type,
% 2.29/1.47 compose: $i > $i > $i ).
% 2.29/1.47
% 2.29/1.47 thf(intersection_decl,type,
% 2.29/1.47 intersection: $i > $i > $i ).
% 2.29/1.47
% 2.29/1.47 thf(cross_product_decl,type,
% 2.29/1.47 cross_product: $i > $i > $i ).
% 2.29/1.47
% 2.29/1.47 thf(complement_decl,type,
% 2.29/1.47 complement: $i > $i ).
% 2.29/1.47
% 2.29/1.47 thf(flip_decl,type,
% 2.29/1.47 flip: $i > $i ).
% 2.29/1.47
% 2.29/1.47 thf(first_decl,type,
% 2.29/1.47 first: $i > $i ).
% 2.29/1.47
% 2.29/1.47 thf(reflexivity,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [X: $i] : ( qmltpeq @ X @ X ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(symmetry,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [X: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [Y: $i] : ( mimplies @ ( qmltpeq @ X @ Y ) @ ( qmltpeq @ Y @ X ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(transitivity,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [X: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [Y: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [Z: $i] : ( mimplies @ ( mand @ ( qmltpeq @ X @ Y ) @ ( qmltpeq @ Y @ Z ) ) @ ( qmltpeq @ X @ Z ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(apply_substitution_1,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( apply @ A @ C ) @ ( apply @ B @ C ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(apply_substitution_2,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( apply @ C @ A ) @ ( apply @ C @ B ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(complement_substitution_1,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( complement @ A ) @ ( complement @ B ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(compose_substitution_1,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( compose @ A @ C ) @ ( compose @ B @ C ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(compose_substitution_2,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( compose @ C @ A ) @ ( compose @ C @ B ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(cross_product_substitution_1,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( cross_product @ A @ C ) @ ( cross_product @ B @ C ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(cross_product_substitution_2,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( cross_product @ C @ A ) @ ( cross_product @ C @ B ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(domain_of_substitution_1,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( domain_of @ A ) @ ( domain_of @ B ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(first_substitution_1,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( first @ A ) @ ( first @ B ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(flip_substitution_1,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( flip @ A ) @ ( flip @ B ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(image_substitution_1,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( image @ A @ C ) @ ( image @ B @ C ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(image_substitution_2,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( image @ C @ A ) @ ( image @ C @ B ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(intersection_substitution_1,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( intersection @ A @ C ) @ ( intersection @ B @ C ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(intersection_substitution_2,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( intersection @ C @ A ) @ ( intersection @ C @ B ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(inverse_substitution_1,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( inverse @ A ) @ ( inverse @ B ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(ordered_pair_substitution_1,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( ordered_pair @ A @ C ) @ ( ordered_pair @ B @ C ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(ordered_pair_substitution_2,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( ordered_pair @ C @ A ) @ ( ordered_pair @ C @ B ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(power_class_substitution_1,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( power_class @ A ) @ ( power_class @ B ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(range_of_substitution_1,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( range_of @ A ) @ ( range_of @ B ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(restrict_substitution_1,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [C: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [D: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( restrict @ A @ C @ D ) @ ( restrict @ B @ C @ D ) ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(restrict_substitution_2,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [C: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [D: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( restrict @ C @ A @ D ) @ ( restrict @ C @ B @ D ) ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(restrict_substitution_3,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [C: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [D: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( restrict @ C @ D @ A ) @ ( restrict @ C @ D @ B ) ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(rotate_substitution_1,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( rotate @ A ) @ ( rotate @ B ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(second_substitution_1,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( second @ A ) @ ( second @ B ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(singleton_substitution_1,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( singleton @ A ) @ ( singleton @ B ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(successor_substitution_1,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( successor @ A ) @ ( successor @ B ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(sum_class_substitution_1,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( sum_class @ A ) @ ( sum_class @ B ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(union_substitution_1,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( union @ A @ C ) @ ( union @ B @ C ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(union_substitution_2,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( union @ C @ A ) @ ( union @ C @ B ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(unordered_pair_substitution_1,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( unordered_pair @ A @ C ) @ ( unordered_pair @ B @ C ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(unordered_pair_substitution_2,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [C: $i] : ( mimplies @ ( qmltpeq @ A @ B ) @ ( qmltpeq @ ( unordered_pair @ C @ A ) @ ( unordered_pair @ C @ B ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(disjoint_substitution_1,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( disjoint @ A @ C ) ) @ ( disjoint @ B @ C ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(disjoint_substitution_2,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( disjoint @ C @ A ) ) @ ( disjoint @ C @ B ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(function_substitution_1,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( function @ A ) ) @ ( function @ B ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(inductive_substitution_1,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( inductive @ A ) ) @ ( inductive @ B ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(member_substitution_1,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( member @ A @ C ) ) @ ( member @ B @ C ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(member_substitution_2,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( member @ C @ A ) ) @ ( member @ C @ B ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(subclass_substitution_1,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( subclass @ A @ C ) ) @ ( subclass @ B @ C ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(subclass_substitution_2,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [A: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [B: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [C: $i] : ( mimplies @ ( mand @ ( qmltpeq @ A @ B ) @ ( subclass @ C @ A ) ) @ ( subclass @ C @ B ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(subclass_defn,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [X: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [Y: $i] :
% 2.29/1.47 ( mequiv @ ( subclass @ X @ Y )
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [U: $i] : ( mimplies @ ( member @ U @ X ) @ ( member @ U @ Y ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(class_elements_are_sets,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [X: $i] : ( subclass @ X @ universal_class ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(extensionality,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [X: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [Y: $i] : ( mequiv @ ( qmltpeq @ X @ Y ) @ ( mand @ ( subclass @ X @ Y ) @ ( subclass @ Y @ X ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(unordered_pair_defn,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [U: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [X: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [Y: $i] : ( mequiv @ ( member @ U @ ( unordered_pair @ X @ Y ) ) @ ( mand @ ( member @ U @ universal_class ) @ ( mor @ ( qmltpeq @ U @ X ) @ ( qmltpeq @ U @ Y ) ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(unordered_pair_0,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [X: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [Y: $i] : ( member @ ( unordered_pair @ X @ Y ) @ universal_class ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(singleton_set_defn,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [X: $i] : ( qmltpeq @ ( singleton @ X ) @ ( unordered_pair @ X @ X ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(ordered_pair_defn,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [X: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [Y: $i] : ( qmltpeq @ ( ordered_pair @ X @ Y ) @ ( unordered_pair @ ( singleton @ X ) @ ( unordered_pair @ X @ ( singleton @ Y ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(cross_product_defn,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [U: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [V: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [X: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [Y: $i] : ( mequiv @ ( member @ ( ordered_pair @ U @ V ) @ ( cross_product @ X @ Y ) ) @ ( mand @ ( member @ U @ X ) @ ( member @ V @ Y ) ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(cross_product_0,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [X: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [Y: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [Z: $i] : ( mimplies @ ( member @ Z @ ( cross_product @ X @ Y ) ) @ ( qmltpeq @ Z @ ( ordered_pair @ ( first @ Z ) @ ( second @ Z ) ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(element_relation_defn,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [X: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [Y: $i] : ( mequiv @ ( member @ ( ordered_pair @ X @ Y ) @ element_relation ) @ ( mand @ ( member @ Y @ universal_class ) @ ( member @ X @ Y ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(element_relation_0,axiom,
% 2.29/1.47 mlocal @ ( subclass @ element_relation @ ( cross_product @ universal_class @ universal_class ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(intersection_0,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [X: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [Y: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [Z: $i] : ( mequiv @ ( member @ Z @ ( intersection @ X @ Y ) ) @ ( mand @ ( member @ Z @ X ) @ ( member @ Z @ Y ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(complement_0,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [X: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [Z: $i] : ( mequiv @ ( member @ Z @ ( complement @ X ) ) @ ( mand @ ( member @ Z @ universal_class ) @ ( mnot @ ( member @ Z @ X ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(restrict_defn,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [X: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [XR: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [Y: $i] : ( qmltpeq @ ( restrict @ XR @ X @ Y ) @ ( intersection @ XR @ ( cross_product @ X @ Y ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(null_class_defn,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [X: $i] : ( mnot @ ( member @ X @ null_class ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(domain_of_0,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [X: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [Z: $i] : ( mequiv @ ( member @ Z @ ( domain_of @ X ) ) @ ( mand @ ( member @ Z @ universal_class ) @ ( mnot @ ( qmltpeq @ ( restrict @ X @ ( singleton @ Z ) @ universal_class ) @ null_class ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(rotate_defn,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [X: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [U: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [V: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [W: $i] : ( mequiv @ ( member @ ( ordered_pair @ ( ordered_pair @ U @ V ) @ W ) @ ( rotate @ X ) ) @ ( mand @ ( member @ ( ordered_pair @ ( ordered_pair @ U @ V ) @ W ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) ) @ ( member @ ( ordered_pair @ ( ordered_pair @ V @ W ) @ U ) @ X ) ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(rotate_0,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [X: $i] : ( subclass @ ( rotate @ X ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(flip_defn,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [U: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [V: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [W: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [X: $i] : ( mequiv @ ( member @ ( ordered_pair @ ( ordered_pair @ U @ V ) @ W ) @ ( flip @ X ) ) @ ( mand @ ( member @ ( ordered_pair @ ( ordered_pair @ U @ V ) @ W ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) ) @ ( member @ ( ordered_pair @ ( ordered_pair @ V @ U ) @ W ) @ X ) ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(flip_0,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [X: $i] : ( subclass @ ( flip @ X ) @ ( cross_product @ ( cross_product @ universal_class @ universal_class ) @ universal_class ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(union_defn,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [X: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [Y: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [Z: $i] : ( mequiv @ ( member @ Z @ ( union @ X @ Y ) ) @ ( mor @ ( member @ Z @ X ) @ ( member @ Z @ Y ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(successor_defn,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [X: $i] : ( qmltpeq @ ( successor @ X ) @ ( union @ X @ ( singleton @ X ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(successor_relation_defn1,axiom,
% 2.29/1.47 mlocal @ ( subclass @ successor_relation @ ( cross_product @ universal_class @ universal_class ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(successor_relation_defn2,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [X: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [Y: $i] : ( mequiv @ ( member @ ( ordered_pair @ X @ Y ) @ successor_relation ) @ ( mand @ ( member @ X @ universal_class ) @ ( mand @ ( member @ Y @ universal_class ) @ ( qmltpeq @ ( successor @ X ) @ Y ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(inverse_defn,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [Y: $i] : ( qmltpeq @ ( inverse @ Y ) @ ( domain_of @ ( flip @ ( cross_product @ Y @ universal_class ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(range_of_defn,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [Z: $i] : ( qmltpeq @ ( range_of @ Z ) @ ( domain_of @ ( inverse @ Z ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(image_defn,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [X: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [XR: $i] : ( qmltpeq @ ( image @ XR @ X ) @ ( range_of @ ( restrict @ XR @ X @ universal_class ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(inductive_defn,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [X: $i] : ( mequiv @ ( inductive @ X ) @ ( mand @ ( member @ null_class @ X ) @ ( subclass @ ( image @ successor_relation @ X ) @ X ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(infinity,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mexists_di
% 2.29/1.47 @ ^ [X: $i] :
% 2.29/1.47 ( mand @ ( member @ X @ universal_class )
% 2.29/1.47 @ ( mand @ ( inductive @ X )
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [Y: $i] : ( mimplies @ ( inductive @ Y ) @ ( subclass @ X @ Y ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(sum_class_defn,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [U: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [X: $i] :
% 2.29/1.47 ( mequiv @ ( member @ U @ ( sum_class @ X ) )
% 2.29/1.47 @ ( mexists_di
% 2.29/1.47 @ ^ [Y: $i] : ( mand @ ( member @ U @ Y ) @ ( member @ Y @ X ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(sum_class_0,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [X: $i] : ( mimplies @ ( member @ X @ universal_class ) @ ( member @ ( sum_class @ X ) @ universal_class ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(power_class_defn,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [U: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [X: $i] : ( mequiv @ ( member @ U @ ( power_class @ X ) ) @ ( mand @ ( member @ U @ universal_class ) @ ( subclass @ U @ X ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(power_class_0,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [U: $i] : ( mimplies @ ( member @ U @ universal_class ) @ ( member @ ( power_class @ U ) @ universal_class ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(compose_defn1,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [XR: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [YR: $i] : ( subclass @ ( compose @ YR @ XR ) @ ( cross_product @ universal_class @ universal_class ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(compose_defn2,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [XR: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [YR: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [U: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [V: $i] : ( mequiv @ ( member @ ( ordered_pair @ U @ V ) @ ( compose @ YR @ XR ) ) @ ( mand @ ( member @ U @ universal_class ) @ ( member @ V @ ( image @ YR @ ( image @ YR @ ( singleton @ U ) ) ) ) ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(function_defn,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [XF: $i] : ( mequiv @ ( function @ XF ) @ ( mand @ ( subclass @ XF @ ( cross_product @ universal_class @ universal_class ) ) @ ( subclass @ ( compose @ XF @ ( inverse @ XF ) ) @ identity_relation ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(replacement,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [X: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [XF: $i] : ( mimplies @ ( mand @ ( member @ X @ universal_class ) @ ( function @ XF ) ) @ ( member @ ( image @ XF @ X ) @ universal_class ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(disjoint_defn,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [X: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [Y: $i] :
% 2.29/1.47 ( mequiv @ ( disjoint @ X @ Y )
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [U: $i] : ( mnot @ ( mand @ ( member @ U @ X ) @ ( member @ U @ Y ) ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(regularity,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [X: $i] :
% 2.29/1.47 ( mimplies @ ( mnot @ ( qmltpeq @ X @ null_class ) )
% 2.29/1.47 @ ( mexists_di
% 2.29/1.47 @ ^ [U: $i] : ( mand @ ( member @ U @ universal_class ) @ ( mand @ ( member @ U @ X ) @ ( disjoint @ U @ X ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(apply_defn,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [XF: $i] :
% 2.29/1.47 ( mforall_di
% 2.29/1.47 @ ^ [Y: $i] : ( qmltpeq @ ( apply @ XF @ Y ) @ ( sum_class @ ( image @ XF @ ( singleton @ Y ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(choice,axiom,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mexists_di
% 2.29/1.47 @ ^ [XF: $i] :
% 2.29/1.47 ( mand @ ( function @ XF )
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [Y: $i] : ( mimplies @ ( member @ Y @ universal_class ) @ ( mor @ ( qmltpeq @ Y @ null_class ) @ ( member @ ( apply @ XF @ Y ) @ Y ) ) ) ) ) ) ) ).
% 2.29/1.47
% 2.29/1.47 thf(reflexivity_0,conjecture,
% 2.29/1.47 ( mlocal
% 2.29/1.47 @ ( mforall_di
% 2.29/1.47 @ ^ [X: $i] : ( qmltpeq @ X @ X ) ) ) ).
% 2.29/1.47 ------------------------
% 2.29/1.49 % (16388)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_16121 for (2999ds/4Mi)
% 2.29/1.49 % (16387)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_16121 for (2999ds/183Mi)
% 2.29/1.49 % (16388)Instruction limit reached!
% 2.29/1.49 % (16388)------------------------------
% 2.29/1.49 % (16388)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.29/1.49 % (16388)Termination reason: Unknown
% 2.29/1.49 % (16388)Termination phase: shuffling
% 2.29/1.49
% 2.29/1.49 % (16388)Memory used [KB]: 1279
% 2.29/1.49 % (16388)Time elapsed: 0.003 s
% 2.29/1.49 % (16388)Instructions burned: 6 (million)
% 2.29/1.49 % (16388)------------------------------
% 2.29/1.49 % (16388)------------------------------
% 2.37/1.50 % (16389)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_16121 for (2999ds/27Mi)
% 2.37/1.50 % (16390)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_16121 for (2999ds/2Mi)
% 2.37/1.50 % (16391)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_16121 for (2999ds/2Mi)
% 2.37/1.50 % (16393)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_16121 for (2999ds/18Mi)
% 2.37/1.50 % (16392)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_16121 for (2999ds/275Mi)
% 2.37/1.50 % (16390)Instruction limit reached!
% 2.37/1.50 % (16390)------------------------------
% 2.37/1.50 % (16390)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.50 % (16390)Termination reason: Unknown
% 2.37/1.50 % (16390)Termination phase: shuffling
% 2.37/1.50
% 2.37/1.50 % (16390)Memory used [KB]: 1151
% 2.37/1.50 % (16390)Time elapsed: 0.003 s
% 2.37/1.50 % (16390)Instructions burned: 2 (million)
% 2.37/1.50 % (16390)------------------------------
% 2.37/1.50 % (16390)------------------------------
% 2.37/1.50 % (16391)Instruction limit reached!
% 2.37/1.50 % (16391)------------------------------
% 2.37/1.50 % (16391)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.50 % (16391)Termination reason: Unknown
% 2.37/1.50 % (16391)Termination phase: shuffling
% 2.37/1.50
% 2.37/1.50 % (16391)Memory used [KB]: 1151
% 2.37/1.50 % (16391)Time elapsed: 0.003 s
% 2.37/1.50 % (16391)Instructions burned: 2 (million)
% 2.37/1.50 % (16391)------------------------------
% 2.37/1.50 % (16391)------------------------------
% 2.37/1.50 % (16394)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_16121 for (2999ds/3Mi)
% 2.37/1.50 % (16394)Instruction limit reached!
% 2.37/1.50 % (16394)------------------------------
% 2.37/1.50 % (16394)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.50 % (16394)Termination reason: Unknown
% 2.37/1.50 % (16394)Termination phase: shuffling
% 2.37/1.50
% 2.37/1.50 % (16394)Memory used [KB]: 1151
% 2.37/1.50 % (16394)Time elapsed: 0.003 s
% 2.37/1.50 % (16394)Instructions burned: 5 (million)
% 2.37/1.50 % (16394)------------------------------
% 2.37/1.50 % (16394)------------------------------
% 2.37/1.51 % (16393)Instruction limit reached!
% 2.37/1.51 % (16393)------------------------------
% 2.37/1.51 % (16393)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.51 % (16393)Termination reason: Unknown
% 2.37/1.51 % (16393)Termination phase: Property scanning
% 2.37/1.51
% 2.37/1.51 % (16393)Memory used [KB]: 1407
% 2.37/1.51 % (16393)Time elapsed: 0.011 s
% 2.37/1.51 % (16393)Instructions burned: 18 (million)
% 2.37/1.51 % (16393)------------------------------
% 2.37/1.51 % (16393)------------------------------
% 2.37/1.51 % (16389)Instruction limit reached!
% 2.37/1.51 % (16389)------------------------------
% 2.37/1.51 % (16389)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.51 % (16389)Termination reason: Unknown
% 2.37/1.51 % (16389)Termination phase: Property scanning
% 2.37/1.51
% 2.37/1.51 % (16389)Memory used [KB]: 1407
% 2.37/1.51 % (16389)Time elapsed: 0.016 s
% 2.37/1.51 % (16389)Instructions burned: 28 (million)
% 2.37/1.51 % (16389)------------------------------
% 2.37/1.51 % (16389)------------------------------
% 2.37/1.51 % (16392)First to succeed.
% 2.37/1.51 % (16392)Refutation found. Thanks to Tanya!
% 2.37/1.51 % SZS status Theorem for DTF2THF_16121
% 2.37/1.51 % SZS output start Proof for DTF2THF_16121
% 2.37/1.51 thf(type_def_5, type, mworld: $tType).
% 2.37/1.51 thf(func_def_0, type, mworld: $tType).
% 2.37/1.51 thf(func_def_1, type, mrel: mworld > mworld > $o).
% 2.37/1.51 thf(func_def_2, type, mactual: mworld).
% 2.37/1.51 thf(func_def_3, type, mlocal: (mworld > $o) > $o).
% 2.37/1.51 thf(func_def_5, type, mnot: (mworld > $o) > mworld > $o).
% 2.37/1.51 thf(func_def_6, type, mand: (mworld > $o) > (mworld > $o) > mworld > $o).
% 2.37/1.51 thf(func_def_7, type, mor: (mworld > $o) > (mworld > $o) > mworld > $o).
% 2.37/1.51 thf(func_def_8, type, mimplies: (mworld > $o) > (mworld > $o) > mworld > $o).
% 2.37/1.51 thf(func_def_9, type, mequiv: (mworld > $o) > (mworld > $o) > mworld > $o).
% 2.37/1.51 thf(func_def_10, type, mbox: (mworld > $o) > mworld > $o).
% 2.37/1.51 thf(func_def_11, type, mdia: (mworld > $o) > mworld > $o).
% 2.37/1.51 thf(func_def_12, type, eiw_di: $i > mworld > $o).
% 2.37/1.51 thf(func_def_13, type, mforall_di: ($i > mworld > $o) > mworld > $o).
% 2.37/1.51 thf(func_def_14, type, mexists_di: ($i > mworld > $o) > mworld > $o).
% 2.37/1.51 thf(func_def_20, type, qmltpeq: $i > $i > mworld > $o).
% 2.37/1.51 thf(func_def_21, type, inductive: $i > mworld > $o).
% 2.37/1.51 thf(func_def_22, type, disjoint: $i > $i > mworld > $o).
% 2.37/1.51 thf(func_def_23, type, function: $i > mworld > $o).
% 2.37/1.51 thf(func_def_24, type, member: $i > $i > mworld > $o).
% 2.37/1.51 thf(func_def_25, type, subclass: $i > $i > mworld > $o).
% 2.37/1.51 thf(func_def_26, type, singleton: $i > $i).
% 2.37/1.51 thf(func_def_27, type, image: $i > $i > $i).
% 2.37/1.51 thf(func_def_28, type, inverse: $i > $i).
% 2.37/1.51 thf(func_def_29, type, rotate: $i > $i).
% 2.37/1.51 thf(func_def_30, type, unordered_pair: $i > $i > $i).
% 2.37/1.51 thf(func_def_31, type, successor: $i > $i).
% 2.37/1.51 thf(func_def_32, type, apply: $i > $i > $i).
% 2.37/1.51 thf(func_def_33, type, power_class: $i > $i).
% 2.37/1.51 thf(func_def_34, type, range_of: $i > $i).
% 2.37/1.51 thf(func_def_35, type, union: $i > $i > $i).
% 2.37/1.51 thf(func_def_36, type, restrict: $i > $i > $i > $i).
% 2.37/1.51 thf(func_def_37, type, second: $i > $i).
% 2.37/1.51 thf(func_def_38, type, ordered_pair: $i > $i > $i).
% 2.37/1.51 thf(func_def_39, type, domain_of: $i > $i).
% 2.37/1.51 thf(func_def_40, type, sum_class: $i > $i).
% 2.37/1.51 thf(func_def_41, type, compose: $i > $i > $i).
% 2.37/1.51 thf(func_def_42, type, intersection: $i > $i > $i).
% 2.37/1.51 thf(func_def_43, type, cross_product: $i > $i > $i).
% 2.37/1.51 thf(func_def_44, type, complement: $i > $i).
% 2.37/1.51 thf(func_def_45, type, flip: $i > $i).
% 2.37/1.51 thf(func_def_46, type, first: $i > $i).
% 2.37/1.51 thf(func_def_62, type, ph1: !>[X0: $tType]:(X0)).
% 2.37/1.51 thf(f293,plain,(
% 2.37/1.51 $false),
% 2.37/1.51 inference(subsumption_resolution,[status(thm)],[f291,f292])).
% 2.37/1.51 thf(f292,plain,(
% 2.37/1.51 ($true = (mlocal @ (mforall_di @ (^[Y0 : $i]: (qmltpeq @ Y0 @ Y0)))))),
% 2.37/1.51 inference(cnf_transformation,[status(thm)],[f245])).
% 2.37/1.51 thf(f245,plain,(
% 2.37/1.51 ($true = (mlocal @ (mforall_di @ (^[Y0 : $i]: (qmltpeq @ Y0 @ Y0)))))),
% 2.37/1.51 inference(fool_elimination,[status(thm)],[f244])).
% 2.37/1.51 thf(f244,plain,(
% 2.37/1.51 (mlocal @ (mforall_di @ (^[X0 : $i] : (qmltpeq @ X0 @ X0))))),
% 2.37/1.51 inference(rectify,[status(thm)],[f13])).
% 2.37/1.51 thf(f13,axiom,(
% 2.37/1.51 (mlocal @ (mforall_di @ (^[X5 : $i] : (qmltpeq @ X5 @ X5))))),
% 2.37/1.51 file('/export/starexec/sandbox2/tmp/tmp.xzz2ub9STI/DTF2THF_16121.p',reflexivity)).
% 2.37/1.51 thf(f291,plain,(
% 2.37/1.51 ($true != (mlocal @ (mforall_di @ (^[Y0 : $i]: (qmltpeq @ Y0 @ Y0)))))),
% 2.37/1.51 inference(cnf_transformation,[status(thm)],[f290])).
% 2.37/1.51 thf(f290,plain,(
% 2.37/1.51 ($true != (mlocal @ (mforall_di @ (^[Y0 : $i]: (qmltpeq @ Y0 @ Y0)))))),
% 2.37/1.51 inference(flattening,[status(thm)],[f211])).
% 2.37/1.51 thf(f211,plain,(
% 2.37/1.51 ~($true = (mlocal @ (mforall_di @ (^[Y0 : $i]: (qmltpeq @ Y0 @ Y0)))))),
% 2.37/1.51 inference(fool_elimination,[status(thm)],[f210])).
% 2.37/1.51 thf(f210,plain,(
% 2.37/1.51 ~(mlocal @ (mforall_di @ (^[X0 : $i] : (qmltpeq @ X0 @ X0))))),
% 2.37/1.51 inference(rectify,[status(thm)],[f97])).
% 2.37/1.51 thf(f97,negated_conjecture,(
% 2.37/1.51 ~(mlocal @ (mforall_di @ (^[X5 : $i] : (qmltpeq @ X5 @ X5))))),
% 2.37/1.51 inference(negated_conjecture,[status(cth)],[f96])).
% 2.37/1.51 thf(f96,conjecture,(
% 2.37/1.51 (mlocal @ (mforall_di @ (^[X5 : $i] : (qmltpeq @ X5 @ X5))))),
% 2.37/1.51 file('/export/starexec/sandbox2/tmp/tmp.xzz2ub9STI/DTF2THF_16121.p',reflexivity_0)).
% 2.37/1.51 % SZS output end Proof for DTF2THF_16121
% 2.37/1.51 % (16392)------------------------------
% 2.37/1.51 % (16392)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.37/1.51 % (16392)Termination reason: Refutation
% 2.37/1.52
% 2.37/1.52 % (16392)Memory used [KB]: 5884
% 2.37/1.52 % (16392)Time elapsed: 0.015 s
% 2.37/1.52 % (16392)Instructions burned: 27 (million)
% 2.37/1.52 % (16392)------------------------------
% 2.37/1.52 % (16392)------------------------------
% 2.37/1.52 % (16386)Success in time 0.037 s
%------------------------------------------------------------------------------