P. 73 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7201-7300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	findcard2d 7201*	Deduction version of findcard2 7199. If you also need y e. Fin (which doesn't come for free due to ssfiexmid 7184), use findcard2sd 7202 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 7202*	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 7203	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 7204	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 7205	An infinite set is not finite. (Contributed by Jim Kingdon, 20-Feb-2022.)
|- ( om ~<_ A -> -. A e. Fin )
Theorem	ominf 7206	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 7059. (Contributed by NM, 2-Jun-1998.)
|- -. om e. Fin
Theorem	isinfinf 7207*	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 7208*	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	fidcen 7209	Equinumerosity of finite sets is decidable. (Contributed by Jim Kingdon, 10-Feb-2026.)
|- ( ( A e. Fin /\ B e. Fin ) -> DECID A ~~ B )
Theorem	tridc 7210*	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 7211*	Lemma for fimax2gtri 7212. 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 7212*	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 7213*	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 7214*	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	elssdc 7215*	Membership in a finite subset of a set with decidable equality is decidable. (Contributed by Jim Kingdon, 11-Feb-2026.)
|- ( ph -> A. x e. B A. y e. B DECID x = y )   &    |- ( ph -> X e. B )   &    |- ( ph -> A C_ B )   &    |- ( ph -> A e. Fin )   =>    |- ( ph -> DECID X e. A )
Theorem	eqsndc 7216*	Decidability of equality between a finite subset of a set with decidable equality, and a singleton whose element is an element of the larger set. (Contributed by Jim Kingdon, 15-Feb-2026.)
|- ( ph -> A. x e. B A. y e. B DECID x = y )   &    |- ( ph -> X e. B )   &    |- ( ph -> A C_ B )   &    |- ( ph -> A e. Fin )   =>    |- ( ph -> DECID A = { X } )
Theorem	infm 7217*	An infinite set is inhabited. (Contributed by Jim Kingdon, 18-Feb-2022.)
|- ( om ~<_ A -> E. x x e. A )
Theorem	infn0 7218	An infinite set is not empty. (Contributed by NM, 23-Oct-2004.)
|- ( om ~<_ A -> A =/= (/) )
Theorem	inffiexmid 7219*	If any given set is either finite or infinite, excluded middle follows. For another example, ~P 1o is not infinite, by pw1ninf 17243, but also cannot be shown to be finite by pw1fin 7223. (Contributed by Jim Kingdon, 15-Jun-2022.)
|- ( x e. Fin \/ om ~<_ x )   =>    |- ( ph \/ -. ph )
Theorem	en2eqpr 7220	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 7221	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 7222	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 7223	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 7224	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 7225	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 7226	Reformulation of ss1o0el1 4336 using 1o instead of { (/) }. (Contributed by BJ, 9-Aug-2024.)
|- ( A C_ 1o -> ( (/) e. A <-> A = 1o ) )
Theorem	pw1dc1 7227	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 7228	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 7229*	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 7230	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 7231*	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 7232	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 7233	The B e. V condition in unsnfi 7232. This is intended to show that unsnfi 7232 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 7234	The -. B e. A condition in unsnfi 7232. This is intended to show that unsnfi 7232 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 7235	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 7236*	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 7237*	Union of complementary parts into whole. This is a case where we can strengthen undifss 3608 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 7238	Union of complementary parts into whole. This is a case where we can strengthen undifss 3608 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 7239	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 7240	A pair is finite if it consists of two unequal sets. For the case where A = B, see snfig 7109. For the cases where one or both is a proper class, see prprc1 3821, prprc2 3822, or prprc 3823. (Contributed by Jim Kingdon, 31-May-2022.)
|- ( ( A e. V /\ B e. W /\ A =/= B ) -> { A , B } e. Fin )
Theorem	prfidceq 7241*	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 7242	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 7243*	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 7244*	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 7245	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	imaf1fi 7246	The image of a finite set under a one-to-one mapping is finite. (Contributed by Jim Kingdon, 28-Mar-2026.)
|- ( ( F : A -1-1-> B /\ X C_ A /\ X e. Fin ) -> ( F " X ) e. Fin )
Theorem	3xpfi 7247	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 7248	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 7249	Corollary of the Pigeonhole Principle using equality. Strengthening of phpm 7173 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 7250*	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 7251*	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	exmidssfi 7252*	Excluded middle is equivalent to any subset of a finite set being finite. Theorem 2.1 of [Bauer], p. 485. (Contributed by Jim Kingdon, 20-Mar-2026.)
|- (EXMID <-> A. x A. y ( ( x e. Fin /\ y C_ x ) -> y e. Fin ) )
Theorem	opabfi 7253*	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 7254*	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 7255	An ordinal which is a singleton is { (/) }. (Contributed by Jim Kingdon, 19-Oct-2021.)
|- ( ( A e. V /\ { A } e. On ) -> A = (/) )
Theorem	fnfi 7256	A version of fnex 5940 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 7257	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 7258	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 7259	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 7260	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 7261	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 7262	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 7263	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 7264	A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 7265. (Contributed by AV, 10-Jan-2020.)
|- ( ( A e. V /\ F : A -1-1-> B ) -> ( F e. Fin <-> ran F e. Fin ) )
Theorem	f1vrnfibi 7265	A one-to-one function which is a set is finite if and only if its range is finite. See also f1dmvrnfibi 7264. (Contributed by AV, 10-Jan-2020.)
|- ( ( F e. V /\ F : A -1-1-> B ) -> ( F e. Fin <-> ran F e. Fin ) )
Theorem	iunfidisj 7266*	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	mapfi 7267	Set exponentiation of finite sets is finite. (Contributed by Jeff Madsen, 19-Jun-2011.)
|- ( ( A e. Fin /\ B e. Fin ) -> ( A ^m B ) e. Fin )
Theorem	elfpw 7268	Membership in a class of finite subsets. (Contributed by Stefan O'Rear, 4-Apr-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
|- ( A e. ( ~P B i^i Fin ) <-> ( A C_ B /\ A e. Fin ) )
Theorem	fissfi 7269*	A finite subset of a finite set is a decidable subset. (Contributed by Jim Kingdon, 18-May-2026.)
|- ( ( S C_ A /\ A e. Fin /\ S e. Fin ) -> A. x e. A DECID x e. S )
Theorem	relresfin 7270	Restricting to a subset of a finite relation's domain gives a finite set. (Contributed by Jim Kingdon, 6-Oct-2026.)
|- ( ph -> E e. Fin )   &    |- ( ph -> Rel E )   &    |- ( ph -> dom E e. Fin )   &    |- ( ph -> B C_ dom E )   &    |- ( ph -> B e. Fin )   =>    |- ( ph -> ( E |` B ) e. Fin )
Theorem	f1finf1o 7271	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 7272	A set with one element is a singleton. (Contributed by FL, 18-Aug-2008.)
|- ( ( A e. B /\ B ~~ 1o ) -> B = { A } )
Theorem	en1eqsnbi 7273	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 7274*	A case where the antecedent of snexg 4323 is not needed. The class { x | ph } is from dcextest 4730. (Contributed by Mario Carneiro and Jim Kingdon, 4-Jul-2022.)
|- { { x | ph } } e. _V
Theorem	preimaf1ofi 7275	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 7276*	Lemma for fidcenum 7280. 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 7277*	Lemma for fidcenum 7280. Induction step for fidcenumlemrk 7278. (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 7278*	Lemma for fidcenum 7280. (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 7279*	Lemma for fidcenum 7280. 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 7280*	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 ) )
2.6.33  Schroeder-Bernstein Theorem
Theorem	sbthlem1 7281*	Lemma for isbth 7291. (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 7282*	Lemma for isbth 7291. (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 7283*	Lemma for isbth 7291. (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 7284*	Lemma for isbth 7291. (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 7285*	Lemma for isbth 7291. (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 7286*	Lemma for isbth 7291. (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 7287*	Lemma for isbth 7291. (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 7288*	Lemma for isbth 7291. (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 7289*	Lemma for isbth 7291. (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 7290*	Lemma for isbth 7291. (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 7291	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 7281 through sbthlemi10 7290; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlemi10 7290. 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 17291. (Contributed by NM, 8-Jun-1998.)
|- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> A ~~ B )
2.6.34  Finitely supported functions
Syntax	cfsupp 7292	Extend class definition to include the predicate to be a finitely supported function.
class finSupp
Definition	df-fsuppfi 7293*	Define the property of a function to be finitely supported (in relation to a given zero). This definition will perhaps be most useful when the domain of the function has decidable equality; consider as an alternative ( R supp Z ) e. Fin (see df-supp 6480) which will behave well when a codomain of the function has decidable equality. If both domain and codomain have decidable equality, the two are equivalent as seen at funisfsuppdc 7298. (Contributed by AV, 23-May-2019.) (Revised by Jim Kingdon, 16-Oct-2026.)
|- finSupp = { <. r , z >. | ( Fun r /\ E. w ( w e. Fin /\ w C_ dom r /\ ( r " ( dom r \ w ) ) C_ { z } ) ) }
Theorem	relfsupp 7294	The property of a function to be finitely supported is a relation. (Contributed by AV, 7-Jun-2019.)
|- Rel finSupp
Theorem	relprcnfsupp 7295	A proper class is never finitely supported. (Contributed by AV, 7-Jun-2019.)
|- ( -. A e. _V -> -. A finSupp Z )
Theorem	isfsuppfi 7296*	The property of a class to be a finitely supported function (in relation to a given zero). (Contributed by AV, 23-May-2019.)
|- ( ( R e. V /\ Z e. U ) -> ( R finSupp Z <-> ( Fun R /\ E. w ( w e. Fin /\ w C_ dom R /\ ( R " ( dom R \ w ) ) C_ { Z } ) ) ) )
Theorem	isfsuppfid 7297	Deduction form of isfsuppfi 7296. (Contributed by SN, 29-Jul-2024.) (Revised by Jim Kingdon, 16-Oct-2026.)
|- ( ph -> R e. V )   &    |- ( ph -> Z e. U )   &    |- ( ph -> Fun R )   &    |- ( ph -> W e. Fin )   &    |- ( ph -> W C_ dom R )   &    |- ( ph -> ( R " ( dom R \ W ) ) C_ { Z } )   =>    |- ( ph -> R finSupp Z )
Theorem	funisfsuppdc 7298*	The property of a function to be finitely supported (in relation to a given zero). This shows that df-fsuppfi 7293 and df-supp 6480 give the same results, in the case of decidable equality. (Contributed by AV, 23-May-2019.) (Revised by Jim Kingdon, 17-Oct-2026.)
|- ( ph -> F e. V )   &    |- ( ph -> F : A --> B )   &    |- ( ph -> Z e. B )   &    |- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> A. x e. B A. y e. B DECID x = y )   =>    |- ( ph -> ( F finSupp Z <-> ( F supp Z ) e. Fin ) )
Theorem	fsuppfund 7299	A finitely supported function is a function. (Contributed by SN, 8-Mar-2025.)
|- ( ph -> F finSupp Z )   =>    |- ( ph -> Fun F )
Theorem	fsuppfiss 7300*	A finitely supported function has a finite subset of its domain such that values outside that subset map to zero. (Contributed by Jim Kingdon, 24-Oct-2026.)
|- ( ph -> F finSupp Z )   =>    |- ( ph -> E. w ( w e. Fin /\ w C_ dom F /\ ( F " ( dom F \ w ) ) C_ { Z } ) )