P. 70 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6901-7000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	findcard 6901*	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 6902*	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 6903*	Variation of findcard2 6902 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 6904*	Deduction version of findcard2 6902. If you also need y e. Fin (which doesn't come for free due to ssfiexmid 6889), use findcard2sd 6905 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 6905*	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 6906	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 6907	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 6908	An infinite set is not finite. (Contributed by Jim Kingdon, 20-Feb-2022.)
|- ( om ~<_ A -> -. A e. Fin )
Theorem	ominf 6909	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 6780. (Contributed by NM, 2-Jun-1998.)
|- -. om e. Fin
Theorem	isinfinf 6910*	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 6911*	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 6912*	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 6913*	Lemma for fimax2gtri 6914. 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 6914*	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 6915*	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 6916*	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 6917*	An infinite set is inhabited. (Contributed by Jim Kingdon, 18-Feb-2022.)
|- ( om ~<_ A -> E. x x e. A )
Theorem	infn0 6918	An infinite set is not empty. (Contributed by NM, 23-Oct-2004.)
|- ( om ~<_ A -> A =/= (/) )
Theorem	inffiexmid 6919*	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 6920	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 6921	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 6922	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 6923	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 6924	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 6925	Reformulation of ss1o0el1 4209 using 1o instead of { (/) }. (Contributed by BJ, 9-Aug-2024.)
|- ( A C_ 1o -> ( (/) e. A <-> A = 1o ) )
Theorem	pw1dc1 6926	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 6927	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 6928*	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 6929	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 6930*	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 6931	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 6932	The B e. V condition in unsnfi 6931. This is intended to show that unsnfi 6931 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 6933	The -. B e. A condition in unsnfi 6931. This is intended to show that unsnfi 6931 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 6934	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 6935*	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 6936*	Union of complementary parts into whole. This is a case where we can strengthen undifss 3515 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 6937	Union of complementary parts into whole. This is a case where we can strengthen undifss 3515 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 6938	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 6939	A pair is finite if it consists of two unequal sets. For the case where A = B, see snfig 6827. For the cases where one or both is a proper class, see prprc1 3712, prprc2 3713, or prprc 3714. (Contributed by Jim Kingdon, 31-May-2022.)
|- ( ( A e. V /\ B e. W /\ A =/= B ) -> { A , B } e. Fin )
Theorem	tpfidisj 6940	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 6941*	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 6942	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 6943	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 6944	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 6945	Corollary of the Pigeonhole Principle using equality. Strengthening of phpm 6878 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 6946*	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 6947*	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 6948	An ordinal which is a singleton is { (/) }. (Contributed by Jim Kingdon, 19-Oct-2021.)
|- ( ( A e. V /\ { A } e. On ) -> A = (/) )
Theorem	fnfi 6949	A version of fnex 5751 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 6950	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 6951	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 6952	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 6953	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 6954	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 6955	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 6956	A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 6957. (Contributed by AV, 10-Jan-2020.)
|- ( ( A e. V /\ F : A -1-1-> B ) -> ( F e. Fin <-> ran F e. Fin ) )
Theorem	f1vrnfibi 6957	A one-to-one function which is a set is finite if and only if its range is finite. See also f1dmvrnfibi 6956. (Contributed by AV, 10-Jan-2020.)
|- ( ( F e. V /\ F : A -1-1-> B ) -> ( F e. Fin <-> ran F e. Fin ) )
Theorem	iunfidisj 6958*	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 6959	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 6960	A set with one element is a singleton. (Contributed by FL, 18-Aug-2008.)
|- ( ( A e. B /\ B ~~ 1o ) -> B = { A } )
Theorem	en1eqsnbi 6961	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 6962*	A case where the antecedent of snexg 4196 is not needed. The class { x | ph } is from dcextest 4592. (Contributed by Mario Carneiro and Jim Kingdon, 4-Jul-2022.)
|- { { x | ph } } e. _V
Theorem	preimaf1ofi 6963	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 6964*	Lemma for fidcenum 6968. 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 6965*	Lemma for fidcenum 6968. Induction step for fidcenumlemrk 6966. (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 6966*	Lemma for fidcenum 6968. (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 6967*	Lemma for fidcenum 6968. 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 6968*	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 6969*	Lemma for isbth 6979. (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 6970*	Lemma for isbth 6979. (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 6971*	Lemma for isbth 6979. (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 6972*	Lemma for isbth 6979. (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 6973*	Lemma for isbth 6979. (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 6974*	Lemma for isbth 6979. (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 6975*	Lemma for isbth 6979. (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 6976*	Lemma for isbth 6979. (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 6977*	Lemma for isbth 6979. (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 6978*	Lemma for isbth 6979. (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 6979	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 6969 through sbthlemi10 6978; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlemi10 6978. 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 15043. (Contributed by NM, 8-Jun-1998.)
|- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> A ~~ B )
2.6.33  Finite intersections
Syntax	cfi 6980	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 6981*	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 6984). (Contributed by FL, 27-Apr-2008.)
|- fi = ( x e. _V |-> { z | E. y e. ( ~P x i^i Fin ) z = |^| y } )
Theorem	fival 6982*	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 6983*	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 6984*	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 6985	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 6986	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 6987	The set of finite intersections of the empty set. (Contributed by Mario Carneiro, 30-Aug-2015.)
|- ( fi ` (/) ) = (/)
Theorem	fieq0 6988	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 6989	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 ) )
Theorem	fiuni 6990	The union of the finite intersections of a set is simply the union of the set itself. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
|- ( A e. V -> U. A = U. ( fi ` A ) )
Theorem	fipwssg 6991	If a set is a family of subsets of some base set, then so is its finite intersection. (Contributed by Stefan O'Rear, 2-Aug-2015.)
|- ( ( A e. V /\ A C_ ~P X ) -> ( fi ` A ) C_ ~P X )
Theorem	fifo 6992*	Describe a surjection from nonempty finite sets to finite intersections. (Contributed by Mario Carneiro, 18-May-2015.)
|- F = ( y e. ( ( ~P A i^i Fin ) \ { (/) } ) |-> |^| y )   =>    |- ( A e. V -> F : ( ( ~P A i^i Fin ) \ { (/) } ) -onto-> ( fi ` A ) )
Theorem	dcfi 6993*	Decidability of a family of propositions indexed by a finite set. (Contributed by Jim Kingdon, 30-Sep-2024.)
|- ( ( A e. Fin /\ A. x e. A DECID ph ) -> DECID A. x e. A ph )
2.6.34  Supremum and infimum
Syntax	csup 6994	Extend class notation to include supremum of class A. Here R is ordinarily a relation that strictly orders class B. For example, R could be 'less than' and B could be the set of real numbers.
class sup ( A , B , R )
Syntax	cinf 6995	Extend class notation to include infimum of class A. Here R is ordinarily a relation that strictly orders class B. For example, R could be 'less than' and B could be the set of real numbers.
class inf ( A , B , R )
Definition	df-sup 6996*	Define the supremum of class A. It is meaningful when R is a relation that strictly orders B and when the supremum exists. (Contributed by NM, 22-May-1999.)
|- sup ( A , B , R ) = U. { x e. B | ( A. y e. A -. x R y /\ A. y e. B ( y R x -> E. z e. A y R z ) ) }
Definition	df-inf 6997	Define the infimum of class A. It is meaningful when R is a relation that strictly orders B and when the infimum exists. For example, R could be 'less than', B could be the set of real numbers, and A could be the set of all positive reals; in this case the infimum is 0. The infimum is defined as the supremum using the converse ordering relation. In the given example, 0 is the supremum of all reals (greatest real number) for which all positive reals are greater. (Contributed by AV, 2-Sep-2020.)
|- inf ( A , B , R ) = sup ( A , B , `' R )
Theorem	supeq1 6998	Equality theorem for supremum. (Contributed by NM, 22-May-1999.)
|- ( B = C -> sup ( B , A , R ) = sup ( C , A , R ) )
Theorem	supeq1d 6999	Equality deduction for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
|- ( ph -> B = C )   =>    |- ( ph -> sup ( B , A , R ) = sup ( C , A , R ) )
Theorem	supeq1i 7000	Equality inference for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
|- B = C   =>    |- sup ( B , A , R ) = sup ( C , A , R )