P. 71 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7001-7100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fisbth 7001	Schroeder-Bernstein Theorem for finite sets. (Contributed by Jim Kingdon, 12-Sep-2021.)
|- ( ( ( A e. Fin /\ B e. Fin ) /\ ( A ~<_ B /\ B ~<_ A ) ) -> A ~~ B )
Theorem	0fin 7002	The empty set is finite. (Contributed by FL, 14-Jul-2008.)
|- (/) e. Fin
Theorem	fin0 7003*	A nonempty finite set has at least one element. (Contributed by Jim Kingdon, 10-Sep-2021.)
|- ( A e. Fin -> ( A =/= (/) <-> E. x x e. A ) )
Theorem	fin0or 7004*	A finite set is either empty or inhabited. (Contributed by Jim Kingdon, 30-Sep-2021.)
|- ( A e. Fin -> ( A = (/) \/ E. x x e. A ) )
Theorem	diffitest 7005*	If subtracting any set from a finite set gives a finite set, any proposition of the form -. ph is decidable. This is not a proof of full excluded middle, but it is close enough to show we won't be able to prove A e. Fin -> ( A \ B ) e. Fin. (Contributed by Jim Kingdon, 8-Sep-2021.)
|- A. a e. Fin A. b ( a \ b ) e. Fin   =>    |- ( -. ph \/ -. -. ph )
Theorem	findcard 7006*	Schema for induction on the cardinality of a finite set. The inductive hypothesis is that the result is true on the given set with any one element removed. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( x = (/) -> ( ph <-> ps ) )   &    |- ( x = ( y \ { z } ) -> ( ph <-> ch ) )   &    |- ( x = y -> ( ph <-> th ) )   &    |- ( x = A -> ( ph <-> ta ) )   &    |- ps   &    |- ( y e. Fin -> ( A. z e. y ch -> th ) )   =>    |- ( A e. Fin -> ta )
Theorem	findcard2 7007*	Schema for induction on the cardinality of a finite set. The inductive step shows that the result is true if one more element is added to the set. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 8-Jul-2010.)
|- ( x = (/) -> ( ph <-> ps ) )   &    |- ( x = y -> ( ph <-> ch ) )   &    |- ( x = ( y u. { z } ) -> ( ph <-> th ) )   &    |- ( x = A -> ( ph <-> ta ) )   &    |- ps   &    |- ( y e. Fin -> ( ch -> th ) )   =>    |- ( A e. Fin -> ta )
Theorem	findcard2s 7008*	Variation of findcard2 7007 requiring that the element added in the induction step not be a member of the original set. (Contributed by Paul Chapman, 30-Nov-2012.)
|- ( x = (/) -> ( ph <-> ps ) )   &    |- ( x = y -> ( ph <-> ch ) )   &    |- ( x = ( y u. { z } ) -> ( ph <-> th ) )   &    |- ( x = A -> ( ph <-> ta ) )   &    |- ps   &    |- ( ( y e. Fin /\ -. z e. y ) -> ( ch -> th ) )   =>    |- ( A e. Fin -> ta )
Theorem	findcard2d 7009*	Deduction version of findcard2 7007. If you also need y e. Fin (which doesn't come for free due to ssfiexmid 6994), use findcard2sd 7010 instead. (Contributed by SO, 16-Jul-2018.)
|- ( x = (/) -> ( ps <-> ch ) )   &    |- ( x = y -> ( ps <-> th ) )   &    |- ( x = ( y u. { z } ) -> ( ps <-> ta ) )   &    |- ( x = A -> ( ps <-> et ) )   &    |- ( ph -> ch )   &    |- ( ( ph /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> ( th -> ta ) )   &    |- ( ph -> A e. Fin )   =>    |- ( ph -> et )
Theorem	findcard2sd 7010*	Deduction form of finite set induction . (Contributed by Jim Kingdon, 14-Sep-2021.)
|- ( x = (/) -> ( ps <-> ch ) )   &    |- ( x = y -> ( ps <-> th ) )   &    |- ( x = ( y u. { z } ) -> ( ps <-> ta ) )   &    |- ( x = A -> ( ps <-> et ) )   &    |- ( ph -> ch )   &    |- ( ( ( ph /\ y e. Fin ) /\ ( y C_ A /\ z e. ( A \ y ) ) ) -> ( th -> ta ) )   &    |- ( ph -> A e. Fin )   =>    |- ( ph -> et )
Theorem	diffisn 7011	Subtracting a singleton from a finite set produces a finite set. (Contributed by Jim Kingdon, 11-Sep-2021.)
|- ( ( A e. Fin /\ B e. A ) -> ( A \ { B } ) e. Fin )
Theorem	diffifi 7012	Subtracting one finite set from another produces a finite set. (Contributed by Jim Kingdon, 8-Sep-2021.)
|- ( ( A e. Fin /\ B e. Fin /\ B C_ A ) -> ( A \ B ) e. Fin )
Theorem	infnfi 7013	An infinite set is not finite. (Contributed by Jim Kingdon, 20-Feb-2022.)
|- ( om ~<_ A -> -. A e. Fin )
Theorem	ominf 7014	The set of natural numbers is not finite. Although we supply this theorem because we can, the more natural way to express " om is infinite" is om ~<_ om which is an instance of domrefg 6876. (Contributed by NM, 2-Jun-1998.)
|- -. om e. Fin
Theorem	isinfinf 7015*	An infinite set contains subsets of arbitrarily large finite cardinality. (Contributed by Jim Kingdon, 15-Jun-2022.)
|- ( om ~<_ A -> A. n e. om E. x ( x C_ A /\ x ~~ n ) )
Theorem	ac6sfi 7016*	Existence of a choice function for finite sets. (Contributed by Jeff Hankins, 26-Jun-2009.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)
|- ( y = ( f ` x ) -> ( ph <-> ps ) )   =>    |- ( ( A e. Fin /\ A. x e. A E. y e. B ph ) -> E. f ( f : A --> B /\ A. x e. A ps ) )
Theorem	tridc 7017*	A trichotomous order is decidable. (Contributed by Jim Kingdon, 5-Sep-2022.)
|- ( ph -> R Po A )   &    |- ( ph -> A. x e. A A. y e. A ( x R y \/ x = y \/ y R x ) )   &    |- ( ph -> B e. A )   &    |- ( ph -> C e. A )   =>    |- ( ph -> DECID B R C )
Theorem	fimax2gtrilemstep 7018*	Lemma for fimax2gtri 7019. The induction step. (Contributed by Jim Kingdon, 5-Sep-2022.)
|- ( ph -> R Po A )   &    |- ( ph -> A. x e. A A. y e. A ( x R y \/ x = y \/ y R x ) )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> A =/= (/) )   &    |- ( ph -> U e. Fin )   &    |- ( ph -> U C_ A )   &    |- ( ph -> Z e. A )   &    |- ( ph -> V e. A )   &    |- ( ph -> -. V e. U )   &    |- ( ph -> A. y e. U -. Z R y )   =>    |- ( ph -> E. x e. A A. y e. ( U u. { V } ) -. x R y )
Theorem	fimax2gtri 7019*	A finite set has a maximum under a trichotomous order. (Contributed by Jim Kingdon, 5-Sep-2022.)
|- ( ph -> R Po A )   &    |- ( ph -> A. x e. A A. y e. A ( x R y \/ x = y \/ y R x ) )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> A =/= (/) )   =>    |- ( ph -> E. x e. A A. y e. A -. x R y )
Theorem	finexdc 7020*	Decidability of existence, over a finite set and defined by a decidable proposition. (Contributed by Jim Kingdon, 12-Jul-2022.)
|- ( ( A e. Fin /\ A. x e. A DECID ph ) -> DECID E. x e. A ph )
Theorem	dfrex2fin 7021*	Relationship between universal and existential quantifiers over a finite set. Remark in Section 2.2.1 of [Pierik], p. 8. Although Pierik does not mention the decidability condition explicitly, it does say "only finitely many x to check" which means there must be some way of checking each value of x. (Contributed by Jim Kingdon, 11-Jul-2022.)
|- ( ( A e. Fin /\ A. x e. A DECID ph ) -> ( E. x e. A ph <-> -. A. x e. A -. ph ) )
Theorem	infm 7022*	An infinite set is inhabited. (Contributed by Jim Kingdon, 18-Feb-2022.)
|- ( om ~<_ A -> E. x x e. A )
Theorem	infn0 7023	An infinite set is not empty. (Contributed by NM, 23-Oct-2004.)
|- ( om ~<_ A -> A =/= (/) )
Theorem	inffiexmid 7024*	If any given set is either finite or infinite, excluded middle follows. (Contributed by Jim Kingdon, 15-Jun-2022.)
|- ( x e. Fin \/ om ~<_ x )   =>    |- ( ph \/ -. ph )
Theorem	en2eqpr 7025	Building a set with two elements. (Contributed by FL, 11-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
|- ( ( C ~~ 2o /\ A e. C /\ B e. C ) -> ( A =/= B -> C = { A , B } ) )
Theorem	exmidpw 7026	Excluded middle is equivalent to the power set of 1o having two elements. Remark of [PradicBrown2022], p. 2. (Contributed by Jim Kingdon, 30-Jun-2022.)
|- (EXMID <-> ~P 1o ~~ 2o )
Theorem	exmidpweq 7027	Excluded middle is equivalent to the power set of 1o being 2o. (Contributed by Jim Kingdon, 28-Jul-2024.)
|- (EXMID <-> ~P 1o = 2o )
Theorem	pw1fin 7028	Excluded middle is equivalent to the power set of 1o being finite. (Contributed by SN and Jim Kingdon, 7-Aug-2024.)
|- (EXMID <-> ~P 1o e. Fin )
Theorem	pw1dc0el 7029	Another equivalent of excluded middle, which is a mere reformulation of the definition. (Contributed by BJ, 9-Aug-2024.)
|- (EXMID <-> A. x e. ~P 1oDECID (/) e. x )
Theorem	exmidpw2en 7030	The power set of a set being equinumerous to set exponentiation with a base of ordinal 2o is equivalent to excluded middle. This is Metamath 100 proof #52. The forward direction uses excluded middle expressed as EXMID to show this equinumerosity. The reverse direction is the one which establishes that power set being equinumerous to set exponentiation implies excluded middle. This resolves the question of whether we will be able to prove this equinumerosity theorem in the negative. (Contributed by Jim Kingdon, 13-Aug-2022.)
|- (EXMID <-> A. x ~P x ~~ ( 2o ^m x ) )
Theorem	ss1o0el1o 7031	Reformulation of ss1o0el1 4252 using 1o instead of { (/) }. (Contributed by BJ, 9-Aug-2024.)
|- ( A C_ 1o -> ( (/) e. A <-> A = 1o ) )
Theorem	pw1dc1 7032	If, in the set of truth values (the powerset of 1o), equality to 1o is decidable, then excluded middle holds (and conversely). (Contributed by BJ and Jim Kingdon, 8-Aug-2024.)
|- (EXMID <-> A. x e. ~P 1oDECID x = 1o )
Theorem	fientri3 7033	Trichotomy of dominance for finite sets. (Contributed by Jim Kingdon, 15-Sep-2021.)
|- ( ( A e. Fin /\ B e. Fin ) -> ( A ~<_ B \/ B ~<_ A ) )
Theorem	nnwetri 7034*	A natural number is well-ordered by _E. More specifically, this order both satisfies We and is trichotomous. (Contributed by Jim Kingdon, 25-Sep-2021.)
|- ( A e. om -> ( _E We A /\ A. x e. A A. y e. A ( x _E y \/ x = y \/ y _E x ) ) )
Theorem	onunsnss 7035	Adding a singleton to create an ordinal. (Contributed by Jim Kingdon, 20-Oct-2021.)
|- ( ( B e. V /\ ( A u. { B } ) e. On ) -> B C_ A )
Theorem	unfiexmid 7036*	If the union of any two finite sets is finite, excluded middle follows. Remark 8.1.17 of [AczelRathjen], p. 74. (Contributed by Mario Carneiro and Jim Kingdon, 5-Mar-2022.)
|- ( ( x e. Fin /\ y e. Fin ) -> ( x u. y ) e. Fin )   =>    |- ( ph \/ -. ph )
Theorem	unsnfi 7037	Adding a singleton to a finite set yields a finite set. (Contributed by Jim Kingdon, 3-Feb-2022.)
|- ( ( A e. Fin /\ B e. V /\ -. B e. A ) -> ( A u. { B } ) e. Fin )
Theorem	unsnfidcex 7038	The B e. V condition in unsnfi 7037. This is intended to show that unsnfi 7037 without that condition would not be provable but it probably would need to be strengthened (for example, to imply included middle) to fully show that. (Contributed by Jim Kingdon, 6-Feb-2022.)
|- ( ( A e. Fin /\ -. B e. A /\ ( A u. { B } ) e. Fin ) -> DECID -. B e. _V )
Theorem	unsnfidcel 7039	The -. B e. A condition in unsnfi 7037. This is intended to show that unsnfi 7037 without that condition would not be provable but it probably would need to be strengthened (for example, to imply included middle) to fully show that. (Contributed by Jim Kingdon, 6-Feb-2022.)
|- ( ( A e. Fin /\ B e. V /\ ( A u. { B } ) e. Fin ) -> DECID -. B e. A )
Theorem	unfidisj 7040	The union of two disjoint finite sets is finite. (Contributed by Jim Kingdon, 25-Feb-2022.)
|- ( ( A e. Fin /\ B e. Fin /\ ( A i^i B ) = (/) ) -> ( A u. B ) e. Fin )
Theorem	undifdcss 7041*	Union of complementary parts into whole and decidability. (Contributed by Jim Kingdon, 17-Jun-2022.)
|- ( A = ( B u. ( A \ B ) ) <-> ( B C_ A /\ A. x e. A DECID x e. B ) )
Theorem	undifdc 7042*	Union of complementary parts into whole. This is a case where we can strengthen undifss 3545 from subset to equality. (Contributed by Jim Kingdon, 17-Jun-2022.)
|- ( ( A. x e. A A. y e. A DECID x = y /\ B e. Fin /\ B C_ A ) -> A = ( B u. ( A \ B ) ) )
Theorem	undiffi 7043	Union of complementary parts into whole. This is a case where we can strengthen undifss 3545 from subset to equality. (Contributed by Jim Kingdon, 2-Mar-2022.)
|- ( ( A e. Fin /\ B e. Fin /\ B C_ A ) -> A = ( B u. ( A \ B ) ) )
Theorem	unfiin 7044	The union of two finite sets is finite if their intersection is. (Contributed by Jim Kingdon, 2-Mar-2022.)
|- ( ( A e. Fin /\ B e. Fin /\ ( A i^i B ) e. Fin ) -> ( A u. B ) e. Fin )
Theorem	prfidisj 7045	A pair is finite if it consists of two unequal sets. For the case where A = B, see snfig 6925. For the cases where one or both is a proper class, see prprc1 3746, prprc2 3747, or prprc 3748. (Contributed by Jim Kingdon, 31-May-2022.)
|- ( ( A e. V /\ B e. W /\ A =/= B ) -> { A , B } e. Fin )
Theorem	prfidceq 7046*	A pair is finite if it consists of elements of a class with decidable equality. (Contributed by Jim Kingdon, 13-Oct-2025.)
|- ( ph -> A e. C )   &    |- ( ph -> B e. C )   &    |- ( ph -> A. x e. C A. y e. C DECID x = y )   =>    |- ( ph -> { A , B } e. Fin )
Theorem	tpfidisj 7047	A triple is finite if it consists of three unequal sets. (Contributed by Jim Kingdon, 1-Oct-2022.)
|- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( ph -> C e. X )   &    |- ( ph -> A =/= B )   &    |- ( ph -> A =/= C )   &    |- ( ph -> B =/= C )   =>    |- ( ph -> { A , B , C } e. Fin )
Theorem	tpfidceq 7048*	A triple is finite if it consists of elements of a class with decidable equality. (Contributed by Jim Kingdon, 13-Oct-2025.)
|- ( ph -> A e. D )   &    |- ( ph -> B e. D )   &    |- ( ph -> C e. D )   &    |- ( ph -> A. x e. D A. y e. D DECID x = y )   =>    |- ( ph -> { A , B , C } e. Fin )
Theorem	fiintim 7049*	If a class is closed under pairwise intersections, then it is closed under nonempty finite intersections. The converse would appear to require an additional condition, such as x and y not being equal, or A having decidable equality. This theorem is applicable to a topology, which (among other axioms) is closed under finite intersections. Some texts use a pairwise intersection and some texts use a finite intersection, but most topology texts assume excluded middle (in which case the two intersection properties would be equivalent). (Contributed by NM, 22-Sep-2002.) (Revised by Jim Kingdon, 14-Jan-2023.)
|- ( A. x e. A A. y e. A ( x i^i y ) e. A -> A. x ( ( x C_ A /\ x =/= (/) /\ x e. Fin ) -> |^| x e. A ) )
Theorem	xpfi 7050	The Cartesian product of two finite sets is finite. Lemma 8.1.16 of [AczelRathjen], p. 74. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Mar-2015.)
|- ( ( A e. Fin /\ B e. Fin ) -> ( A X. B ) e. Fin )
Theorem	3xpfi 7051	The Cartesian product of three finite sets is a finite set. (Contributed by Alexander van der Vekens, 11-Mar-2018.)
|- ( V e. Fin -> ( ( V X. V ) X. V ) e. Fin )
Theorem	fisseneq 7052	A finite set is equal to its subset if they are equinumerous. (Contributed by FL, 11-Aug-2008.)
|- ( ( B e. Fin /\ A C_ B /\ A ~~ B ) -> A = B )
Theorem	phpeqd 7053	Corollary of the Pigeonhole Principle using equality. Strengthening of phpm 6983 expressed without negation. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- ( ph -> A e. Fin )   &    |- ( ph -> B C_ A )   &    |- ( ph -> A ~~ B )   =>    |- ( ph -> A = B )
Theorem	ssfirab 7054*	A subset of a finite set is finite if it is defined by a decidable property. (Contributed by Jim Kingdon, 27-May-2022.)
|- ( ph -> A e. Fin )   &    |- ( ph -> A. x e. A DECID ps )   =>    |- ( ph -> { x e. A | ps } e. Fin )
Theorem	ssfidc 7055*	A subset of a finite set is finite if membership in the subset is decidable. (Contributed by Jim Kingdon, 27-May-2022.)
|- ( ( A e. Fin /\ B C_ A /\ A. x e. A DECID x e. B ) -> B e. Fin )
Theorem	opabfi 7056*	Finiteness of an ordered pair abstraction which is a decidable subset of finite sets. (Contributed by Jim Kingdon, 16-Sep-2025.)
|- S = { <. x , y >. | ( ( x e. A /\ y e. B ) /\ ps ) }   &    |- ( ph -> A e. Fin )   &    |- ( ph -> B e. Fin )   &    |- ( ph -> A. x e. A A. y e. B DECID ps )   =>    |- ( ph -> S e. Fin )
Theorem	infidc 7057*	The intersection of two sets is finite if one of them is and the other is decidable. (Contributed by Jim Kingdon, 24-May-2025.)
|- ( ( A e. Fin /\ A. x e. A DECID x e. B ) -> ( A i^i B ) e. Fin )
Theorem	snon0 7058	An ordinal which is a singleton is { (/) }. (Contributed by Jim Kingdon, 19-Oct-2021.)
|- ( ( A e. V /\ { A } e. On ) -> A = (/) )
Theorem	fnfi 7059	A version of fnex 5824 for finite sets. (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.)
|- ( ( F Fn A /\ A e. Fin ) -> F e. Fin )
Theorem	fundmfi 7060	The domain of a finite function is finite. (Contributed by Jim Kingdon, 5-Feb-2022.)
|- ( ( A e. Fin /\ Fun A ) -> dom A e. Fin )
Theorem	fundmfibi 7061	A function is finite if and only if its domain is finite. (Contributed by AV, 10-Jan-2020.)
|- ( Fun F -> ( F e. Fin <-> dom F e. Fin ) )
Theorem	resfnfinfinss 7062	The restriction of a function to a finite subset of its domain is finite. (Contributed by Alexander van der Vekens, 3-Feb-2018.)
|- ( ( F Fn A /\ B e. Fin /\ B C_ A ) -> ( F |` B ) e. Fin )
Theorem	residfi 7063	A restricted identity function is finite iff the restricting class is finite. (Contributed by AV, 10-Jan-2020.)
|- ( ( _I |` A ) e. Fin <-> A e. Fin )
Theorem	relcnvfi 7064	If a relation is finite, its converse is as well. (Contributed by Jim Kingdon, 5-Feb-2022.)
|- ( ( Rel A /\ A e. Fin ) -> `' A e. Fin )
Theorem	funrnfi 7065	The range of a finite relation is finite if its converse is a function. (Contributed by Jim Kingdon, 5-Feb-2022.)
|- ( ( Rel A /\ Fun `' A /\ A e. Fin ) -> ran A e. Fin )
Theorem	f1ofi 7066	If a 1-1 and onto function has a finite domain, its range is finite. (Contributed by Jim Kingdon, 21-Feb-2022.)
|- ( ( A e. Fin /\ F : A -1-1-onto-> B ) -> B e. Fin )
Theorem	f1dmvrnfibi 7067	A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 7068. (Contributed by AV, 10-Jan-2020.)
|- ( ( A e. V /\ F : A -1-1-> B ) -> ( F e. Fin <-> ran F e. Fin ) )
Theorem	f1vrnfibi 7068	A one-to-one function which is a set is finite if and only if its range is finite. See also f1dmvrnfibi 7067. (Contributed by AV, 10-Jan-2020.)
|- ( ( F e. V /\ F : A -1-1-> B ) -> ( F e. Fin <-> ran F e. Fin ) )
Theorem	iunfidisj 7069*	The finite union of disjoint finite sets is finite. Note that B depends on x, i.e. can be thought of as B ( x ). (Contributed by NM, 23-Mar-2006.) (Revised by Jim Kingdon, 7-Oct-2022.)
|- ( ( A e. Fin /\ A. x e. A B e. Fin /\ Disj x e. A B ) -> U_ x e. A B e. Fin )
Theorem	f1finf1o 7070	Any injection from one finite set to another of equal size must be a bijection. (Contributed by Jeff Madsen, 5-Jun-2010.)
|- ( ( A ~~ B /\ B e. Fin ) -> ( F : A -1-1-> B <-> F : A -1-1-onto-> B ) )
Theorem	en1eqsn 7071	A set with one element is a singleton. (Contributed by FL, 18-Aug-2008.)
|- ( ( A e. B /\ B ~~ 1o ) -> B = { A } )
Theorem	en1eqsnbi 7072	A set containing an element has exactly one element iff it is a singleton. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.)
|- ( A e. B -> ( B ~~ 1o <-> B = { A } ) )
Theorem	snexxph 7073*	A case where the antecedent of snexg 4239 is not needed. The class { x | ph } is from dcextest 4642. (Contributed by Mario Carneiro and Jim Kingdon, 4-Jul-2022.)
|- { { x | ph } } e. _V
Theorem	preimaf1ofi 7074	The preimage of a finite set under a one-to-one, onto function is finite. (Contributed by Jim Kingdon, 24-Sep-2022.)
|- ( ph -> C C_ B )   &    |- ( ph -> F : A -1-1-onto-> B )   &    |- ( ph -> C e. Fin )   =>    |- ( ph -> ( `' F " C ) e. Fin )
Theorem	fidcenumlemim 7075*	Lemma for fidcenum 7079. Forward direction. (Contributed by Jim Kingdon, 19-Oct-2022.)
|- ( A e. Fin -> ( A. x e. A A. y e. A DECID x = y /\ E. n e. om E. f f : n -onto-> A ) )
Theorem	fidcenumlemrks 7076*	Lemma for fidcenum 7079. Induction step for fidcenumlemrk 7077. (Contributed by Jim Kingdon, 20-Oct-2022.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : N -onto-> A )   &    |- ( ph -> J e. om )   &    |- ( ph -> suc J C_ N )   &    |- ( ph -> ( X e. ( F " J ) \/ -. X e. ( F " J ) ) )   &    |- ( ph -> X e. A )   =>    |- ( ph -> ( X e. ( F " suc J ) \/ -. X e. ( F " suc J ) ) )
Theorem	fidcenumlemrk 7077*	Lemma for fidcenum 7079. (Contributed by Jim Kingdon, 20-Oct-2022.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : N -onto-> A )   &    |- ( ph -> K e. om )   &    |- ( ph -> K C_ N )   &    |- ( ph -> X e. A )   =>    |- ( ph -> ( X e. ( F " K ) \/ -. X e. ( F " K ) ) )
Theorem	fidcenumlemr 7078*	Lemma for fidcenum 7079. Reverse direction (put into deduction form). (Contributed by Jim Kingdon, 19-Oct-2022.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> F : N -onto-> A )   &    |- ( ph -> N e. om )   =>    |- ( ph -> A e. Fin )
Theorem	fidcenum 7079*	A set is finite if and only if it has decidable equality and is finitely enumerable. Proposition 8.1.11 of [AczelRathjen], p. 72. The definition of "finitely enumerable" as E. n e. om E. f f : n -onto-> A is Definition 8.1.4 of [AczelRathjen], p. 71. (Contributed by Jim Kingdon, 19-Oct-2022.)
|- ( A e. Fin <-> ( A. x e. A A. y e. A DECID x = y /\ E. n e. om E. f f : n -onto-> A ) )
2.6.32  Schroeder-Bernstein Theorem
Theorem	sbthlem1 7080*	Lemma for isbth 7090. (Contributed by NM, 22-Mar-1998.)
|- A e. _V   &    |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }   =>    |- U. D C_ ( A \ ( g " ( B \ ( f " U. D ) ) ) )
Theorem	sbthlem2 7081*	Lemma for isbth 7090. (Contributed by NM, 22-Mar-1998.)
|- A e. _V   &    |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }   =>    |- ( ran g C_ A -> ( A \ ( g " ( B \ ( f " U. D ) ) ) ) C_ U. D )
Theorem	sbthlemi3 7082*	Lemma for isbth 7090. (Contributed by NM, 22-Mar-1998.)
|- A e. _V   &    |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }   =>    |- ( (EXMID /\ ran g C_ A ) -> ( g " ( B \ ( f " U. D ) ) ) = ( A \ U. D ) )
Theorem	sbthlemi4 7083*	Lemma for isbth 7090. (Contributed by NM, 27-Mar-1998.)
|- A e. _V   &    |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }   =>    |- ( (EXMID /\ ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) -> ( `' g " ( A \ U. D ) ) = ( B \ ( f " U. D ) ) )
Theorem	sbthlemi5 7084*	Lemma for isbth 7090. (Contributed by NM, 22-Mar-1998.)
|- A e. _V   &    |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }   &    |- H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )   =>    |- ( (EXMID /\ ( dom f = A /\ ran g C_ A ) ) -> dom H = A )
Theorem	sbthlemi6 7085*	Lemma for isbth 7090. (Contributed by NM, 27-Mar-1998.)
|- A e. _V   &    |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }   &    |- H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )   =>    |- ( ( (EXMID /\ ran f C_ B ) /\ ( ( dom g = B /\ ran g C_ A ) /\ Fun `' g ) ) -> ran H = B )
Theorem	sbthlem7 7086*	Lemma for isbth 7090. (Contributed by NM, 27-Mar-1998.)
|- A e. _V   &    |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }   &    |- H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )   =>    |- ( ( Fun f /\ Fun `' g ) -> Fun H )
Theorem	sbthlemi8 7087*	Lemma for isbth 7090. (Contributed by NM, 27-Mar-1998.)
|- A e. _V   &    |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }   &    |- H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )   =>    |- ( ( (EXMID /\ Fun `' f ) /\ ( ( ( Fun g /\ dom g = B ) /\ ran g C_ A ) /\ Fun `' g ) ) -> Fun `' H )
Theorem	sbthlemi9 7088*	Lemma for isbth 7090. (Contributed by NM, 28-Mar-1998.)
|- A e. _V   &    |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }   &    |- H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )   =>    |- ( (EXMID /\ f : A -1-1-> B /\ g : B -1-1-> A ) -> H : A -1-1-onto-> B )
Theorem	sbthlemi10 7089*	Lemma for isbth 7090. (Contributed by NM, 28-Mar-1998.)
|- A e. _V   &    |- D = { x | ( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) }   &    |- H = ( ( f |` U. D ) u. ( `' g |` ( A \ U. D ) ) )   &    |- B e. _V   =>    |- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> A ~~ B )
Theorem	isbth 7090	Schroeder-Bernstein Theorem. Theorem 18 of [Suppes] p. 95. This theorem states that if set A is smaller (has lower cardinality) than B and vice-versa, then A and B are equinumerous (have the same cardinality). The interesting thing is that this can be proved without invoking the Axiom of Choice, as we do here, but the proof as you can see is quite difficult. (The theorem can be proved more easily if we allow AC.) The main proof consists of lemmas sbthlem1 7080 through sbthlemi10 7089; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlemi10 7089. We follow closely the proof in Suppes, which you should consult to understand our proof at a higher level. Note that Suppes' proof, which is credited to J. M. Whitaker, does not require the Axiom of Infinity. The proof does require the law of the excluded middle which cannot be avoided as shown at exmidsbthr 16134. (Contributed by NM, 8-Jun-1998.)
|- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> A ~~ B )
2.6.33  Finite intersections
Syntax	cfi 7091	Extend class notation with the function whose value is the class of finite intersections of the elements of a given set.
class fi
Definition	df-fi 7092*	Function whose value is the class of finite intersections of the elements of the argument. Note that the empty intersection being the universal class, hence a proper class, it cannot be an element of that class. Therefore, the function value is the class of nonempty finite intersections of elements of the argument (see elfi2 7095). (Contributed by FL, 27-Apr-2008.)
|- fi = ( x e. _V |-> { z | E. y e. ( ~P x i^i Fin ) z = |^| y } )
Theorem	fival 7093*	The set of all the finite intersections of the elements of A. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
|- ( A e. V -> ( fi ` A ) = { y | E. x e. ( ~P A i^i Fin ) y = |^| x } )
Theorem	elfi 7094*	Specific properties of an element of ( fi ` B ). (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
|- ( ( A e. V /\ B e. W ) -> ( A e. ( fi ` B ) <-> E. x e. ( ~P B i^i Fin ) A = |^| x ) )
Theorem	elfi2 7095*	The empty intersection need not be considered in the set of finite intersections. (Contributed by Mario Carneiro, 21-Mar-2015.)
|- ( B e. V -> ( A e. ( fi ` B ) <-> E. x e. ( ( ~P B i^i Fin ) \ { (/) } ) A = |^| x ) )
Theorem	elfir 7096	Sufficient condition for an element of ( fi ` B ). (Contributed by Mario Carneiro, 24-Nov-2013.)
|- ( ( B e. V /\ ( A C_ B /\ A =/= (/) /\ A e. Fin ) ) -> |^| A e. ( fi ` B ) )
Theorem	ssfii 7097	Any element of a set A is the intersection of a finite subset of A. (Contributed by FL, 27-Apr-2008.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
|- ( A e. V -> A C_ ( fi ` A ) )
Theorem	fi0 7098	The set of finite intersections of the empty set. (Contributed by Mario Carneiro, 30-Aug-2015.)
|- ( fi ` (/) ) = (/)
Theorem	fieq0 7099	A set is empty iff the class of all the finite intersections of that set is empty. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
|- ( A e. V -> ( A = (/) <-> ( fi ` A ) = (/) ) )
Theorem	fiss 7100	Subset relationship for function fi. (Contributed by Jeff Hankins, 7-Oct-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
|- ( ( B e. V /\ A C_ B ) -> ( fi ` A ) C_ ( fi ` B ) )