P. 69 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6801-6900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	snnen2og 6801	A singleton { A } is never equinumerous with the ordinal number 2. If A is a proper class, see snnen2oprc 6802. (Contributed by Jim Kingdon, 1-Sep-2021.)
|- ( A e. V -> -. { A } ~~ 2o )
Theorem	snnen2oprc 6802	A singleton { A } is never equinumerous with the ordinal number 2. If A is a set, see snnen2og 6801. (Contributed by Jim Kingdon, 1-Sep-2021.)
|- ( -. A e. _V -> -. { A } ~~ 2o )
Theorem	1nen2 6803	One and two are not equinumerous. (Contributed by Jim Kingdon, 25-Jan-2022.)
|- -. 1o ~~ 2o
Theorem	phplem4dom 6804	Dominance of successors implies dominance of the original natural numbers. (Contributed by Jim Kingdon, 1-Sep-2021.)
|- ( ( A e. om /\ B e. om ) -> ( suc A ~<_ suc B -> A ~<_ B ) )
Theorem	php5dom 6805	A natural number does not dominate its successor. (Contributed by Jim Kingdon, 1-Sep-2021.)
|- ( A e. om -> -. suc A ~<_ A )
Theorem	nndomo 6806	Cardinal ordering agrees with natural number ordering. Example 3 of [Enderton] p. 146. (Contributed by NM, 17-Jun-1998.)
|- ( ( A e. om /\ B e. om ) -> ( A ~<_ B <-> A C_ B ) )
Theorem	phpm 6807*	Pigeonhole Principle. A natural number is not equinumerous to a proper subset of itself. By "proper subset" here we mean that there is an element which is in the natural number and not in the subset, or in symbols E. x x e. ( A \ B ) (which is stronger than not being equal in the absence of excluded middle). Theorem (Pigeonhole Principle) of [Enderton] p. 134. The theorem is so-called because you can't put n + 1 pigeons into n holes (if each hole holds only one pigeon). The proof consists of lemmas phplem1 6794 through phplem4 6797, nneneq 6799, and this final piece of the proof. (Contributed by NM, 29-May-1998.)
|- ( ( A e. om /\ B C_ A /\ E. x x e. ( A \ B ) ) -> -. A ~~ B )
Theorem	phpelm 6808	Pigeonhole Principle. A natural number is not equinumerous to an element of itself. (Contributed by Jim Kingdon, 6-Sep-2021.)
|- ( ( A e. om /\ B e. A ) -> -. A ~~ B )
Theorem	phplem4on 6809	Equinumerosity of successors of an ordinal and a natural number implies equinumerosity of the originals. (Contributed by Jim Kingdon, 5-Sep-2021.)
|- ( ( A e. On /\ B e. om ) -> ( suc A ~~ suc B -> A ~~ B ) )
2.6.31  Finite sets
Theorem	fict 6810	A finite set is dominated by om. Also see finct 7054. (Contributed by Thierry Arnoux, 27-Mar-2018.)
|- ( A e. Fin -> A ~<_ om )
Theorem	fidceq 6811	Equality of members of a finite set is decidable. This may be counterintuitive: cannot any two sets be elements of a finite set? Well, to show, for example, that { B , C } is finite would require showing it is equinumerous to 1o or to 2o but to show that you'd need to know B = C or -. B = C, respectively. (Contributed by Jim Kingdon, 5-Sep-2021.)
|- ( ( A e. Fin /\ B e. A /\ C e. A ) -> DECID B = C )
Theorem	fidifsnen 6812	All decrements of a finite set are equinumerous. (Contributed by Jim Kingdon, 9-Sep-2021.)
|- ( ( X e. Fin /\ A e. X /\ B e. X ) -> ( X \ { A } ) ~~ ( X \ { B } ) )
Theorem	fidifsnid 6813	If we remove a single element from a finite set then put it back in, we end up with the original finite set. This strengthens difsnss 3702 from subset to equality when the set is finite. (Contributed by Jim Kingdon, 9-Sep-2021.)
|- ( ( A e. Fin /\ B e. A ) -> ( ( A \ { B } ) u. { B } ) = A )
Theorem	nnfi 6814	Natural numbers are finite sets. (Contributed by Stefan O'Rear, 21-Mar-2015.)
|- ( A e. om -> A e. Fin )
Theorem	enfi 6815	Equinumerous sets have the same finiteness. (Contributed by NM, 22-Aug-2008.)
|- ( A ~~ B -> ( A e. Fin <-> B e. Fin ) )
Theorem	enfii 6816	A set equinumerous to a finite set is finite. (Contributed by Mario Carneiro, 12-Mar-2015.)
|- ( ( B e. Fin /\ A ~~ B ) -> A e. Fin )
Theorem	ssfilem 6817*	Lemma for ssfiexmid 6818. (Contributed by Jim Kingdon, 3-Feb-2022.)
|- { z e. { (/) } | ph } e. Fin   =>    |- ( ph \/ -. ph )
Theorem	ssfiexmid 6818*	If any subset of a finite set is finite, excluded middle follows. One direction of Theorem 2.1 of [Bauer], p. 485. (Contributed by Jim Kingdon, 19-May-2020.)
|- A. x A. y ( ( x e. Fin /\ y C_ x ) -> y e. Fin )   =>    |- ( ph \/ -. ph )
Theorem	infiexmid 6819*	If the intersection of any finite set and any other set is finite, excluded middle follows. (Contributed by Jim Kingdon, 5-Feb-2022.)
|- ( x e. Fin -> ( x i^i y ) e. Fin )   =>    |- ( ph \/ -. ph )
Theorem	domfiexmid 6820*	If any set dominated by a finite set is finite, excluded middle follows. (Contributed by Jim Kingdon, 3-Feb-2022.)
|- ( ( x e. Fin /\ y ~<_ x ) -> y e. Fin )   =>    |- ( ph \/ -. ph )
Theorem	dif1en 6821	If a set A is equinumerous to the successor of a natural number M, then A with an element removed is equinumerous to M. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.)
|- ( ( M e. om /\ A ~~ suc M /\ X e. A ) -> ( A \ { X } ) ~~ M )
Theorem	dif1enen 6822	Subtracting one element from each of two equinumerous finite sets. (Contributed by Jim Kingdon, 5-Jun-2022.)
|- ( ph -> A e. Fin )   &    |- ( ph -> A ~~ B )   &    |- ( ph -> C e. A )   &    |- ( ph -> D e. B )   =>    |- ( ph -> ( A \ { C } ) ~~ ( B \ { D } ) )
Theorem	fiunsnnn 6823	Adding one element to a finite set which is equinumerous to a natural number. (Contributed by Jim Kingdon, 13-Sep-2021.)
|- ( ( ( A e. Fin /\ B e. ( _V \ A ) ) /\ ( N e. om /\ A ~~ N ) ) -> ( A u. { B } ) ~~ suc N )
Theorem	php5fin 6824	A finite set is not equinumerous to a set which adds one element. (Contributed by Jim Kingdon, 13-Sep-2021.)
|- ( ( A e. Fin /\ B e. ( _V \ A ) ) -> -. A ~~ ( A u. { B } ) )
Theorem	fisbth 6825	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 6826	The empty set is finite. (Contributed by FL, 14-Jul-2008.)
|- (/) e. Fin
Theorem	fin0 6827*	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 6828*	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 6829*	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 6830*	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 6831*	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 6832*	Variation of findcard2 6831 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 6833*	Deduction version of findcard2 6831. If you also need y e. Fin (which doesn't come for free due to ssfiexmid 6818), use findcard2sd 6834 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 6834*	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 6835	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 6836	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 6837	An infinite set is not finite. (Contributed by Jim Kingdon, 20-Feb-2022.)
|- ( om ~<_ A -> -. A e. Fin )
Theorem	ominf 6838	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 6709. (Contributed by NM, 2-Jun-1998.)
|- -. om e. Fin
Theorem	isinfinf 6839*	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 6840*	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 6841*	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 6842*	Lemma for fimax2gtri 6843. 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 6843*	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 6844*	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 6845*	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 6846*	An infinite set is inhabited. (Contributed by Jim Kingdon, 18-Feb-2022.)
|- ( om ~<_ A -> E. x x e. A )
Theorem	infn0 6847	An infinite set is not empty. (Contributed by NM, 23-Oct-2004.)
|- ( om ~<_ A -> A =/= (/) )
Theorem	inffiexmid 6848*	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 6849	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 6850	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 6851	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 6852	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 6853	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	ss1o0el1o 6854	Reformulation of ss1o0el1 4158 using 1o instead of { (/) }. (Contributed by BJ, 9-Aug-2024.)
|- ( A C_ 1o -> ( (/) e. A <-> A = 1o ) )
Theorem	pw1dc1 6855	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 6856	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 6857*	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 6858	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 6859*	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 6860	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 6861	The B e. V condition in unsnfi 6860. This is intended to show that unsnfi 6860 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 6862	The -. B e. A condition in unsnfi 6860. This is intended to show that unsnfi 6860 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 6863	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 6864*	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 6865*	Union of complementary parts into whole. This is a case where we can strengthen undifss 3474 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 6866	Union of complementary parts into whole. This is a case where we can strengthen undifss 3474 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 6867	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 6868	A pair is finite if it consists of two unequal sets. For the case where A = B, see snfig 6756. For the cases where one or both is a proper class, see prprc1 3667, prprc2 3668, or prprc 3669. (Contributed by Jim Kingdon, 31-May-2022.)
|- ( ( A e. V /\ B e. W /\ A =/= B ) -> { A , B } e. Fin )
Theorem	tpfidisj 6869	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	fiintim 6870*	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 6871	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 6872	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 6873	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 6874	Corollary of the Pigeonhole Principle using equality. Strengthening of phpm 6807 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 6875*	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 6876*	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	snon0 6877	An ordinal which is a singleton is { (/) }. (Contributed by Jim Kingdon, 19-Oct-2021.)
|- ( ( A e. V /\ { A } e. On ) -> A = (/) )
Theorem	fnfi 6878	A version of fnex 5688 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 6879	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 6880	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 6881	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	relcnvfi 6882	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 6883	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 6884	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 6885	A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 6886. (Contributed by AV, 10-Jan-2020.)
|- ( ( A e. V /\ F : A -1-1-> B ) -> ( F e. Fin <-> ran F e. Fin ) )
Theorem	f1vrnfibi 6886	A one-to-one function which is a set is finite if and only if its range is finite. See also f1dmvrnfibi 6885. (Contributed by AV, 10-Jan-2020.)
|- ( ( F e. V /\ F : A -1-1-> B ) -> ( F e. Fin <-> ran F e. Fin ) )
Theorem	iunfidisj 6887*	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 6888	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 6889	A set with one element is a singleton. (Contributed by FL, 18-Aug-2008.)
|- ( ( A e. B /\ B ~~ 1o ) -> B = { A } )
Theorem	en1eqsnbi 6890	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 6891*	A case where the antecedent of snexg 4145 is not needed. The class { x | ph } is from dcextest 4539. (Contributed by Mario Carneiro and Jim Kingdon, 4-Jul-2022.)
|- { { x | ph } } e. _V
Theorem	preimaf1ofi 6892	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 6893*	Lemma for fidcenum 6897. 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 6894*	Lemma for fidcenum 6897. Induction step for fidcenumlemrk 6895. (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 6895*	Lemma for fidcenum 6897. (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 6896*	Lemma for fidcenum 6897. 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 6897*	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 6898*	Lemma for isbth 6908. (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 6899*	Lemma for isbth 6908. (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 6900*	Lemma for isbth 6908. (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 ) )