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	residfi 7001	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 7002	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 7003	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 7004	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 7005	A one-to-one function whose domain is a set is finite if and only if its range is finite. See also f1vrnfibi 7006. (Contributed by AV, 10-Jan-2020.)
|- ( ( A e. V /\ F : A -1-1-> B ) -> ( F e. Fin <-> ran F e. Fin ) )
Theorem	f1vrnfibi 7006	A one-to-one function which is a set is finite if and only if its range is finite. See also f1dmvrnfibi 7005. (Contributed by AV, 10-Jan-2020.)
|- ( ( F e. V /\ F : A -1-1-> B ) -> ( F e. Fin <-> ran F e. Fin ) )
Theorem	iunfidisj 7007*	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 7008	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 7009	A set with one element is a singleton. (Contributed by FL, 18-Aug-2008.)
|- ( ( A e. B /\ B ~~ 1o ) -> B = { A } )
Theorem	en1eqsnbi 7010	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 7011*	A case where the antecedent of snexg 4214 is not needed. The class { x | ph } is from dcextest 4614. (Contributed by Mario Carneiro and Jim Kingdon, 4-Jul-2022.)
|- { { x | ph } } e. _V
Theorem	preimaf1ofi 7012	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 7013*	Lemma for fidcenum 7017. 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 7014*	Lemma for fidcenum 7017. Induction step for fidcenumlemrk 7015. (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 7015*	Lemma for fidcenum 7017. (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 7016*	Lemma for fidcenum 7017. 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 7017*	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 7018*	Lemma for isbth 7028. (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 7019*	Lemma for isbth 7028. (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 7020*	Lemma for isbth 7028. (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 7021*	Lemma for isbth 7028. (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 7022*	Lemma for isbth 7028. (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 7023*	Lemma for isbth 7028. (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 7024*	Lemma for isbth 7028. (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 7025*	Lemma for isbth 7028. (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 7026*	Lemma for isbth 7028. (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 7027*	Lemma for isbth 7028. (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 7028	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 7018 through sbthlemi10 7027; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlemi10 7027. 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 15583. (Contributed by NM, 8-Jun-1998.)
|- ( (EXMID /\ ( A ~<_ B /\ B ~<_ A ) ) -> A ~~ B )
2.6.33  Finite intersections
Syntax	cfi 7029	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 7030*	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 7033). (Contributed by FL, 27-Apr-2008.)
|- fi = ( x e. _V |-> { z | E. y e. ( ~P x i^i Fin ) z = |^| y } )
Theorem	fival 7031*	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 7032*	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 7033*	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 7034	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 7035	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 7036	The set of finite intersections of the empty set. (Contributed by Mario Carneiro, 30-Aug-2015.)
|- ( fi ` (/) ) = (/)
Theorem	fieq0 7037	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 7038	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 7039	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 7040	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 7041*	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 7042*	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 7043	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 7044	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 7045*	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 7046	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 7047	Equality theorem for supremum. (Contributed by NM, 22-May-1999.)
|- ( B = C -> sup ( B , A , R ) = sup ( C , A , R ) )
Theorem	supeq1d 7048	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 7049	Equality inference for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
|- B = C   =>    |- sup ( B , A , R ) = sup ( C , A , R )
Theorem	supeq2 7050	Equality theorem for supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( B = C -> sup ( A , B , R ) = sup ( A , C , R ) )
Theorem	supeq3 7051	Equality theorem for supremum. (Contributed by Scott Fenton, 13-Jun-2018.)
|- ( R = S -> sup ( A , B , R ) = sup ( A , B , S ) )
Theorem	supeq123d 7052	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 7053	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 7054*	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 8101) 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 7055*	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 7056*	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 7057*	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 7058*	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 7059*	A supremum belongs to its base class (closure law). See also supubti 7060 and suplubti 7061. (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 7060*	A supremum is an upper bound. See also supclti 7059 and suplubti 7061. This proof demonstrates how to expand an iota-based definition (df-iota 5216) using riotacl2 5888. (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 7061*	A supremum is the least upper bound. See also supclti 7059 and supubti 7060. (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 7062*	Bidirectional form of suplubti 7061. (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 7063*	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 7064	The supremum under an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
|- sup ( B , (/) , R ) = (/)
Theorem	supmaxti 7065*	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 7066*	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 7067*	Lemma for isoti 7068. (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 7068*	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 7069*	Lemma for supisoti 7071. (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 7070*	Lemma for supisoti 7071. (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 7071*	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 7072	Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
|- ( B = C -> inf ( B , A , R ) = inf ( C , A , R ) )
Theorem	infeq1d 7073	Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
|- ( ph -> B = C )   =>    |- ( ph -> inf ( B , A , R ) = inf ( C , A , R ) )
Theorem	infeq1i 7074	Equality inference for infimum. (Contributed by AV, 2-Sep-2020.)
|- B = C   =>    |- inf ( B , A , R ) = inf ( C , A , R )
Theorem	infeq2 7075	Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
|- ( B = C -> inf ( A , B , R ) = inf ( A , C , R ) )
Theorem	infeq3 7076	Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
|- ( R = S -> inf ( A , B , R ) = inf ( A , B , S ) )
Theorem	infeq123d 7077	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 7078	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 7079*	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 7080*	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 7081*	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 7082*	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 7083*	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 7084*	An infimum belongs to its base class (closure law). See also inflbti 7085 and infglbti 7086. (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 7085*	An infimum is a lower bound. See also infclti 7084 and infglbti 7086. (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 7086*	An infimum is the greatest lower bound. See also infclti 7084 and inflbti 7085. (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 7087*	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 7088*	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 7089*	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 7090*	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 7091*	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 7092	The infimum regarding an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
|- inf ( B , (/) , R ) = (/)
Theorem	infisoti 7093*	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 7094	Existence of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( A e. C -> sup ( B , A , R ) e. _V )
Theorem	infex2g 7095	Existence of infimum. (Contributed by Jim Kingdon, 1-Oct-2024.)
|- ( A e. C -> inf ( B , A , R ) e. _V )
2.6.35  Ordinal isomorphism
Theorem	ordiso2 7096	Generalize ordiso 7097 to proper classes. (Contributed by Mario Carneiro, 24-Jun-2015.)
|- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ Ord B ) -> A = B )
Theorem	ordiso 7097*	Order-isomorphic ordinal numbers are equal. (Contributed by Jeff Hankins, 16-Oct-2009.) (Proof shortened by Mario Carneiro, 24-Jun-2015.)
|- ( ( A e. On /\ B e. On ) -> ( A = B <-> E. f f Isom _E , _E ( A , B ) ) )
2.6.36  Disjoint union
2.6.36.1  Disjoint union
Syntax	cdju 7098	Extend class notation to include disjoint union of two classes.
class ( A ⊔ B )
Definition	df-dju 7099	Disjoint union of two classes. This is a way of creating a class which contains elements corresponding to each element of A or B, tagging each one with whether it came from A or B. (Contributed by Jim Kingdon, 20-Jun-2022.)
|- ( A ⊔ B ) = ( ( { (/) } X. A ) u. ( { 1o } X. B ) )
Theorem	djueq12 7100	Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
|- ( ( A = B /\ C = D ) -> ( A ⊔ C ) = ( B ⊔ D ) )