TPTP Problem File: SEU770^2.p
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% File : SEU770^2 : TPTP v8.2.0. Released v3.7.0.
% Domain : Set Theory
% Problem : Binary Relations on a Set
% Version : Especial > Reduced > Especial.
% English : (! A:i.! B:i.! phi:i>(i>o).(! x:i.in x A ->
% (? y:i.in y B & phi x y)) -> (! R:i.in R (breln1Set B) ->
% reflwellordering B R -> (! x:i.in x A ->
% singleton (dsetconstr B (^ y:i.phi x y & (! z:i.in z B ->
% phi x z -> in (kpair y z) R))))))
% Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% Source : [Bro08]
% Names : ZFC272l [Bro08]
% Status : Theorem
% Rating : 1.00 v8.2.0, 0.92 v8.1.0, 1.00 v3.7.0
% Syntax : Number of formulae : 32 ( 11 unt; 20 typ; 11 def)
% Number of atoms : 70 ( 14 equ; 0 cnn)
% Maximal formula atoms : 14 ( 5 avg)
% Number of connectives : 162 ( 1 ~; 1 |; 8 &; 118 @)
% ( 0 <=>; 34 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 3 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 32 ( 32 >; 0 *; 0 +; 0 <<)
% Number of symbols : 21 ( 20 usr; 7 con; 0-2 aty)
% Number of variables : 50 ( 16 ^; 32 !; 2 ?; 50 :)
% SPC : TH0_THM_EQU_NAR
% Comments : http://mathgate.info/detsetitem.php?id=498
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thf(in_type,type,
in: $i > $i > $o ).
thf(emptyset_type,type,
emptyset: $i ).
thf(powerset_type,type,
powerset: $i > $i ).
thf(dsetconstr_type,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(dsetconstrI_type,type,
dsetconstrI: $o ).
thf(dsetconstrI,definition,
( dsetconstrI
= ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
( ( in @ Xx @ A )
=> ( ( Xphi @ Xx )
=> ( in @ Xx
@ ( dsetconstr @ A
@ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
thf(dsetconstrEL_type,type,
dsetconstrEL: $o ).
thf(dsetconstrEL,definition,
( dsetconstrEL
= ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
( ( in @ Xx
@ ( dsetconstr @ A
@ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
=> ( in @ Xx @ A ) ) ) ) ).
thf(dsetconstrER_type,type,
dsetconstrER: $o ).
thf(dsetconstrER,definition,
( dsetconstrER
= ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
( ( in @ Xx
@ ( dsetconstr @ A
@ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
=> ( Xphi @ Xx ) ) ) ) ).
thf(nonempty_type,type,
nonempty: $i > $o ).
thf(nonempty,definition,
( nonempty
= ( ^ [Xx: $i] : Xx != emptyset ) ) ).
thf(nonemptyI_type,type,
nonemptyI: $o ).
thf(nonemptyI,definition,
( nonemptyI
= ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
( ( in @ Xx @ A )
=> ( ( Xphi @ Xx )
=> ( nonempty
@ ( dsetconstr @ A
@ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
thf(powersetI_type,type,
powersetI: $o ).
thf(powersetI,definition,
( powersetI
= ( ! [A: $i,B: $i] :
( ! [Xx: $i] :
( ( in @ Xx @ B )
=> ( in @ Xx @ A ) )
=> ( in @ B @ ( powerset @ A ) ) ) ) ) ).
thf(kpair_type,type,
kpair: $i > $i > $i ).
thf(singleton_type,type,
singleton: $i > $o ).
thf(ex1_type,type,
ex1: $i > ( $i > $o ) > $o ).
thf(ex1,definition,
( ex1
= ( ^ [A: $i,Xphi: $i > $o] :
( singleton
@ ( dsetconstr @ A
@ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
thf(ex1I_type,type,
ex1I: $o ).
thf(ex1I,definition,
( ex1I
= ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
( ( in @ Xx @ A )
=> ( ( Xphi @ Xx )
=> ( ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ( Xphi @ Xy )
=> ( Xy = Xx ) ) )
=> ( ex1 @ A
@ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
thf(breln1Set_type,type,
breln1Set: $i > $i ).
thf(transitive_type,type,
transitive: $i > $i > $o ).
thf(antisymmetric_type,type,
antisymmetric: $i > $i > $o ).
thf(antisymmetric,definition,
( antisymmetric
= ( ^ [A: $i,R: $i] :
! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
& ( in @ ( kpair @ Xy @ Xx ) @ R ) )
=> ( Xx = Xy ) ) ) ) ) ) ).
thf(reflexive_type,type,
reflexive: $i > $i > $o ).
thf(refllinearorder_type,type,
refllinearorder: $i > $i > $o ).
thf(refllinearorder,definition,
( refllinearorder
= ( ^ [A: $i,R: $i] :
( ( reflexive @ A @ R )
& ( transitive @ A @ R )
& ( antisymmetric @ A @ R )
& ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ R )
| ( in @ ( kpair @ Xy @ Xx ) @ R ) ) ) ) ) ) ) ).
thf(reflwellordering_type,type,
reflwellordering: $i > $i > $o ).
thf(reflwellordering,definition,
( reflwellordering
= ( ^ [A: $i,R: $i] :
( ( refllinearorder @ A @ R )
& ! [X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ( ( nonempty @ X )
=> ? [Xx: $i] :
( ( in @ Xx @ X )
& ! [Xy: $i] :
( ( in @ Xy @ X )
=> ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ) ).
thf(choice2fnsingleton,conjecture,
( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( nonemptyI
=> ( powersetI
=> ( ex1I
=> ! [A: $i,B: $i,Xphi: $i > $i > $o] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ? [Xy: $i] :
( ( in @ Xy @ B )
& ( Xphi @ Xx @ Xy ) ) )
=> ! [R: $i] :
( ( in @ R @ ( breln1Set @ B ) )
=> ( ( reflwellordering @ B @ R )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( singleton
@ ( dsetconstr @ B
@ ^ [Xy: $i] :
( ( Xphi @ Xx @ Xy )
& ! [Xz: $i] :
( ( in @ Xz @ B )
=> ( ( Xphi @ Xx @ Xz )
=> ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
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