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	snon0 7001	An ordinal which is a singleton is { (/) }. (Contributed by Jim Kingdon, 19-Oct-2021.)
|- ( ( A e. V /\ { A } e. On ) -> A = (/) )
Theorem	fnfi 7002	A version of fnex 5784 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 7003	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 7004	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 7005	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 7006	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 7007	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 7008	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 7009	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 7010	A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 7011. (Contributed by AV, 10-Jan-2020.)
|- ( ( A e. V /\ F : A -1-1-> B ) -> ( F e. Fin <-> ran F e. Fin ) )
Theorem	f1vrnfibi 7011	A one-to-one function which is a set is finite if and only if its range is finite. See also f1dmvrnfibi 7010. (Contributed by AV, 10-Jan-2020.)
|- ( ( F e. V /\ F : A -1-1-> B ) -> ( F e. Fin <-> ran F e. Fin ) )
Theorem	iunfidisj 7012*	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 7013	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 7014	A set with one element is a singleton. (Contributed by FL, 18-Aug-2008.)
|- ( ( A e. B /\ B ~~ 1o ) -> B = { A } )
Theorem	en1eqsnbi 7015	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 7016*	A case where the antecedent of snexg 4217 is not needed. The class { x | ph } is from dcextest 4617. (Contributed by Mario Carneiro and Jim Kingdon, 4-Jul-2022.)
|- { { x | ph } } e. _V
Theorem	preimaf1ofi 7017	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 7018*	Lemma for fidcenum 7022. 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 7019*	Lemma for fidcenum 7022. Induction step for fidcenumlemrk 7020. (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 7020*	Lemma for fidcenum 7022. (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 7021*	Lemma for fidcenum 7022. 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 7022*	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 7023*	Lemma for isbth 7033. (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 7024*	Lemma for isbth 7033. (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 7025*	Lemma for isbth 7033. (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 7026*	Lemma for isbth 7033. (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 7027*	Lemma for isbth 7033. (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 7028*	Lemma for isbth 7033. (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 7029*	Lemma for isbth 7033. (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 7030*	Lemma for isbth 7033. (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 7031*	Lemma for isbth 7033. (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 7032*	Lemma for isbth 7033. (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 7033	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 7023 through sbthlemi10 7032; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlemi10 7032. 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 15667. (Contributed by NM, 8-Jun-1998.)
|- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> A ~~ B )
2.6.33  Finite intersections
Syntax	cfi 7034	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 7035*	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 7038). (Contributed by FL, 27-Apr-2008.)
|- fi = ( x e. _V |-> { z | E. y e. ( ~P x i^i Fin ) z = |^| y } )
Theorem	fival 7036*	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 7037*	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 7038*	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 7039	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 7040	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 7041	The set of finite intersections of the empty set. (Contributed by Mario Carneiro, 30-Aug-2015.)
|- ( fi ` (/) ) = (/)
Theorem	fieq0 7042	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 7043	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 7044	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 7045	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 7046*	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 7047*	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 7048	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 7049	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 7050*	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 7051	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 7052	Equality theorem for supremum. (Contributed by NM, 22-May-1999.)
|- ( B = C -> sup ( B , A , R ) = sup ( C , A , R ) )
Theorem	supeq1d 7053	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 7054	Equality inference for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
|- B = C   =>    |- sup ( B , A , R ) = sup ( C , A , R )
Theorem	supeq2 7055	Equality theorem for supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( B = C -> sup ( A , B , R ) = sup ( A , C , R ) )
Theorem	supeq3 7056	Equality theorem for supremum. (Contributed by Scott Fenton, 13-Jun-2018.)
|- ( R = S -> sup ( A , B , R ) = sup ( A , B , S ) )
Theorem	supeq123d 7057	Equality deduction for supremum. (Contributed by Stefan O'Rear, 20-Jan-2015.)
|- ( ph -> A = D )   &    |- ( ph -> B = E )   &    |- ( ph -> C = F )   =>    |- ( ph -> sup ( A , B , C ) = sup ( D , E , F ) )
Theorem	nfsup 7058	Hypothesis builder for supremum. (Contributed by Mario Carneiro, 20-Mar-2014.)
|- F/_ x A   &    |- F/_ x B   &    |- F/_ x R   =>    |- F/_ x sup ( A , B , R )
Theorem	supmoti 7059*	Any class B has at most one supremum in A (where R is interpreted as 'less than'). The hypothesis is satisfied by real numbers (see lttri3 8106) or other orders which correspond to tight apartnesses. (Contributed by Jim Kingdon, 23-Nov-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   =>    |- ( ph -> E* x e. A ( A. y e. B -. x R y /\ A. y e. A ( y R x -> E. z e. B y R z ) ) )
Theorem	supeuti 7060*	A supremum is unique. Similar to Theorem I.26 of [Apostol] p. 24 (but for suprema in general). (Contributed by Jim Kingdon, 23-Nov-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> E. x e. A ( A. y e. B -. x R y /\ A. y e. A ( y R x -> E. z e. B y R z ) ) )   =>    |- ( ph -> E! x e. A ( A. y e. B -. x R y /\ A. y e. A ( y R x -> E. z e. B y R z ) ) )
Theorem	supval2ti 7061*	Alternate expression for the supremum. (Contributed by Jim Kingdon, 23-Nov-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> E. x e. A ( A. y e. B -. x R y /\ A. y e. A ( y R x -> E. z e. B y R z ) ) )   =>    |- ( ph -> sup ( B , A , R ) = ( iota_ x e. A ( A. y e. B -. x R y /\ A. y e. A ( y R x -> E. z e. B y R z ) ) ) )
Theorem	eqsupti 7062*	Sufficient condition for an element to be equal to the supremum. (Contributed by Jim Kingdon, 23-Nov-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   =>    |- ( ph -> ( ( C e. A /\ A. y e. B -. C R y /\ A. y e. A ( y R C -> E. z e. B y R z ) ) -> sup ( B , A , R ) = C ) )
Theorem	eqsuptid 7063*	Sufficient condition for an element to be equal to the supremum. (Contributed by Jim Kingdon, 24-Nov-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> C e. A )   &    |- ( ( ph /\ y e. B ) -> -. C R y )   &    |- ( ( ph /\ ( y e. A /\ y R C ) ) -> E. z e. B y R z )   =>    |- ( ph -> sup ( B , A , R ) = C )
Theorem	supclti 7064*	A supremum belongs to its base class (closure law). See also supubti 7065 and suplubti 7066. (Contributed by Jim Kingdon, 24-Nov-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> E. x e. A ( A. y e. B -. x R y /\ A. y e. A ( y R x -> E. z e. B y R z ) ) )   =>    |- ( ph -> sup ( B , A , R ) e. A )
Theorem	supubti 7065*	A supremum is an upper bound. See also supclti 7064 and suplubti 7066. This proof demonstrates how to expand an iota-based definition (df-iota 5219) using riotacl2 5891. (Contributed by Jim Kingdon, 24-Nov-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> E. x e. A ( A. y e. B -. x R y /\ A. y e. A ( y R x -> E. z e. B y R z ) ) )   =>    |- ( ph -> ( C e. B -> -. sup ( B , A , R ) R C ) )
Theorem	suplubti 7066*	A supremum is the least upper bound. See also supclti 7064 and supubti 7065. (Contributed by Jim Kingdon, 24-Nov-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> E. x e. A ( A. y e. B -. x R y /\ A. y e. A ( y R x -> E. z e. B y R z ) ) )   =>    |- ( ph -> ( ( C e. A /\ C R sup ( B , A , R ) ) -> E. z e. B C R z ) )
Theorem	suplub2ti 7067*	Bidirectional form of suplubti 7066. (Contributed by Jim Kingdon, 17-Jan-2022.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> E. x e. A ( A. y e. B -. x R y /\ A. y e. A ( y R x -> E. z e. B y R z ) ) )   &    |- ( ph -> R Or A )   &    |- ( ph -> B C_ A )   =>    |- ( ( ph /\ C e. A ) -> ( C R sup ( B , A , R ) <-> E. z e. B C R z ) )
Theorem	supelti 7068*	Supremum membership in a set. (Contributed by Jim Kingdon, 16-Jan-2022.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> E. x e. C ( A. y e. B -. x R y /\ A. y e. A ( y R x -> E. z e. B y R z ) ) )   &    |- ( ph -> C C_ A )   =>    |- ( ph -> sup ( B , A , R ) e. C )
Theorem	sup00 7069	The supremum under an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
|- sup ( B , (/) , R ) = (/)
Theorem	supmaxti 7070*	The greatest element of a set is its supremum. Note that the converse is not true; the supremum might not be an element of the set considered. (Contributed by Jim Kingdon, 24-Nov-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> C e. A )   &    |- ( ph -> C e. B )   &    |- ( ( ph /\ y e. B ) -> -. C R y )   =>    |- ( ph -> sup ( B , A , R ) = C )
Theorem	supsnti 7071*	The supremum of a singleton. (Contributed by Jim Kingdon, 26-Nov-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> B e. A )   =>    |- ( ph -> sup ( { B } , A , R ) = B )
Theorem	isotilem 7072*	Lemma for isoti 7073. (Contributed by Jim Kingdon, 26-Nov-2021.)
|- ( F Isom R , S ( A , B ) -> ( A. x e. B A. y e. B ( x = y <-> ( -. x S y /\ -. y S x ) ) -> A. u e. A A. v e. A ( u = v <-> ( -. u R v /\ -. v R u ) ) ) )
Theorem	isoti 7073*	An isomorphism preserves tightness. (Contributed by Jim Kingdon, 26-Nov-2021.)
|- ( F Isom R , S ( A , B ) -> ( A. u e. A A. v e. A ( u = v <-> ( -. u R v /\ -. v R u ) ) <-> A. u e. B A. v e. B ( u = v <-> ( -. u S v /\ -. v S u ) ) ) )
Theorem	supisolem 7074*	Lemma for supisoti 7076. (Contributed by Mario Carneiro, 24-Dec-2016.)
|- ( ph -> F Isom R , S ( A , B ) )   &    |- ( ph -> C C_ A )   =>    |- ( ( ph /\ D e. A ) -> ( ( A. y e. C -. D R y /\ A. y e. A ( y R D -> E. z e. C y R z ) ) <-> ( A. w e. ( F " C ) -. ( F ` D ) S w /\ A. w e. B ( w S ( F ` D ) -> E. v e. ( F " C ) w S v ) ) ) )
Theorem	supisoex 7075*	Lemma for supisoti 7076. (Contributed by Mario Carneiro, 24-Dec-2016.)
|- ( ph -> F Isom R , S ( A , B ) )   &    |- ( ph -> C C_ A )   &    |- ( ph -> E. x e. A ( A. y e. C -. x R y /\ A. y e. A ( y R x -> E. z e. C y R z ) ) )   =>    |- ( ph -> E. u e. B ( A. w e. ( F " C ) -. u S w /\ A. w e. B ( w S u -> E. v e. ( F " C ) w S v ) ) )
Theorem	supisoti 7076*	Image of a supremum under an isomorphism. (Contributed by Jim Kingdon, 26-Nov-2021.)
|- ( ph -> F Isom R , S ( A , B ) )   &    |- ( ph -> C C_ A )   &    |- ( ph -> E. x e. A ( A. y e. C -. x R y /\ A. y e. A ( y R x -> E. z e. C y R z ) ) )   &    |- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   =>    |- ( ph -> sup ( ( F " C ) , B , S ) = ( F ` sup ( C , A , R ) ) )
Theorem	infeq1 7077	Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
|- ( B = C -> inf ( B , A , R ) = inf ( C , A , R ) )
Theorem	infeq1d 7078	Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
|- ( ph -> B = C )   =>    |- ( ph -> inf ( B , A , R ) = inf ( C , A , R ) )
Theorem	infeq1i 7079	Equality inference for infimum. (Contributed by AV, 2-Sep-2020.)
|- B = C   =>    |- inf ( B , A , R ) = inf ( C , A , R )
Theorem	infeq2 7080	Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
|- ( B = C -> inf ( A , B , R ) = inf ( A , C , R ) )
Theorem	infeq3 7081	Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
|- ( R = S -> inf ( A , B , R ) = inf ( A , B , S ) )
Theorem	infeq123d 7082	Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
|- ( ph -> A = D )   &    |- ( ph -> B = E )   &    |- ( ph -> C = F )   =>    |- ( ph -> inf ( A , B , C ) = inf ( D , E , F ) )
Theorem	nfinf 7083	Hypothesis builder for infimum. (Contributed by AV, 2-Sep-2020.)
|- F/_ x A   &    |- F/_ x B   &    |- F/_ x R   =>    |- F/_ xinf ( A , B , R )
Theorem	cnvinfex 7084*	Two ways of expressing existence of an infimum (one in terms of converse). (Contributed by Jim Kingdon, 17-Dec-2021.)
|- ( ph -> E. x e. A ( A. y e. B -. y R x /\ A. y e. A ( x R y -> E. z e. B z R y ) ) )   =>    |- ( ph -> E. x e. A ( A. y e. B -. x `' R y /\ A. y e. A ( y `' R x -> E. z e. B y `' R z ) ) )
Theorem	cnvti 7085*	If a relation satisfies a condition corresponding to tightness of an apartness generated by an order, so does its converse. (Contributed by Jim Kingdon, 17-Dec-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   =>    |- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u `' R v /\ -. v `' R u ) ) )
Theorem	eqinfti 7086*	Sufficient condition for an element to be equal to the infimum. (Contributed by Jim Kingdon, 16-Dec-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   =>    |- ( ph -> ( ( C e. A /\ A. y e. B -. y R C /\ A. y e. A ( C R y -> E. z e. B z R y ) ) -> inf ( B , A , R ) = C ) )
Theorem	eqinftid 7087*	Sufficient condition for an element to be equal to the infimum. (Contributed by Jim Kingdon, 16-Dec-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> C e. A )   &    |- ( ( ph /\ y e. B ) -> -. y R C )   &    |- ( ( ph /\ ( y e. A /\ C R y ) ) -> E. z e. B z R y )   =>    |- ( ph -> inf ( B , A , R ) = C )
Theorem	infvalti 7088*	Alternate expression for the infimum. (Contributed by Jim Kingdon, 17-Dec-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> E. x e. A ( A. y e. B -. y R x /\ A. y e. A ( x R y -> E. z e. B z R y ) ) )   =>    |- ( ph -> inf ( B , A , R ) = ( iota_ x e. A ( A. y e. B -. y R x /\ A. y e. A ( x R y -> E. z e. B z R y ) ) ) )
Theorem	infclti 7089*	An infimum belongs to its base class (closure law). See also inflbti 7090 and infglbti 7091. (Contributed by Jim Kingdon, 17-Dec-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> E. x e. A ( A. y e. B -. y R x /\ A. y e. A ( x R y -> E. z e. B z R y ) ) )   =>    |- ( ph -> inf ( B , A , R ) e. A )
Theorem	inflbti 7090*	An infimum is a lower bound. See also infclti 7089 and infglbti 7091. (Contributed by Jim Kingdon, 18-Dec-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> E. x e. A ( A. y e. B -. y R x /\ A. y e. A ( x R y -> E. z e. B z R y ) ) )   =>    |- ( ph -> ( C e. B -> -. C Rinf ( B , A , R ) ) )
Theorem	infglbti 7091*	An infimum is the greatest lower bound. See also infclti 7089 and inflbti 7090. (Contributed by Jim Kingdon, 18-Dec-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> E. x e. A ( A. y e. B -. y R x /\ A. y e. A ( x R y -> E. z e. B z R y ) ) )   =>    |- ( ph -> ( ( C e. A /\ inf ( B , A , R ) R C ) -> E. z e. B z R C ) )
Theorem	infnlbti 7092*	A lower bound is not greater than the infimum. (Contributed by Jim Kingdon, 18-Dec-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> E. x e. A ( A. y e. B -. y R x /\ A. y e. A ( x R y -> E. z e. B z R y ) ) )   =>    |- ( ph -> ( ( C e. A /\ A. z e. B -. z R C ) -> -. inf ( B , A , R ) R C ) )
Theorem	infminti 7093*	The smallest element of a set is its infimum. Note that the converse is not true; the infimum might not be an element of the set considered. (Contributed by Jim Kingdon, 18-Dec-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> C e. A )   &    |- ( ph -> C e. B )   &    |- ( ( ph /\ y e. B ) -> -. y R C )   =>    |- ( ph -> inf ( B , A , R ) = C )
Theorem	infmoti 7094*	Any class B has at most one infimum in A (where R is interpreted as 'less than'). (Contributed by Jim Kingdon, 18-Dec-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   =>    |- ( ph -> E* x e. A ( A. y e. B -. y R x /\ A. y e. A ( x R y -> E. z e. B z R y ) ) )
Theorem	infeuti 7095*	An infimum is unique. (Contributed by Jim Kingdon, 19-Dec-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> E. x e. A ( A. y e. B -. y R x /\ A. y e. A ( x R y -> E. z e. B z R y ) ) )   =>    |- ( ph -> E! x e. A ( A. y e. B -. y R x /\ A. y e. A ( x R y -> E. z e. B z R y ) ) )
Theorem	infsnti 7096*	The infimum of a singleton. (Contributed by Jim Kingdon, 19-Dec-2021.)
|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   &    |- ( ph -> B e. A )   =>    |- ( ph -> inf ( { B } , A , R ) = B )
Theorem	inf00 7097	The infimum regarding an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
|- inf ( B , (/) , R ) = (/)
Theorem	infisoti 7098*	Image of an infimum under an isomorphism. (Contributed by Jim Kingdon, 19-Dec-2021.)
|- ( ph -> F Isom R , S ( A , B ) )   &    |- ( ph -> C C_ A )   &    |- ( ph -> E. x e. A ( A. y e. C -. y R x /\ A. y e. A ( x R y -> E. z e. C z R y ) ) )   &    |- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )   =>    |- ( ph -> inf ( ( F " C ) , B , S ) = ( F ` inf ( C , A , R ) ) )
Theorem	supex2g 7099	Existence of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( A e. C -> sup ( B , A , R ) e. _V )
Theorem	infex2g 7100	Existence of infimum. (Contributed by Jim Kingdon, 1-Oct-2024.)
|- ( A e. C -> inf ( B , A , R ) e. _V )