%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : TOP001-2 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n007.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:42:35 AM UTC 2025
% Result : Unsatisfiable 1.74s 1.94s
% Output : Proof 1.74s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 79
% Syntax : Number of formulae : 173 ( 37 unt; 0 typ; 0 def)
% Number of atoms : 867 ( 0 equ)
% Maximal formula atoms : 12 ( 5 avg)
% Number of connectives : 1214 ( 558 ~; 576 |; 0 &)
% ( 80 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 6 avg)
% Maximal term depth : 9 ( 2 avg)
% Number of FOOLs : 38 ( 38 fml; 0 var)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of predicates : 10 ( 9 usr; 1 prp; 0-3 aty)
% Number of functors : 7 ( 7 usr; 2 con; 0-3 aty)
% Number of variables : 480 ( 444 !; 0 ?; 480 :)
% Comments :
%------------------------------------------------------------------------------
tff(element_of_collection_type,type,
element_of_collection: ( $i * $i ) > $o ).
tff(f_type,type,
f: $i ).
tff(f10_type,type,
f10: ( $i * $i * $i ) > $i ).
tff(in_1st_set_type,type,
in_1st_set: ( $i * $i ) > $i ).
tff(cx_type,type,
cx: $i ).
tff(f1_type,type,
f1: ( $i * $i ) > $i ).
tff(union_of_members_type,type,
union_of_members: $i > $i ).
tff(top_of_basis_type,type,
top_of_basis: $i > $i ).
tff(element_of_set_type,type,
element_of_set: ( $i * $i ) > $o ).
tff(subset_sets_type,type,
subset_sets: ( $i * $i ) > $o ).
tff(equal_sets_type,type,
equal_sets: ( $i * $i ) > $o ).
tff(basis_type,type,
basis: ( $i * $i ) > $o ).
tff(1,plain,
( ~ subset_sets(union_of_members(top_of_basis(f)),cx)
<=> ~ subset_sets(union_of_members(top_of_basis(f)),cx) ),
inference(rewrite,[status(thm)],[]) ).
tff(2,axiom,
~ subset_sets(union_of_members(top_of_basis(f)),cx),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',lemma_1a_2) ).
tff(3,plain,
~ subset_sets(union_of_members(top_of_basis(f)),cx),
inference(modus_ponens,[status(thm)],[2,1]) ).
tff(4,plain,
^ [Y: $i,X: $i] :
refl(( ( subset_sets(X,Y)
| element_of_set(in_1st_set(X,Y),X) )
<=> ( subset_sets(X,Y)
| element_of_set(in_1st_set(X,Y),X) ) )),
inference(bind,[status(th)],[]) ).
tff(5,plain,
( ! [Y: $i,X: $i] :
( subset_sets(X,Y)
| element_of_set(in_1st_set(X,Y),X) )
<=> ! [Y: $i,X: $i] :
( subset_sets(X,Y)
| element_of_set(in_1st_set(X,Y),X) ) ),
inference(quant_intro,[status(thm)],[4]) ).
tff(6,plain,
( ! [Y: $i,X: $i] :
( subset_sets(X,Y)
| element_of_set(in_1st_set(X,Y),X) )
<=> ! [Y: $i,X: $i] :
( subset_sets(X,Y)
| element_of_set(in_1st_set(X,Y),X) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(7,axiom,
! [Y: $i,X: $i] :
( subset_sets(X,Y)
| element_of_set(in_1st_set(X,Y),X) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',set_theory_4) ).
tff(8,plain,
! [Y: $i,X: $i] :
( subset_sets(X,Y)
| element_of_set(in_1st_set(X,Y),X) ),
inference(modus_ponens,[status(thm)],[7,6]) ).
tff(9,plain,
! [Y: $i,X: $i] :
( subset_sets(X,Y)
| element_of_set(in_1st_set(X,Y),X) ),
inference(skolemize,[status(sab)],[8]) ).
tff(10,plain,
! [Y: $i,X: $i] :
( subset_sets(X,Y)
| element_of_set(in_1st_set(X,Y),X) ),
inference(modus_ponens,[status(thm)],[9,5]) ).
tff(11,plain,
( ( ~ ! [Y: $i,X: $i] :
( subset_sets(X,Y)
| element_of_set(in_1st_set(X,Y),X) )
| subset_sets(union_of_members(top_of_basis(f)),cx)
| element_of_set(in_1st_set(union_of_members(top_of_basis(f)),cx),union_of_members(top_of_basis(f))) )
<=> ( ~ ! [Y: $i,X: $i] :
( subset_sets(X,Y)
| element_of_set(in_1st_set(X,Y),X) )
| subset_sets(union_of_members(top_of_basis(f)),cx)
| element_of_set(in_1st_set(union_of_members(top_of_basis(f)),cx),union_of_members(top_of_basis(f))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(12,plain,
( ~ ! [Y: $i,X: $i] :
( subset_sets(X,Y)
| element_of_set(in_1st_set(X,Y),X) )
| subset_sets(union_of_members(top_of_basis(f)),cx)
| element_of_set(in_1st_set(union_of_members(top_of_basis(f)),cx),union_of_members(top_of_basis(f))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(13,plain,
( ~ ! [Y: $i,X: $i] :
( subset_sets(X,Y)
| element_of_set(in_1st_set(X,Y),X) )
| subset_sets(union_of_members(top_of_basis(f)),cx)
| element_of_set(in_1st_set(union_of_members(top_of_basis(f)),cx),union_of_members(top_of_basis(f))) ),
inference(modus_ponens,[status(thm)],[12,11]) ).
tff(14,plain,
element_of_set(in_1st_set(union_of_members(top_of_basis(f)),cx),union_of_members(top_of_basis(f))),
inference(unit_resolution,[status(thm)],[13,10,3]) ).
tff(15,plain,
^ [Vf: $i,U: $i] :
refl(( ( ~ element_of_set(U,union_of_members(Vf))
| element_of_set(U,f1(Vf,U)) )
<=> ( ~ element_of_set(U,union_of_members(Vf))
| element_of_set(U,f1(Vf,U)) ) )),
inference(bind,[status(th)],[]) ).
tff(16,plain,
( ! [Vf: $i,U: $i] :
( ~ element_of_set(U,union_of_members(Vf))
| element_of_set(U,f1(Vf,U)) )
<=> ! [Vf: $i,U: $i] :
( ~ element_of_set(U,union_of_members(Vf))
| element_of_set(U,f1(Vf,U)) ) ),
inference(quant_intro,[status(thm)],[15]) ).
tff(17,plain,
( ! [Vf: $i,U: $i] :
( ~ element_of_set(U,union_of_members(Vf))
| element_of_set(U,f1(Vf,U)) )
<=> ! [Vf: $i,U: $i] :
( ~ element_of_set(U,union_of_members(Vf))
| element_of_set(U,f1(Vf,U)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(18,axiom,
! [Vf: $i,U: $i] :
( ~ element_of_set(U,union_of_members(Vf))
| element_of_set(U,f1(Vf,U)) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',union_of_members_1) ).
tff(19,plain,
! [Vf: $i,U: $i] :
( ~ element_of_set(U,union_of_members(Vf))
| element_of_set(U,f1(Vf,U)) ),
inference(modus_ponens,[status(thm)],[18,17]) ).
tff(20,plain,
! [Vf: $i,U: $i] :
( ~ element_of_set(U,union_of_members(Vf))
| element_of_set(U,f1(Vf,U)) ),
inference(skolemize,[status(sab)],[19]) ).
tff(21,plain,
! [Vf: $i,U: $i] :
( ~ element_of_set(U,union_of_members(Vf))
| element_of_set(U,f1(Vf,U)) ),
inference(modus_ponens,[status(thm)],[20,16]) ).
tff(22,plain,
( ( ~ ! [Vf: $i,U: $i] :
( ~ element_of_set(U,union_of_members(Vf))
| element_of_set(U,f1(Vf,U)) )
| ~ element_of_set(in_1st_set(union_of_members(top_of_basis(f)),cx),union_of_members(top_of_basis(f)))
| element_of_set(in_1st_set(union_of_members(top_of_basis(f)),cx),f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx))) )
<=> ( ~ ! [Vf: $i,U: $i] :
( ~ element_of_set(U,union_of_members(Vf))
| element_of_set(U,f1(Vf,U)) )
| ~ element_of_set(in_1st_set(union_of_members(top_of_basis(f)),cx),union_of_members(top_of_basis(f)))
| element_of_set(in_1st_set(union_of_members(top_of_basis(f)),cx),f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(23,plain,
( ~ ! [Vf: $i,U: $i] :
( ~ element_of_set(U,union_of_members(Vf))
| element_of_set(U,f1(Vf,U)) )
| ~ element_of_set(in_1st_set(union_of_members(top_of_basis(f)),cx),union_of_members(top_of_basis(f)))
| element_of_set(in_1st_set(union_of_members(top_of_basis(f)),cx),f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(24,plain,
( ~ ! [Vf: $i,U: $i] :
( ~ element_of_set(U,union_of_members(Vf))
| element_of_set(U,f1(Vf,U)) )
| ~ element_of_set(in_1st_set(union_of_members(top_of_basis(f)),cx),union_of_members(top_of_basis(f)))
| element_of_set(in_1st_set(union_of_members(top_of_basis(f)),cx),f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx))) ),
inference(modus_ponens,[status(thm)],[23,22]) ).
tff(25,plain,
element_of_set(in_1st_set(union_of_members(top_of_basis(f)),cx),f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx))),
inference(unit_resolution,[status(thm)],[24,21,14]) ).
tff(26,plain,
^ [Y: $i,X: $i] :
refl(( ( subset_sets(X,Y)
| ~ element_of_set(in_1st_set(X,Y),Y) )
<=> ( subset_sets(X,Y)
| ~ element_of_set(in_1st_set(X,Y),Y) ) )),
inference(bind,[status(th)],[]) ).
tff(27,plain,
( ! [Y: $i,X: $i] :
( subset_sets(X,Y)
| ~ element_of_set(in_1st_set(X,Y),Y) )
<=> ! [Y: $i,X: $i] :
( subset_sets(X,Y)
| ~ element_of_set(in_1st_set(X,Y),Y) ) ),
inference(quant_intro,[status(thm)],[26]) ).
tff(28,plain,
( ! [Y: $i,X: $i] :
( subset_sets(X,Y)
| ~ element_of_set(in_1st_set(X,Y),Y) )
<=> ! [Y: $i,X: $i] :
( subset_sets(X,Y)
| ~ element_of_set(in_1st_set(X,Y),Y) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(29,axiom,
! [Y: $i,X: $i] :
( subset_sets(X,Y)
| ~ element_of_set(in_1st_set(X,Y),Y) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',set_theory_5) ).
tff(30,plain,
! [Y: $i,X: $i] :
( subset_sets(X,Y)
| ~ element_of_set(in_1st_set(X,Y),Y) ),
inference(modus_ponens,[status(thm)],[29,28]) ).
tff(31,plain,
! [Y: $i,X: $i] :
( subset_sets(X,Y)
| ~ element_of_set(in_1st_set(X,Y),Y) ),
inference(skolemize,[status(sab)],[30]) ).
tff(32,plain,
! [Y: $i,X: $i] :
( subset_sets(X,Y)
| ~ element_of_set(in_1st_set(X,Y),Y) ),
inference(modus_ponens,[status(thm)],[31,27]) ).
tff(33,plain,
( ( ~ ! [Y: $i,X: $i] :
( subset_sets(X,Y)
| ~ element_of_set(in_1st_set(X,Y),Y) )
| subset_sets(union_of_members(top_of_basis(f)),cx)
| ~ element_of_set(in_1st_set(union_of_members(top_of_basis(f)),cx),cx) )
<=> ( ~ ! [Y: $i,X: $i] :
( subset_sets(X,Y)
| ~ element_of_set(in_1st_set(X,Y),Y) )
| subset_sets(union_of_members(top_of_basis(f)),cx)
| ~ element_of_set(in_1st_set(union_of_members(top_of_basis(f)),cx),cx) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(34,plain,
( ~ ! [Y: $i,X: $i] :
( subset_sets(X,Y)
| ~ element_of_set(in_1st_set(X,Y),Y) )
| subset_sets(union_of_members(top_of_basis(f)),cx)
| ~ element_of_set(in_1st_set(union_of_members(top_of_basis(f)),cx),cx) ),
inference(quant_inst,[status(thm)],[]) ).
tff(35,plain,
( ~ ! [Y: $i,X: $i] :
( subset_sets(X,Y)
| ~ element_of_set(in_1st_set(X,Y),Y) )
| subset_sets(union_of_members(top_of_basis(f)),cx)
| ~ element_of_set(in_1st_set(union_of_members(top_of_basis(f)),cx),cx) ),
inference(modus_ponens,[status(thm)],[34,33]) ).
tff(36,plain,
~ element_of_set(in_1st_set(union_of_members(top_of_basis(f)),cx),cx),
inference(unit_resolution,[status(thm)],[35,32,3]) ).
tff(37,plain,
^ [Y: $i,U: $i,X: $i] :
refl(( ( element_of_set(U,Y)
| ~ element_of_set(U,X)
| ~ subset_sets(X,Y) )
<=> ( element_of_set(U,Y)
| ~ element_of_set(U,X)
| ~ subset_sets(X,Y) ) )),
inference(bind,[status(th)],[]) ).
tff(38,plain,
( ! [Y: $i,U: $i,X: $i] :
( element_of_set(U,Y)
| ~ element_of_set(U,X)
| ~ subset_sets(X,Y) )
<=> ! [Y: $i,U: $i,X: $i] :
( element_of_set(U,Y)
| ~ element_of_set(U,X)
| ~ subset_sets(X,Y) ) ),
inference(quant_intro,[status(thm)],[37]) ).
tff(39,plain,
( ! [Y: $i,U: $i,X: $i] :
( element_of_set(U,Y)
| ~ element_of_set(U,X)
| ~ subset_sets(X,Y) )
<=> ! [Y: $i,U: $i,X: $i] :
( element_of_set(U,Y)
| ~ element_of_set(U,X)
| ~ subset_sets(X,Y) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(40,plain,
^ [Y: $i,U: $i,X: $i] :
trans(monotonicity(rewrite(( ( ~ subset_sets(X,Y)
| ~ element_of_set(U,X) )
<=> ( ~ element_of_set(U,X)
| ~ subset_sets(X,Y) ) )),
( ( ~ subset_sets(X,Y)
| ~ element_of_set(U,X)
| element_of_set(U,Y) )
<=> ( ~ element_of_set(U,X)
| ~ subset_sets(X,Y)
| element_of_set(U,Y) ) )),
rewrite(( ( ~ element_of_set(U,X)
| ~ subset_sets(X,Y)
| element_of_set(U,Y) )
<=> ( element_of_set(U,Y)
| ~ element_of_set(U,X)
| ~ subset_sets(X,Y) ) )),
( ( ~ subset_sets(X,Y)
| ~ element_of_set(U,X)
| element_of_set(U,Y) )
<=> ( element_of_set(U,Y)
| ~ element_of_set(U,X)
| ~ subset_sets(X,Y) ) )),
inference(bind,[status(th)],[]) ).
tff(41,plain,
( ! [Y: $i,U: $i,X: $i] :
( ~ subset_sets(X,Y)
| ~ element_of_set(U,X)
| element_of_set(U,Y) )
<=> ! [Y: $i,U: $i,X: $i] :
( element_of_set(U,Y)
| ~ element_of_set(U,X)
| ~ subset_sets(X,Y) ) ),
inference(quant_intro,[status(thm)],[40]) ).
tff(42,axiom,
! [Y: $i,U: $i,X: $i] :
( ~ subset_sets(X,Y)
| ~ element_of_set(U,X)
| element_of_set(U,Y) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',set_theory_2) ).
tff(43,plain,
! [Y: $i,U: $i,X: $i] :
( element_of_set(U,Y)
| ~ element_of_set(U,X)
| ~ subset_sets(X,Y) ),
inference(modus_ponens,[status(thm)],[42,41]) ).
tff(44,plain,
! [Y: $i,U: $i,X: $i] :
( element_of_set(U,Y)
| ~ element_of_set(U,X)
| ~ subset_sets(X,Y) ),
inference(modus_ponens,[status(thm)],[43,39]) ).
tff(45,plain,
! [Y: $i,U: $i,X: $i] :
( element_of_set(U,Y)
| ~ element_of_set(U,X)
| ~ subset_sets(X,Y) ),
inference(skolemize,[status(sab)],[44]) ).
tff(46,plain,
! [Y: $i,U: $i,X: $i] :
( element_of_set(U,Y)
| ~ element_of_set(U,X)
| ~ subset_sets(X,Y) ),
inference(modus_ponens,[status(thm)],[45,38]) ).
tff(47,plain,
( ( ~ ! [Y: $i,U: $i,X: $i] :
( element_of_set(U,Y)
| ~ element_of_set(U,X)
| ~ subset_sets(X,Y) )
| element_of_set(in_1st_set(union_of_members(top_of_basis(f)),cx),cx)
| ~ element_of_set(in_1st_set(union_of_members(top_of_basis(f)),cx),f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)))
| ~ subset_sets(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx) )
<=> ( ~ ! [Y: $i,U: $i,X: $i] :
( element_of_set(U,Y)
| ~ element_of_set(U,X)
| ~ subset_sets(X,Y) )
| element_of_set(in_1st_set(union_of_members(top_of_basis(f)),cx),cx)
| ~ element_of_set(in_1st_set(union_of_members(top_of_basis(f)),cx),f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)))
| ~ subset_sets(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(48,plain,
( ~ ! [Y: $i,U: $i,X: $i] :
( element_of_set(U,Y)
| ~ element_of_set(U,X)
| ~ subset_sets(X,Y) )
| element_of_set(in_1st_set(union_of_members(top_of_basis(f)),cx),cx)
| ~ element_of_set(in_1st_set(union_of_members(top_of_basis(f)),cx),f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)))
| ~ subset_sets(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx) ),
inference(quant_inst,[status(thm)],[]) ).
tff(49,plain,
( ~ ! [Y: $i,U: $i,X: $i] :
( element_of_set(U,Y)
| ~ element_of_set(U,X)
| ~ subset_sets(X,Y) )
| element_of_set(in_1st_set(union_of_members(top_of_basis(f)),cx),cx)
| ~ element_of_set(in_1st_set(union_of_members(top_of_basis(f)),cx),f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)))
| ~ subset_sets(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx) ),
inference(modus_ponens,[status(thm)],[48,47]) ).
tff(50,plain,
~ subset_sets(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),
inference(unit_resolution,[status(thm)],[49,46,36,25]) ).
tff(51,plain,
( ( ~ ! [Y: $i,X: $i] :
( subset_sets(X,Y)
| element_of_set(in_1st_set(X,Y),X) )
| subset_sets(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx))) )
<=> ( ~ ! [Y: $i,X: $i] :
( subset_sets(X,Y)
| element_of_set(in_1st_set(X,Y),X) )
| subset_sets(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(52,plain,
( ~ ! [Y: $i,X: $i] :
( subset_sets(X,Y)
| element_of_set(in_1st_set(X,Y),X) )
| subset_sets(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(53,plain,
( ~ ! [Y: $i,X: $i] :
( subset_sets(X,Y)
| element_of_set(in_1st_set(X,Y),X) )
| subset_sets(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx))) ),
inference(modus_ponens,[status(thm)],[52,51]) ).
tff(54,plain,
element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx))),
inference(unit_resolution,[status(thm)],[53,10,50]) ).
tff(55,plain,
^ [Vf: $i,U: $i] :
refl(( ( ~ element_of_set(U,union_of_members(Vf))
| element_of_collection(f1(Vf,U),Vf) )
<=> ( ~ element_of_set(U,union_of_members(Vf))
| element_of_collection(f1(Vf,U),Vf) ) )),
inference(bind,[status(th)],[]) ).
tff(56,plain,
( ! [Vf: $i,U: $i] :
( ~ element_of_set(U,union_of_members(Vf))
| element_of_collection(f1(Vf,U),Vf) )
<=> ! [Vf: $i,U: $i] :
( ~ element_of_set(U,union_of_members(Vf))
| element_of_collection(f1(Vf,U),Vf) ) ),
inference(quant_intro,[status(thm)],[55]) ).
tff(57,plain,
( ! [Vf: $i,U: $i] :
( ~ element_of_set(U,union_of_members(Vf))
| element_of_collection(f1(Vf,U),Vf) )
<=> ! [Vf: $i,U: $i] :
( ~ element_of_set(U,union_of_members(Vf))
| element_of_collection(f1(Vf,U),Vf) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(58,axiom,
! [Vf: $i,U: $i] :
( ~ element_of_set(U,union_of_members(Vf))
| element_of_collection(f1(Vf,U),Vf) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',union_of_members_2) ).
tff(59,plain,
! [Vf: $i,U: $i] :
( ~ element_of_set(U,union_of_members(Vf))
| element_of_collection(f1(Vf,U),Vf) ),
inference(modus_ponens,[status(thm)],[58,57]) ).
tff(60,plain,
! [Vf: $i,U: $i] :
( ~ element_of_set(U,union_of_members(Vf))
| element_of_collection(f1(Vf,U),Vf) ),
inference(skolemize,[status(sab)],[59]) ).
tff(61,plain,
! [Vf: $i,U: $i] :
( ~ element_of_set(U,union_of_members(Vf))
| element_of_collection(f1(Vf,U),Vf) ),
inference(modus_ponens,[status(thm)],[60,56]) ).
tff(62,plain,
( ( ~ ! [Vf: $i,U: $i] :
( ~ element_of_set(U,union_of_members(Vf))
| element_of_collection(f1(Vf,U),Vf) )
| ~ element_of_set(in_1st_set(union_of_members(top_of_basis(f)),cx),union_of_members(top_of_basis(f)))
| element_of_collection(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),top_of_basis(f)) )
<=> ( ~ ! [Vf: $i,U: $i] :
( ~ element_of_set(U,union_of_members(Vf))
| element_of_collection(f1(Vf,U),Vf) )
| ~ element_of_set(in_1st_set(union_of_members(top_of_basis(f)),cx),union_of_members(top_of_basis(f)))
| element_of_collection(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),top_of_basis(f)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(63,plain,
( ~ ! [Vf: $i,U: $i] :
( ~ element_of_set(U,union_of_members(Vf))
| element_of_collection(f1(Vf,U),Vf) )
| ~ element_of_set(in_1st_set(union_of_members(top_of_basis(f)),cx),union_of_members(top_of_basis(f)))
| element_of_collection(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),top_of_basis(f)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(64,plain,
( ~ ! [Vf: $i,U: $i] :
( ~ element_of_set(U,union_of_members(Vf))
| element_of_collection(f1(Vf,U),Vf) )
| ~ element_of_set(in_1st_set(union_of_members(top_of_basis(f)),cx),union_of_members(top_of_basis(f)))
| element_of_collection(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),top_of_basis(f)) ),
inference(modus_ponens,[status(thm)],[63,62]) ).
tff(65,plain,
element_of_collection(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),top_of_basis(f)),
inference(unit_resolution,[status(thm)],[64,61,14]) ).
tff(66,plain,
^ [Vf: $i,Uu1: $i,U: $i] :
refl(( ( ~ element_of_collection(Uu1,Vf)
| ~ element_of_set(U,Uu1)
| element_of_set(U,union_of_members(Vf)) )
<=> ( ~ element_of_collection(Uu1,Vf)
| ~ element_of_set(U,Uu1)
| element_of_set(U,union_of_members(Vf)) ) )),
inference(bind,[status(th)],[]) ).
tff(67,plain,
( ! [Vf: $i,Uu1: $i,U: $i] :
( ~ element_of_collection(Uu1,Vf)
| ~ element_of_set(U,Uu1)
| element_of_set(U,union_of_members(Vf)) )
<=> ! [Vf: $i,Uu1: $i,U: $i] :
( ~ element_of_collection(Uu1,Vf)
| ~ element_of_set(U,Uu1)
| element_of_set(U,union_of_members(Vf)) ) ),
inference(quant_intro,[status(thm)],[66]) ).
tff(68,plain,
( ! [Vf: $i,Uu1: $i,U: $i] :
( ~ element_of_collection(Uu1,Vf)
| ~ element_of_set(U,Uu1)
| element_of_set(U,union_of_members(Vf)) )
<=> ! [Vf: $i,Uu1: $i,U: $i] :
( ~ element_of_collection(Uu1,Vf)
| ~ element_of_set(U,Uu1)
| element_of_set(U,union_of_members(Vf)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(69,plain,
^ [Vf: $i,Uu1: $i,U: $i] :
trans(monotonicity(rewrite(( ( element_of_set(U,union_of_members(Vf))
| ~ element_of_set(U,Uu1) )
<=> ( ~ element_of_set(U,Uu1)
| element_of_set(U,union_of_members(Vf)) ) )),
( ( element_of_set(U,union_of_members(Vf))
| ~ element_of_set(U,Uu1)
| ~ element_of_collection(Uu1,Vf) )
<=> ( ~ element_of_set(U,Uu1)
| element_of_set(U,union_of_members(Vf))
| ~ element_of_collection(Uu1,Vf) ) )),
rewrite(( ( ~ element_of_set(U,Uu1)
| element_of_set(U,union_of_members(Vf))
| ~ element_of_collection(Uu1,Vf) )
<=> ( ~ element_of_collection(Uu1,Vf)
| ~ element_of_set(U,Uu1)
| element_of_set(U,union_of_members(Vf)) ) )),
( ( element_of_set(U,union_of_members(Vf))
| ~ element_of_set(U,Uu1)
| ~ element_of_collection(Uu1,Vf) )
<=> ( ~ element_of_collection(Uu1,Vf)
| ~ element_of_set(U,Uu1)
| element_of_set(U,union_of_members(Vf)) ) )),
inference(bind,[status(th)],[]) ).
tff(70,plain,
( ! [Vf: $i,Uu1: $i,U: $i] :
( element_of_set(U,union_of_members(Vf))
| ~ element_of_set(U,Uu1)
| ~ element_of_collection(Uu1,Vf) )
<=> ! [Vf: $i,Uu1: $i,U: $i] :
( ~ element_of_collection(Uu1,Vf)
| ~ element_of_set(U,Uu1)
| element_of_set(U,union_of_members(Vf)) ) ),
inference(quant_intro,[status(thm)],[69]) ).
tff(71,axiom,
! [Vf: $i,Uu1: $i,U: $i] :
( element_of_set(U,union_of_members(Vf))
| ~ element_of_set(U,Uu1)
| ~ element_of_collection(Uu1,Vf) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',union_of_members_3) ).
tff(72,plain,
! [Vf: $i,Uu1: $i,U: $i] :
( ~ element_of_collection(Uu1,Vf)
| ~ element_of_set(U,Uu1)
| element_of_set(U,union_of_members(Vf)) ),
inference(modus_ponens,[status(thm)],[71,70]) ).
tff(73,plain,
! [Vf: $i,Uu1: $i,U: $i] :
( ~ element_of_collection(Uu1,Vf)
| ~ element_of_set(U,Uu1)
| element_of_set(U,union_of_members(Vf)) ),
inference(modus_ponens,[status(thm)],[72,68]) ).
tff(74,plain,
! [Vf: $i,Uu1: $i,U: $i] :
( ~ element_of_collection(Uu1,Vf)
| ~ element_of_set(U,Uu1)
| element_of_set(U,union_of_members(Vf)) ),
inference(skolemize,[status(sab)],[73]) ).
tff(75,plain,
! [Vf: $i,Uu1: $i,U: $i] :
( ~ element_of_collection(Uu1,Vf)
| ~ element_of_set(U,Uu1)
| element_of_set(U,union_of_members(Vf)) ),
inference(modus_ponens,[status(thm)],[74,67]) ).
tff(76,plain,
( ( ~ ! [Vf: $i,Uu1: $i,U: $i] :
( ~ element_of_collection(Uu1,Vf)
| ~ element_of_set(U,Uu1)
| element_of_set(U,union_of_members(Vf)) )
| ~ element_of_collection(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),top_of_basis(f))
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)))
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),union_of_members(top_of_basis(f))) )
<=> ( ~ ! [Vf: $i,Uu1: $i,U: $i] :
( ~ element_of_collection(Uu1,Vf)
| ~ element_of_set(U,Uu1)
| element_of_set(U,union_of_members(Vf)) )
| ~ element_of_collection(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),top_of_basis(f))
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)))
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),union_of_members(top_of_basis(f))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(77,plain,
( ~ ! [Vf: $i,Uu1: $i,U: $i] :
( ~ element_of_collection(Uu1,Vf)
| ~ element_of_set(U,Uu1)
| element_of_set(U,union_of_members(Vf)) )
| ~ element_of_collection(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),top_of_basis(f))
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)))
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),union_of_members(top_of_basis(f))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(78,plain,
( ~ ! [Vf: $i,Uu1: $i,U: $i] :
( ~ element_of_collection(Uu1,Vf)
| ~ element_of_set(U,Uu1)
| element_of_set(U,union_of_members(Vf)) )
| ~ element_of_collection(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),top_of_basis(f))
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)))
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),union_of_members(top_of_basis(f))) ),
inference(modus_ponens,[status(thm)],[77,76]) ).
tff(79,plain,
element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),union_of_members(top_of_basis(f))),
inference(unit_resolution,[status(thm)],[78,75,65,54]) ).
tff(80,plain,
( ( ~ ! [Vf: $i,U: $i] :
( ~ element_of_set(U,union_of_members(Vf))
| element_of_set(U,f1(Vf,U)) )
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),union_of_members(top_of_basis(f)))
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx))) )
<=> ( ~ ! [Vf: $i,U: $i] :
( ~ element_of_set(U,union_of_members(Vf))
| element_of_set(U,f1(Vf,U)) )
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),union_of_members(top_of_basis(f)))
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(81,plain,
( ~ ! [Vf: $i,U: $i] :
( ~ element_of_set(U,union_of_members(Vf))
| element_of_set(U,f1(Vf,U)) )
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),union_of_members(top_of_basis(f)))
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(82,plain,
( ~ ! [Vf: $i,U: $i] :
( ~ element_of_set(U,union_of_members(Vf))
| element_of_set(U,f1(Vf,U)) )
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),union_of_members(top_of_basis(f)))
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx))) ),
inference(modus_ponens,[status(thm)],[81,80]) ).
tff(83,plain,
element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx))),
inference(unit_resolution,[status(thm)],[82,21,79]) ).
tff(84,plain,
( ( ~ ! [Y: $i,X: $i] :
( subset_sets(X,Y)
| ~ element_of_set(in_1st_set(X,Y),Y) )
| subset_sets(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),cx) )
<=> ( ~ ! [Y: $i,X: $i] :
( subset_sets(X,Y)
| ~ element_of_set(in_1st_set(X,Y),Y) )
| subset_sets(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),cx) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(85,plain,
( ~ ! [Y: $i,X: $i] :
( subset_sets(X,Y)
| ~ element_of_set(in_1st_set(X,Y),Y) )
| subset_sets(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),cx) ),
inference(quant_inst,[status(thm)],[]) ).
tff(86,plain,
( ~ ! [Y: $i,X: $i] :
( subset_sets(X,Y)
| ~ element_of_set(in_1st_set(X,Y),Y) )
| subset_sets(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),cx) ),
inference(modus_ponens,[status(thm)],[85,84]) ).
tff(87,plain,
~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),cx),
inference(unit_resolution,[status(thm)],[86,32,50]) ).
tff(88,plain,
( ( ~ ! [Y: $i,U: $i,X: $i] :
( element_of_set(U,Y)
| ~ element_of_set(U,X)
| ~ subset_sets(X,Y) )
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),cx)
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)))
| ~ subset_sets(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx) )
<=> ( ~ ! [Y: $i,U: $i,X: $i] :
( element_of_set(U,Y)
| ~ element_of_set(U,X)
| ~ subset_sets(X,Y) )
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),cx)
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)))
| ~ subset_sets(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(89,plain,
( ~ ! [Y: $i,U: $i,X: $i] :
( element_of_set(U,Y)
| ~ element_of_set(U,X)
| ~ subset_sets(X,Y) )
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),cx)
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)))
| ~ subset_sets(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx) ),
inference(quant_inst,[status(thm)],[]) ).
tff(90,plain,
( ~ ! [Y: $i,U: $i,X: $i] :
( element_of_set(U,Y)
| ~ element_of_set(U,X)
| ~ subset_sets(X,Y) )
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),cx)
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)))
| ~ subset_sets(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx) ),
inference(modus_ponens,[status(thm)],[89,88]) ).
tff(91,plain,
~ subset_sets(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),
inference(unit_resolution,[status(thm)],[90,46,87,83]) ).
tff(92,plain,
( ( ~ ! [Y: $i,X: $i] :
( subset_sets(X,Y)
| element_of_set(in_1st_set(X,Y),X) )
| subset_sets(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx)
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx))) )
<=> ( ~ ! [Y: $i,X: $i] :
( subset_sets(X,Y)
| element_of_set(in_1st_set(X,Y),X) )
| subset_sets(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx)
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(93,plain,
( ~ ! [Y: $i,X: $i] :
( subset_sets(X,Y)
| element_of_set(in_1st_set(X,Y),X) )
| subset_sets(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx)
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx))) ),
inference(quant_inst,[status(thm)],[]) ).
tff(94,plain,
( ~ ! [Y: $i,X: $i] :
( subset_sets(X,Y)
| element_of_set(in_1st_set(X,Y),X) )
| subset_sets(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx)
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx))) ),
inference(modus_ponens,[status(thm)],[93,92]) ).
tff(95,plain,
element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx))),
inference(unit_resolution,[status(thm)],[94,10,91]) ).
tff(96,plain,
( ( ~ ! [Vf: $i,U: $i] :
( ~ element_of_set(U,union_of_members(Vf))
| element_of_collection(f1(Vf,U),Vf) )
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),union_of_members(top_of_basis(f)))
| element_of_collection(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),top_of_basis(f)) )
<=> ( ~ ! [Vf: $i,U: $i] :
( ~ element_of_set(U,union_of_members(Vf))
| element_of_collection(f1(Vf,U),Vf) )
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),union_of_members(top_of_basis(f)))
| element_of_collection(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),top_of_basis(f)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(97,plain,
( ~ ! [Vf: $i,U: $i] :
( ~ element_of_set(U,union_of_members(Vf))
| element_of_collection(f1(Vf,U),Vf) )
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),union_of_members(top_of_basis(f)))
| element_of_collection(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),top_of_basis(f)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(98,plain,
( ~ ! [Vf: $i,U: $i] :
( ~ element_of_set(U,union_of_members(Vf))
| element_of_collection(f1(Vf,U),Vf) )
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx),union_of_members(top_of_basis(f)))
| element_of_collection(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),top_of_basis(f)) ),
inference(modus_ponens,[status(thm)],[97,96]) ).
tff(99,plain,
element_of_collection(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),top_of_basis(f)),
inference(unit_resolution,[status(thm)],[98,61,79]) ).
tff(100,plain,
^ [Vf: $i,U: $i,X: $i] :
refl(( ( ~ element_of_set(X,U)
| ~ element_of_collection(U,top_of_basis(Vf))
| element_of_collection(f10(Vf,U,X),Vf) )
<=> ( ~ element_of_set(X,U)
| ~ element_of_collection(U,top_of_basis(Vf))
| element_of_collection(f10(Vf,U,X),Vf) ) )),
inference(bind,[status(th)],[]) ).
tff(101,plain,
( ! [Vf: $i,U: $i,X: $i] :
( ~ element_of_set(X,U)
| ~ element_of_collection(U,top_of_basis(Vf))
| element_of_collection(f10(Vf,U,X),Vf) )
<=> ! [Vf: $i,U: $i,X: $i] :
( ~ element_of_set(X,U)
| ~ element_of_collection(U,top_of_basis(Vf))
| element_of_collection(f10(Vf,U,X),Vf) ) ),
inference(quant_intro,[status(thm)],[100]) ).
tff(102,plain,
( ! [Vf: $i,U: $i,X: $i] :
( ~ element_of_set(X,U)
| ~ element_of_collection(U,top_of_basis(Vf))
| element_of_collection(f10(Vf,U,X),Vf) )
<=> ! [Vf: $i,U: $i,X: $i] :
( ~ element_of_set(X,U)
| ~ element_of_collection(U,top_of_basis(Vf))
| element_of_collection(f10(Vf,U,X),Vf) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(103,plain,
^ [Vf: $i,U: $i,X: $i] :
trans(monotonicity(rewrite(( ( ~ element_of_collection(U,top_of_basis(Vf))
| ~ element_of_set(X,U) )
<=> ( ~ element_of_set(X,U)
| ~ element_of_collection(U,top_of_basis(Vf)) ) )),
( ( ~ element_of_collection(U,top_of_basis(Vf))
| ~ element_of_set(X,U)
| element_of_collection(f10(Vf,U,X),Vf) )
<=> ( ~ element_of_set(X,U)
| ~ element_of_collection(U,top_of_basis(Vf))
| element_of_collection(f10(Vf,U,X),Vf) ) )),
rewrite(( ( ~ element_of_set(X,U)
| ~ element_of_collection(U,top_of_basis(Vf))
| element_of_collection(f10(Vf,U,X),Vf) )
<=> ( ~ element_of_set(X,U)
| ~ element_of_collection(U,top_of_basis(Vf))
| element_of_collection(f10(Vf,U,X),Vf) ) )),
( ( ~ element_of_collection(U,top_of_basis(Vf))
| ~ element_of_set(X,U)
| element_of_collection(f10(Vf,U,X),Vf) )
<=> ( ~ element_of_set(X,U)
| ~ element_of_collection(U,top_of_basis(Vf))
| element_of_collection(f10(Vf,U,X),Vf) ) )),
inference(bind,[status(th)],[]) ).
tff(104,plain,
( ! [Vf: $i,U: $i,X: $i] :
( ~ element_of_collection(U,top_of_basis(Vf))
| ~ element_of_set(X,U)
| element_of_collection(f10(Vf,U,X),Vf) )
<=> ! [Vf: $i,U: $i,X: $i] :
( ~ element_of_set(X,U)
| ~ element_of_collection(U,top_of_basis(Vf))
| element_of_collection(f10(Vf,U,X),Vf) ) ),
inference(quant_intro,[status(thm)],[103]) ).
tff(105,axiom,
! [Vf: $i,U: $i,X: $i] :
( ~ element_of_collection(U,top_of_basis(Vf))
| ~ element_of_set(X,U)
| element_of_collection(f10(Vf,U,X),Vf) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',topology_generated_38) ).
tff(106,plain,
! [Vf: $i,U: $i,X: $i] :
( ~ element_of_set(X,U)
| ~ element_of_collection(U,top_of_basis(Vf))
| element_of_collection(f10(Vf,U,X),Vf) ),
inference(modus_ponens,[status(thm)],[105,104]) ).
tff(107,plain,
! [Vf: $i,U: $i,X: $i] :
( ~ element_of_set(X,U)
| ~ element_of_collection(U,top_of_basis(Vf))
| element_of_collection(f10(Vf,U,X),Vf) ),
inference(modus_ponens,[status(thm)],[106,102]) ).
tff(108,plain,
! [Vf: $i,U: $i,X: $i] :
( ~ element_of_set(X,U)
| ~ element_of_collection(U,top_of_basis(Vf))
| element_of_collection(f10(Vf,U,X),Vf) ),
inference(skolemize,[status(sab)],[107]) ).
tff(109,plain,
! [Vf: $i,U: $i,X: $i] :
( ~ element_of_set(X,U)
| ~ element_of_collection(U,top_of_basis(Vf))
| element_of_collection(f10(Vf,U,X),Vf) ),
inference(modus_ponens,[status(thm)],[108,101]) ).
tff(110,plain,
( ( ~ ! [Vf: $i,U: $i,X: $i] :
( ~ element_of_set(X,U)
| ~ element_of_collection(U,top_of_basis(Vf))
| element_of_collection(f10(Vf,U,X),Vf) )
| ~ element_of_collection(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),top_of_basis(f))
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)))
| element_of_collection(f10(f,f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx)),f) )
<=> ( ~ ! [Vf: $i,U: $i,X: $i] :
( ~ element_of_set(X,U)
| ~ element_of_collection(U,top_of_basis(Vf))
| element_of_collection(f10(Vf,U,X),Vf) )
| ~ element_of_collection(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),top_of_basis(f))
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)))
| element_of_collection(f10(f,f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx)),f) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(111,plain,
( ( ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)))
| ~ element_of_collection(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),top_of_basis(f))
| element_of_collection(f10(f,f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx)),f) )
<=> ( ~ element_of_collection(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),top_of_basis(f))
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)))
| element_of_collection(f10(f,f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx)),f) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(112,plain,
( ( ~ ! [Vf: $i,U: $i,X: $i] :
( ~ element_of_set(X,U)
| ~ element_of_collection(U,top_of_basis(Vf))
| element_of_collection(f10(Vf,U,X),Vf) )
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)))
| ~ element_of_collection(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),top_of_basis(f))
| element_of_collection(f10(f,f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx)),f) )
<=> ( ~ ! [Vf: $i,U: $i,X: $i] :
( ~ element_of_set(X,U)
| ~ element_of_collection(U,top_of_basis(Vf))
| element_of_collection(f10(Vf,U,X),Vf) )
| ~ element_of_collection(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),top_of_basis(f))
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)))
| element_of_collection(f10(f,f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx)),f) ) ),
inference(monotonicity,[status(thm)],[111]) ).
tff(113,plain,
( ( ~ ! [Vf: $i,U: $i,X: $i] :
( ~ element_of_set(X,U)
| ~ element_of_collection(U,top_of_basis(Vf))
| element_of_collection(f10(Vf,U,X),Vf) )
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)))
| ~ element_of_collection(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),top_of_basis(f))
| element_of_collection(f10(f,f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx)),f) )
<=> ( ~ ! [Vf: $i,U: $i,X: $i] :
( ~ element_of_set(X,U)
| ~ element_of_collection(U,top_of_basis(Vf))
| element_of_collection(f10(Vf,U,X),Vf) )
| ~ element_of_collection(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),top_of_basis(f))
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)))
| element_of_collection(f10(f,f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx)),f) ) ),
inference(transitivity,[status(thm)],[112,110]) ).
tff(114,plain,
( ~ ! [Vf: $i,U: $i,X: $i] :
( ~ element_of_set(X,U)
| ~ element_of_collection(U,top_of_basis(Vf))
| element_of_collection(f10(Vf,U,X),Vf) )
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)))
| ~ element_of_collection(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),top_of_basis(f))
| element_of_collection(f10(f,f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx)),f) ),
inference(quant_inst,[status(thm)],[]) ).
tff(115,plain,
( ~ ! [Vf: $i,U: $i,X: $i] :
( ~ element_of_set(X,U)
| ~ element_of_collection(U,top_of_basis(Vf))
| element_of_collection(f10(Vf,U,X),Vf) )
| ~ element_of_collection(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),top_of_basis(f))
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)))
| element_of_collection(f10(f,f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx)),f) ),
inference(modus_ponens,[status(thm)],[114,113]) ).
tff(116,plain,
element_of_collection(f10(f,f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx)),f),
inference(unit_resolution,[status(thm)],[115,109,99,95]) ).
tff(117,plain,
^ [Vf: $i,U: $i,X: $i] :
refl(( ( ~ element_of_set(X,U)
| element_of_set(X,f10(Vf,U,X))
| ~ element_of_collection(U,top_of_basis(Vf)) )
<=> ( ~ element_of_set(X,U)
| element_of_set(X,f10(Vf,U,X))
| ~ element_of_collection(U,top_of_basis(Vf)) ) )),
inference(bind,[status(th)],[]) ).
tff(118,plain,
( ! [Vf: $i,U: $i,X: $i] :
( ~ element_of_set(X,U)
| element_of_set(X,f10(Vf,U,X))
| ~ element_of_collection(U,top_of_basis(Vf)) )
<=> ! [Vf: $i,U: $i,X: $i] :
( ~ element_of_set(X,U)
| element_of_set(X,f10(Vf,U,X))
| ~ element_of_collection(U,top_of_basis(Vf)) ) ),
inference(quant_intro,[status(thm)],[117]) ).
tff(119,plain,
( ! [Vf: $i,U: $i,X: $i] :
( ~ element_of_set(X,U)
| element_of_set(X,f10(Vf,U,X))
| ~ element_of_collection(U,top_of_basis(Vf)) )
<=> ! [Vf: $i,U: $i,X: $i] :
( ~ element_of_set(X,U)
| element_of_set(X,f10(Vf,U,X))
| ~ element_of_collection(U,top_of_basis(Vf)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(120,plain,
^ [Vf: $i,U: $i,X: $i] :
trans(monotonicity(rewrite(( ( ~ element_of_collection(U,top_of_basis(Vf))
| ~ element_of_set(X,U) )
<=> ( ~ element_of_set(X,U)
| ~ element_of_collection(U,top_of_basis(Vf)) ) )),
( ( ~ element_of_collection(U,top_of_basis(Vf))
| ~ element_of_set(X,U)
| element_of_set(X,f10(Vf,U,X)) )
<=> ( ~ element_of_set(X,U)
| ~ element_of_collection(U,top_of_basis(Vf))
| element_of_set(X,f10(Vf,U,X)) ) )),
rewrite(( ( ~ element_of_set(X,U)
| ~ element_of_collection(U,top_of_basis(Vf))
| element_of_set(X,f10(Vf,U,X)) )
<=> ( ~ element_of_set(X,U)
| element_of_set(X,f10(Vf,U,X))
| ~ element_of_collection(U,top_of_basis(Vf)) ) )),
( ( ~ element_of_collection(U,top_of_basis(Vf))
| ~ element_of_set(X,U)
| element_of_set(X,f10(Vf,U,X)) )
<=> ( ~ element_of_set(X,U)
| element_of_set(X,f10(Vf,U,X))
| ~ element_of_collection(U,top_of_basis(Vf)) ) )),
inference(bind,[status(th)],[]) ).
tff(121,plain,
( ! [Vf: $i,U: $i,X: $i] :
( ~ element_of_collection(U,top_of_basis(Vf))
| ~ element_of_set(X,U)
| element_of_set(X,f10(Vf,U,X)) )
<=> ! [Vf: $i,U: $i,X: $i] :
( ~ element_of_set(X,U)
| element_of_set(X,f10(Vf,U,X))
| ~ element_of_collection(U,top_of_basis(Vf)) ) ),
inference(quant_intro,[status(thm)],[120]) ).
tff(122,axiom,
! [Vf: $i,U: $i,X: $i] :
( ~ element_of_collection(U,top_of_basis(Vf))
| ~ element_of_set(X,U)
| element_of_set(X,f10(Vf,U,X)) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',topology_generated_37) ).
tff(123,plain,
! [Vf: $i,U: $i,X: $i] :
( ~ element_of_set(X,U)
| element_of_set(X,f10(Vf,U,X))
| ~ element_of_collection(U,top_of_basis(Vf)) ),
inference(modus_ponens,[status(thm)],[122,121]) ).
tff(124,plain,
! [Vf: $i,U: $i,X: $i] :
( ~ element_of_set(X,U)
| element_of_set(X,f10(Vf,U,X))
| ~ element_of_collection(U,top_of_basis(Vf)) ),
inference(modus_ponens,[status(thm)],[123,119]) ).
tff(125,plain,
! [Vf: $i,U: $i,X: $i] :
( ~ element_of_set(X,U)
| element_of_set(X,f10(Vf,U,X))
| ~ element_of_collection(U,top_of_basis(Vf)) ),
inference(skolemize,[status(sab)],[124]) ).
tff(126,plain,
! [Vf: $i,U: $i,X: $i] :
( ~ element_of_set(X,U)
| element_of_set(X,f10(Vf,U,X))
| ~ element_of_collection(U,top_of_basis(Vf)) ),
inference(modus_ponens,[status(thm)],[125,118]) ).
tff(127,plain,
( ( ~ ! [Vf: $i,U: $i,X: $i] :
( ~ element_of_set(X,U)
| element_of_set(X,f10(Vf,U,X))
| ~ element_of_collection(U,top_of_basis(Vf)) )
| ~ element_of_collection(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),top_of_basis(f))
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)))
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f10(f,f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx))) )
<=> ( ~ ! [Vf: $i,U: $i,X: $i] :
( ~ element_of_set(X,U)
| element_of_set(X,f10(Vf,U,X))
| ~ element_of_collection(U,top_of_basis(Vf)) )
| ~ element_of_collection(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),top_of_basis(f))
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)))
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f10(f,f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(128,plain,
( ( ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)))
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f10(f,f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx)))
| ~ element_of_collection(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),top_of_basis(f)) )
<=> ( ~ element_of_collection(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),top_of_basis(f))
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)))
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f10(f,f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx))) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(129,plain,
( ( ~ ! [Vf: $i,U: $i,X: $i] :
( ~ element_of_set(X,U)
| element_of_set(X,f10(Vf,U,X))
| ~ element_of_collection(U,top_of_basis(Vf)) )
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)))
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f10(f,f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx)))
| ~ element_of_collection(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),top_of_basis(f)) )
<=> ( ~ ! [Vf: $i,U: $i,X: $i] :
( ~ element_of_set(X,U)
| element_of_set(X,f10(Vf,U,X))
| ~ element_of_collection(U,top_of_basis(Vf)) )
| ~ element_of_collection(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),top_of_basis(f))
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)))
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f10(f,f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx))) ) ),
inference(monotonicity,[status(thm)],[128]) ).
tff(130,plain,
( ( ~ ! [Vf: $i,U: $i,X: $i] :
( ~ element_of_set(X,U)
| element_of_set(X,f10(Vf,U,X))
| ~ element_of_collection(U,top_of_basis(Vf)) )
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)))
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f10(f,f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx)))
| ~ element_of_collection(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),top_of_basis(f)) )
<=> ( ~ ! [Vf: $i,U: $i,X: $i] :
( ~ element_of_set(X,U)
| element_of_set(X,f10(Vf,U,X))
| ~ element_of_collection(U,top_of_basis(Vf)) )
| ~ element_of_collection(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),top_of_basis(f))
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)))
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f10(f,f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx))) ) ),
inference(transitivity,[status(thm)],[129,127]) ).
tff(131,plain,
( ~ ! [Vf: $i,U: $i,X: $i] :
( ~ element_of_set(X,U)
| element_of_set(X,f10(Vf,U,X))
| ~ element_of_collection(U,top_of_basis(Vf)) )
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)))
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f10(f,f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx)))
| ~ element_of_collection(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),top_of_basis(f)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(132,plain,
( ~ ! [Vf: $i,U: $i,X: $i] :
( ~ element_of_set(X,U)
| element_of_set(X,f10(Vf,U,X))
| ~ element_of_collection(U,top_of_basis(Vf)) )
| ~ element_of_collection(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),top_of_basis(f))
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)))
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f10(f,f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx))) ),
inference(modus_ponens,[status(thm)],[131,130]) ).
tff(133,plain,
element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f10(f,f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx))),
inference(unit_resolution,[status(thm)],[132,126,99,95]) ).
tff(134,plain,
( ( ~ ! [Y: $i,X: $i] :
( subset_sets(X,Y)
| ~ element_of_set(in_1st_set(X,Y),Y) )
| subset_sets(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx)
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),cx) )
<=> ( ~ ! [Y: $i,X: $i] :
( subset_sets(X,Y)
| ~ element_of_set(in_1st_set(X,Y),Y) )
| subset_sets(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx)
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),cx) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(135,plain,
( ~ ! [Y: $i,X: $i] :
( subset_sets(X,Y)
| ~ element_of_set(in_1st_set(X,Y),Y) )
| subset_sets(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx)
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),cx) ),
inference(quant_inst,[status(thm)],[]) ).
tff(136,plain,
( ~ ! [Y: $i,X: $i] :
( subset_sets(X,Y)
| ~ element_of_set(in_1st_set(X,Y),Y) )
| subset_sets(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx)
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),cx) ),
inference(modus_ponens,[status(thm)],[135,134]) ).
tff(137,plain,
~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),cx),
inference(unit_resolution,[status(thm)],[136,32,91]) ).
tff(138,plain,
( basis(cx,f)
<=> basis(cx,f) ),
inference(rewrite,[status(thm)],[]) ).
tff(139,axiom,
basis(cx,f),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',lemma_1a_1) ).
tff(140,plain,
basis(cx,f),
inference(modus_ponens,[status(thm)],[139,138]) ).
tff(141,plain,
^ [Vf: $i,X: $i] :
refl(( ( ~ basis(X,Vf)
| equal_sets(union_of_members(Vf),X) )
<=> ( ~ basis(X,Vf)
| equal_sets(union_of_members(Vf),X) ) )),
inference(bind,[status(th)],[]) ).
tff(142,plain,
( ! [Vf: $i,X: $i] :
( ~ basis(X,Vf)
| equal_sets(union_of_members(Vf),X) )
<=> ! [Vf: $i,X: $i] :
( ~ basis(X,Vf)
| equal_sets(union_of_members(Vf),X) ) ),
inference(quant_intro,[status(thm)],[141]) ).
tff(143,plain,
( ! [Vf: $i,X: $i] :
( ~ basis(X,Vf)
| equal_sets(union_of_members(Vf),X) )
<=> ! [Vf: $i,X: $i] :
( ~ basis(X,Vf)
| equal_sets(union_of_members(Vf),X) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(144,axiom,
! [Vf: $i,X: $i] :
( ~ basis(X,Vf)
| equal_sets(union_of_members(Vf),X) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',basis_for_topology_28) ).
tff(145,plain,
! [Vf: $i,X: $i] :
( ~ basis(X,Vf)
| equal_sets(union_of_members(Vf),X) ),
inference(modus_ponens,[status(thm)],[144,143]) ).
tff(146,plain,
! [Vf: $i,X: $i] :
( ~ basis(X,Vf)
| equal_sets(union_of_members(Vf),X) ),
inference(skolemize,[status(sab)],[145]) ).
tff(147,plain,
! [Vf: $i,X: $i] :
( ~ basis(X,Vf)
| equal_sets(union_of_members(Vf),X) ),
inference(modus_ponens,[status(thm)],[146,142]) ).
tff(148,plain,
( ( ~ ! [Vf: $i,X: $i] :
( ~ basis(X,Vf)
| equal_sets(union_of_members(Vf),X) )
| ~ basis(cx,f)
| equal_sets(union_of_members(f),cx) )
<=> ( ~ ! [Vf: $i,X: $i] :
( ~ basis(X,Vf)
| equal_sets(union_of_members(Vf),X) )
| ~ basis(cx,f)
| equal_sets(union_of_members(f),cx) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(149,plain,
( ~ ! [Vf: $i,X: $i] :
( ~ basis(X,Vf)
| equal_sets(union_of_members(Vf),X) )
| ~ basis(cx,f)
| equal_sets(union_of_members(f),cx) ),
inference(quant_inst,[status(thm)],[]) ).
tff(150,plain,
( ~ ! [Vf: $i,X: $i] :
( ~ basis(X,Vf)
| equal_sets(union_of_members(Vf),X) )
| ~ basis(cx,f)
| equal_sets(union_of_members(f),cx) ),
inference(modus_ponens,[status(thm)],[149,148]) ).
tff(151,plain,
equal_sets(union_of_members(f),cx),
inference(unit_resolution,[status(thm)],[150,147,140]) ).
tff(152,plain,
^ [Y: $i,X: $i] :
refl(( ( ~ equal_sets(X,Y)
| subset_sets(X,Y) )
<=> ( ~ equal_sets(X,Y)
| subset_sets(X,Y) ) )),
inference(bind,[status(th)],[]) ).
tff(153,plain,
( ! [Y: $i,X: $i] :
( ~ equal_sets(X,Y)
| subset_sets(X,Y) )
<=> ! [Y: $i,X: $i] :
( ~ equal_sets(X,Y)
| subset_sets(X,Y) ) ),
inference(quant_intro,[status(thm)],[152]) ).
tff(154,plain,
( ! [Y: $i,X: $i] :
( ~ equal_sets(X,Y)
| subset_sets(X,Y) )
<=> ! [Y: $i,X: $i] :
( ~ equal_sets(X,Y)
| subset_sets(X,Y) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(155,axiom,
! [Y: $i,X: $i] :
( ~ equal_sets(X,Y)
| subset_sets(X,Y) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',set_theory_3) ).
tff(156,plain,
! [Y: $i,X: $i] :
( ~ equal_sets(X,Y)
| subset_sets(X,Y) ),
inference(modus_ponens,[status(thm)],[155,154]) ).
tff(157,plain,
! [Y: $i,X: $i] :
( ~ equal_sets(X,Y)
| subset_sets(X,Y) ),
inference(skolemize,[status(sab)],[156]) ).
tff(158,plain,
! [Y: $i,X: $i] :
( ~ equal_sets(X,Y)
| subset_sets(X,Y) ),
inference(modus_ponens,[status(thm)],[157,153]) ).
tff(159,plain,
( ( ~ ! [Y: $i,X: $i] :
( ~ equal_sets(X,Y)
| subset_sets(X,Y) )
| ~ equal_sets(union_of_members(f),cx)
| subset_sets(union_of_members(f),cx) )
<=> ( ~ ! [Y: $i,X: $i] :
( ~ equal_sets(X,Y)
| subset_sets(X,Y) )
| ~ equal_sets(union_of_members(f),cx)
| subset_sets(union_of_members(f),cx) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(160,plain,
( ~ ! [Y: $i,X: $i] :
( ~ equal_sets(X,Y)
| subset_sets(X,Y) )
| ~ equal_sets(union_of_members(f),cx)
| subset_sets(union_of_members(f),cx) ),
inference(quant_inst,[status(thm)],[]) ).
tff(161,plain,
( ~ ! [Y: $i,X: $i] :
( ~ equal_sets(X,Y)
| subset_sets(X,Y) )
| ~ equal_sets(union_of_members(f),cx)
| subset_sets(union_of_members(f),cx) ),
inference(modus_ponens,[status(thm)],[160,159]) ).
tff(162,plain,
subset_sets(union_of_members(f),cx),
inference(unit_resolution,[status(thm)],[161,158,151]) ).
tff(163,plain,
( ( ~ ! [Y: $i,U: $i,X: $i] :
( element_of_set(U,Y)
| ~ element_of_set(U,X)
| ~ subset_sets(X,Y) )
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),cx)
| ~ subset_sets(union_of_members(f),cx)
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),union_of_members(f)) )
<=> ( ~ ! [Y: $i,U: $i,X: $i] :
( element_of_set(U,Y)
| ~ element_of_set(U,X)
| ~ subset_sets(X,Y) )
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),cx)
| ~ subset_sets(union_of_members(f),cx)
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),union_of_members(f)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(164,plain,
( ( element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),cx)
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),union_of_members(f))
| ~ subset_sets(union_of_members(f),cx) )
<=> ( element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),cx)
| ~ subset_sets(union_of_members(f),cx)
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),union_of_members(f)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(165,plain,
( ( ~ ! [Y: $i,U: $i,X: $i] :
( element_of_set(U,Y)
| ~ element_of_set(U,X)
| ~ subset_sets(X,Y) )
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),cx)
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),union_of_members(f))
| ~ subset_sets(union_of_members(f),cx) )
<=> ( ~ ! [Y: $i,U: $i,X: $i] :
( element_of_set(U,Y)
| ~ element_of_set(U,X)
| ~ subset_sets(X,Y) )
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),cx)
| ~ subset_sets(union_of_members(f),cx)
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),union_of_members(f)) ) ),
inference(monotonicity,[status(thm)],[164]) ).
tff(166,plain,
( ( ~ ! [Y: $i,U: $i,X: $i] :
( element_of_set(U,Y)
| ~ element_of_set(U,X)
| ~ subset_sets(X,Y) )
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),cx)
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),union_of_members(f))
| ~ subset_sets(union_of_members(f),cx) )
<=> ( ~ ! [Y: $i,U: $i,X: $i] :
( element_of_set(U,Y)
| ~ element_of_set(U,X)
| ~ subset_sets(X,Y) )
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),cx)
| ~ subset_sets(union_of_members(f),cx)
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),union_of_members(f)) ) ),
inference(transitivity,[status(thm)],[165,163]) ).
tff(167,plain,
( ~ ! [Y: $i,U: $i,X: $i] :
( element_of_set(U,Y)
| ~ element_of_set(U,X)
| ~ subset_sets(X,Y) )
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),cx)
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),union_of_members(f))
| ~ subset_sets(union_of_members(f),cx) ),
inference(quant_inst,[status(thm)],[]) ).
tff(168,plain,
( ~ ! [Y: $i,U: $i,X: $i] :
( element_of_set(U,Y)
| ~ element_of_set(U,X)
| ~ subset_sets(X,Y) )
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),cx)
| ~ subset_sets(union_of_members(f),cx)
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),union_of_members(f)) ),
inference(modus_ponens,[status(thm)],[167,166]) ).
tff(169,plain,
~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),union_of_members(f)),
inference(unit_resolution,[status(thm)],[168,46,162,137]) ).
tff(170,plain,
( ( ~ ! [Vf: $i,Uu1: $i,U: $i] :
( ~ element_of_collection(Uu1,Vf)
| ~ element_of_set(U,Uu1)
| element_of_set(U,union_of_members(Vf)) )
| ~ element_of_collection(f10(f,f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx)),f)
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f10(f,f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx)))
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),union_of_members(f)) )
<=> ( ~ ! [Vf: $i,Uu1: $i,U: $i] :
( ~ element_of_collection(Uu1,Vf)
| ~ element_of_set(U,Uu1)
| element_of_set(U,union_of_members(Vf)) )
| ~ element_of_collection(f10(f,f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx)),f)
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f10(f,f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx)))
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),union_of_members(f)) ) ),
inference(rewrite,[status(thm)],[]) ).
tff(171,plain,
( ~ ! [Vf: $i,Uu1: $i,U: $i] :
( ~ element_of_collection(Uu1,Vf)
| ~ element_of_set(U,Uu1)
| element_of_set(U,union_of_members(Vf)) )
| ~ element_of_collection(f10(f,f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx)),f)
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f10(f,f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx)))
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),union_of_members(f)) ),
inference(quant_inst,[status(thm)],[]) ).
tff(172,plain,
( ~ ! [Vf: $i,Uu1: $i,U: $i] :
( ~ element_of_collection(Uu1,Vf)
| ~ element_of_set(U,Uu1)
| element_of_set(U,union_of_members(Vf)) )
| ~ element_of_collection(f10(f,f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx)),f)
| ~ element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),f10(f,f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx)))
| element_of_set(in_1st_set(f1(top_of_basis(f),in_1st_set(f1(top_of_basis(f),in_1st_set(union_of_members(top_of_basis(f)),cx)),cx)),cx),union_of_members(f)) ),
inference(modus_ponens,[status(thm)],[171,170]) ).
tff(173,plain,
$false,
inference(unit_resolution,[status(thm)],[172,75,169,133,116]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : TOP001-2 : TPTP v9.0.0. Released v1.0.0.
% 0.03/0.12 % Command : run_E %s %d THM
% 0.12/0.32 % Computer : n007.cluster.edu
% 0.12/0.32 % Model : x86_64 x86_64
% 0.12/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.32 % Memory : 8042.1875MB
% 0.12/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.32 % CPULimit : 300
% 0.12/0.32 % WCLimit : 300
% 0.12/0.32 % DateTime : Fri Jun 20 13:53:20 EDT 2025
% 0.12/0.32 % CPUTime :
% 1.74/1.94 % SZS status Unsatisfiable
% 1.74/1.94 % SZS output start Proof
% See solution above
% 1.74/1.96 % E exiting
%------------------------------------------------------------------------------