P. 72 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7101-7200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	infnlbti 7101*	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 7102*	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 7103*	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 7104*	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 7105*	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 7106	The infimum regarding an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
|- inf ( B , (/) , R ) = (/)
Theorem	infisoti 7107*	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 7108	Existence of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( A e. C -> sup ( B , A , R ) e. _V )
Theorem	infex2g 7109	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 7110	Generalize ordiso 7111 to proper classes. (Contributed by Mario Carneiro, 24-Jun-2015.)
|- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ Ord B ) -> A = B )
Theorem	ordiso 7111*	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 7112	Extend class notation to include disjoint union of two classes.
class ( A ⊔ B )
Definition	df-dju 7113	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 7114	Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
|- ( ( A = B /\ C = D ) -> ( A ⊔ C ) = ( B ⊔ D ) )
Theorem	djueq1 7115	Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
|- ( A = B -> ( A ⊔ C ) = ( B ⊔ C ) )
Theorem	djueq2 7116	Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
|- ( A = B -> ( C ⊔ A ) = ( C ⊔ B ) )
Theorem	nfdju 7117	Bound-variable hypothesis builder for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
|- F/_ x A   &    |- F/_ x B   =>    |- F/_ x ( A ⊔ B )
Theorem	djuex 7118	The disjoint union of sets is a set. See also the more precise djuss 7145. (Contributed by AV, 28-Jun-2022.)
|- ( ( A e. V /\ B e. W ) -> ( A ⊔ B ) e. _V )
Theorem	djuexb 7119	The disjoint union of two classes is a set iff both classes are sets. (Contributed by Jim Kingdon, 6-Sep-2023.)
|- ( ( A e. _V /\ B e. _V ) <-> ( A ⊔ B ) e. _V )
2.6.36.2  Left and right injections of a disjoint union In this section, we define the left and right injections of a disjoint union and prove their main properties. These injections are restrictions of the "template" functions inl and inr, which appear in most applications in the form (inl |` A ) and (inr |` B ).
Syntax	cinl 7120	Extend class notation to include left injection of a disjoint union.
class inl
Syntax	cinr 7121	Extend class notation to include right injection of a disjoint union.
class inr
Definition	df-inl 7122	Left injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.)
|- inl = ( x e. _V |-> <. (/) , x >. )
Definition	df-inr 7123	Right injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.)
|- inr = ( x e. _V |-> <. 1o , x >. )
Theorem	djulclr 7124	Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) (Revised by BJ, 6-Jul-2022.)
|- ( C e. A -> ( (inl |` A ) ` C ) e. ( A ⊔ B ) )
Theorem	djurclr 7125	Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) (Revised by BJ, 6-Jul-2022.)
|- ( C e. B -> ( (inr |` B ) ` C ) e. ( A ⊔ B ) )
Theorem	djulcl 7126	Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)
|- ( C e. A -> (inl ` C ) e. ( A ⊔ B ) )
Theorem	djurcl 7127	Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)
|- ( C e. B -> (inr ` C ) e. ( A ⊔ B ) )
Theorem	djuf1olem 7128*	Lemma for djulf1o 7133 and djurf1o 7134. (Contributed by BJ and Jim Kingdon, 4-Jul-2022.)
|- X e. _V   &    |- F = ( x e. A |-> <. X , x >. )   =>    |- F : A -1-1-onto-> ( { X } X. A )
Theorem	djuf1olemr 7129*	Lemma for djulf1or 7131 and djurf1or 7132. For a version of this lemma with F defined on A and no restriction in the conclusion, see djuf1olem 7128. (Contributed by BJ and Jim Kingdon, 4-Jul-2022.)
|- X e. _V   &    |- F = ( x e. _V |-> <. X , x >. )   =>    |- ( F |` A ) : A -1-1-onto-> ( { X } X. A )
Theorem	djulclb 7130	Left biconditional closure of disjoint union. (Contributed by Jim Kingdon, 2-Jul-2022.)
|- ( C e. V -> ( C e. A <-> (inl ` C ) e. ( A ⊔ B ) ) )
Theorem	djulf1or 7131	The left injection function on all sets is one to one and onto. (Contributed by BJ and Jim Kingdon, 22-Jun-2022.)
|- (inl |` A ) : A -1-1-onto-> ( { (/) } X. A )
Theorem	djurf1or 7132	The right injection function on all sets is one to one and onto. (Contributed by BJ and Jim Kingdon, 22-Jun-2022.)
|- (inr |` A ) : A -1-1-onto-> ( { 1o } X. A )
Theorem	djulf1o 7133	The left injection function on all sets is one to one and onto. (Contributed by Jim Kingdon, 22-Jun-2022.)
|- inl : _V -1-1-onto-> ( { (/) } X. _V )
Theorem	djurf1o 7134	The right injection function on all sets is one to one and onto. (Contributed by Jim Kingdon, 22-Jun-2022.)
|- inr : _V -1-1-onto-> ( { 1o } X. _V )
Theorem	inresflem 7135*	Lemma for inlresf1 7136 and inrresf1 7137. (Contributed by BJ, 4-Jul-2022.)
|- F : A -1-1-onto-> ( { X } X. A )   &    |- ( x e. A -> ( F ` x ) e. B )   =>    |- F : A -1-1-> B
Theorem	inlresf1 7136	The left injection restricted to the left class of a disjoint union is an injective function from the left class into the disjoint union. (Contributed by AV, 28-Jun-2022.)
|- (inl |` A ) : A -1-1-> ( A ⊔ B )
Theorem	inrresf1 7137	The right injection restricted to the right class of a disjoint union is an injective function from the right class into the disjoint union. (Contributed by AV, 28-Jun-2022.)
|- (inr |` B ) : B -1-1-> ( A ⊔ B )
Theorem	djuinr 7138	The ranges of any left and right injections are disjoint. Remark: the extra generality offered by the two restrictions makes the theorem more readily usable (e.g., by djudom 7168 and djufun 7179) while the simpler statement |- ( ran inl i^i ran inr ) = (/) is easily recovered from it by substituting _V for both A and B as done in casefun 7160). (Contributed by BJ and Jim Kingdon, 21-Jun-2022.)
|- ( ran (inl |` A ) i^i ran (inr |` B ) ) = (/)
Theorem	djuin 7139	The images of any classes under right and left injection produce disjoint sets. (Contributed by Jim Kingdon, 21-Jun-2022.) (Proof shortened by BJ, 9-Jul-2023.)
|- ( (inl " A ) i^i (inr " B ) ) = (/)
Theorem	inl11 7140	Left injection is one-to-one. (Contributed by Jim Kingdon, 12-Jul-2023.)
|- ( ( A e. V /\ B e. W ) -> ( (inl ` A ) = (inl ` B ) <-> A = B ) )
Theorem	djuunr 7141	The disjoint union of two classes is the union of the images of those two classes under right and left injection. (Contributed by Jim Kingdon, 22-Jun-2022.) (Proof shortened by BJ, 6-Jul-2022.)
|- ( ran (inl |` A ) u. ran (inr |` B ) ) = ( A ⊔ B )
Theorem	djuun 7142	The disjoint union of two classes is the union of the images of those two classes under right and left injection. (Contributed by Jim Kingdon, 22-Jun-2022.) (Proof shortened by BJ, 9-Jul-2023.)
|- ( (inl " A ) u. (inr " B ) ) = ( A ⊔ B )
Theorem	eldju 7143*	Element of a disjoint union. (Contributed by BJ and Jim Kingdon, 23-Jun-2022.)
|- ( C e. ( A ⊔ B ) <-> ( E. x e. A C = ( (inl |` A ) ` x ) \/ E. x e. B C = ( (inr |` B ) ` x ) ) )
Theorem	djur 7144*	A member of a disjoint union can be mapped from one of the classes which produced it. (Contributed by Jim Kingdon, 23-Jun-2022.) Upgrade implication to biconditional and shorten proof. (Revised by BJ, 14-Jul-2023.)
|- ( C e. ( A ⊔ B ) <-> ( E. x e. A C = (inl ` x ) \/ E. x e. B C = (inr ` x ) ) )
2.6.36.3  Universal property of the disjoint union
Theorem	djuss 7145	A disjoint union is a subset of a Cartesian product. (Contributed by AV, 25-Jun-2022.)
|- ( A ⊔ B ) C_ ( { (/) , 1o } X. ( A u. B ) )
Theorem	eldju1st 7146	The first component of an element of a disjoint union is either (/) or 1o. (Contributed by AV, 26-Jun-2022.)
|- ( X e. ( A ⊔ B ) -> ( ( 1st ` X ) = (/) \/ ( 1st ` X ) = 1o ) )
Theorem	eldju2ndl 7147	The second component of an element of a disjoint union is an element of the left class of the disjoint union if its first component is the empty set. (Contributed by AV, 26-Jun-2022.)
|- ( ( X e. ( A ⊔ B ) /\ ( 1st ` X ) = (/) ) -> ( 2nd ` X ) e. A )
Theorem	eldju2ndr 7148	The second component of an element of a disjoint union is an element of the right class of the disjoint union if its first component is not the empty set. (Contributed by AV, 26-Jun-2022.)
|- ( ( X e. ( A ⊔ B ) /\ ( 1st ` X ) =/= (/) ) -> ( 2nd ` X ) e. B )
Theorem	1stinl 7149	The first component of the value of a left injection is the empty set. (Contributed by AV, 27-Jun-2022.)
|- ( X e. V -> ( 1st ` (inl ` X ) ) = (/) )
Theorem	2ndinl 7150	The second component of the value of a left injection is its argument. (Contributed by AV, 27-Jun-2022.)
|- ( X e. V -> ( 2nd ` (inl ` X ) ) = X )
Theorem	1stinr 7151	The first component of the value of a right injection is 1o. (Contributed by AV, 27-Jun-2022.)
|- ( X e. V -> ( 1st ` (inr ` X ) ) = 1o )
Theorem	2ndinr 7152	The second component of the value of a right injection is its argument. (Contributed by AV, 27-Jun-2022.)
|- ( X e. V -> ( 2nd ` (inr ` X ) ) = X )
Theorem	djune 7153	Left and right injection never produce equal values. (Contributed by Jim Kingdon, 2-Jul-2022.)
|- ( ( A e. V /\ B e. W ) -> (inl ` A ) =/= (inr ` B ) )
Theorem	updjudhf 7154*	The mapping of an element of the disjoint union to the value of the corresponding function is a function. (Contributed by AV, 26-Jun-2022.)
|- ( ph -> F : A --> C )   &    |- ( ph -> G : B --> C )   &    |- H = ( x e. ( A ⊔ B ) |-> if ( ( 1st ` x ) = (/) , ( F ` ( 2nd ` x ) ) , ( G ` ( 2nd ` x ) ) ) )   =>    |- ( ph -> H : ( A ⊔ B ) --> C )
Theorem	updjudhcoinlf 7155*	The composition of the mapping of an element of the disjoint union to the value of the corresponding function and the left injection equals the first function. (Contributed by AV, 27-Jun-2022.)
|- ( ph -> F : A --> C )   &    |- ( ph -> G : B --> C )   &    |- H = ( x e. ( A ⊔ B ) |-> if ( ( 1st ` x ) = (/) , ( F ` ( 2nd ` x ) ) , ( G ` ( 2nd ` x ) ) ) )   =>    |- ( ph -> ( H o. (inl |` A ) ) = F )
Theorem	updjudhcoinrg 7156*	The composition of the mapping of an element of the disjoint union to the value of the corresponding function and the right injection equals the second function. (Contributed by AV, 27-Jun-2022.)
|- ( ph -> F : A --> C )   &    |- ( ph -> G : B --> C )   &    |- H = ( x e. ( A ⊔ B ) |-> if ( ( 1st ` x ) = (/) , ( F ` ( 2nd ` x ) ) , ( G ` ( 2nd ` x ) ) ) )   =>    |- ( ph -> ( H o. (inr |` B ) ) = G )
Theorem	updjud 7157*	Universal property of the disjoint union. (Proposed by BJ, 25-Jun-2022.) (Contributed by AV, 28-Jun-2022.)
|- ( ph -> F : A --> C )   &    |- ( ph -> G : B --> C )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   =>    |- ( ph -> E! h ( h : ( A ⊔ B ) --> C /\ ( h o. (inl |` A ) ) = F /\ ( h o. (inr |` B ) ) = G ) )
Syntax	cdjucase 7158	Syntax for the "case" construction.
class case ( R , S )
Definition	df-case 7159	The "case" construction: if F : A --> X and G : B --> X are functions, then case ( F , G ) : ( A ⊔ B ) --> X is the natural function obtained by a definition by cases, hence the name. It is the unique function whose existence is asserted by the universal property of disjoint unions updjud 7157. The definition is adapted to make sense also for binary relations (where the universal property also holds). (Contributed by MC and BJ, 10-Jul-2022.)
|- case ( R , S ) = ( ( R o. `'inl ) u. ( S o. `'inr ) )
Theorem	casefun 7160	The "case" construction of two functions is a function. (Contributed by BJ, 10-Jul-2022.)
|- ( ph -> Fun F )   &    |- ( ph -> Fun G )   =>    |- ( ph -> Fun case ( F , G ) )
Theorem	casedm 7161	The domain of the "case" construction is the disjoint union of the domains. TODO (although less important): |- ran case ( F , G ) = ( ran F u. ran G ). (Contributed by BJ, 10-Jul-2022.)
|- dom case ( F , G ) = ( dom F ⊔ dom G )
Theorem	caserel 7162	The "case" construction of two relations is a relation, with bounds on its domain and codomain. Typically, the "case" construction is used when both relations have a common codomain. (Contributed by BJ, 10-Jul-2022.)
|- case ( R , S ) C_ ( ( dom R ⊔ dom S ) X. ( ran R u. ran S ) )
Theorem	casef 7163	The "case" construction of two functions is a function on the disjoint union of their domains. (Contributed by BJ, 10-Jul-2022.)
|- ( ph -> F : A --> X )   &    |- ( ph -> G : B --> X )   =>    |- ( ph -> case ( F , G ) : ( A ⊔ B ) --> X )
Theorem	caseinj 7164	The "case" construction of two injective relations with disjoint ranges is an injective relation. (Contributed by BJ, 10-Jul-2022.)
|- ( ph -> Fun `' R )   &    |- ( ph -> Fun `' S )   &    |- ( ph -> ( ran R i^i ran S ) = (/) )   =>    |- ( ph -> Fun `'case ( R , S ) )
Theorem	casef1 7165	The "case" construction of two injective functions with disjoint ranges is an injective function. (Contributed by BJ, 10-Jul-2022.)
|- ( ph -> F : A -1-1-> X )   &    |- ( ph -> G : B -1-1-> X )   &    |- ( ph -> ( ran F i^i ran G ) = (/) )   =>    |- ( ph -> case ( F , G ) : ( A ⊔ B ) -1-1-> X )
Theorem	caseinl 7166	Applying the "case" construction to a left injection. (Contributed by Jim Kingdon, 15-Mar-2023.)
|- ( ph -> F Fn B )   &    |- ( ph -> Fun G )   &    |- ( ph -> A e. B )   =>    |- ( ph -> (case ( F , G ) ` (inl ` A ) ) = ( F ` A ) )
Theorem	caseinr 7167	Applying the "case" construction to a right injection. (Contributed by Jim Kingdon, 12-Jul-2023.)
|- ( ph -> Fun F )   &    |- ( ph -> G Fn B )   &    |- ( ph -> A e. B )   =>    |- ( ph -> (case ( F , G ) ` (inr ` A ) ) = ( G ` A ) )
2.6.36.4  Dominance and equinumerosity properties of disjoint union
Theorem	djudom 7168	Dominance law for disjoint union. (Contributed by Jim Kingdon, 25-Jul-2022.)
|- ( ( A ~<_ B /\ C ~<_ D ) -> ( A ⊔ C ) ~<_ ( B ⊔ D ) )
Theorem	omp1eomlem 7169*	Lemma for omp1eom 7170. (Contributed by Jim Kingdon, 11-Jul-2023.)
|- F = ( x e. om |-> if ( x = (/) , (inr ` x ) , (inl ` U. x ) ) )   &    |- S = ( x e. om |-> suc x )   &    |- G = case ( S , ( _I |` 1o ) )   =>    |- F : om -1-1-onto-> ( om ⊔ 1o )
Theorem	omp1eom 7170	Adding one to om. (Contributed by Jim Kingdon, 10-Jul-2023.)
|- ( om ⊔ 1o ) ~~ om
Theorem	endjusym 7171	Reversing right and left operands of a disjoint union produces an equinumerous result. (Contributed by Jim Kingdon, 10-Jul-2023.)
|- ( ( A e. V /\ B e. W ) -> ( A ⊔ B ) ~~ ( B ⊔ A ) )
Theorem	eninl 7172	Equinumerosity of a set and its image under left injection. (Contributed by Jim Kingdon, 30-Jul-2023.)
|- ( A e. V -> (inl " A ) ~~ A )
Theorem	eninr 7173	Equinumerosity of a set and its image under right injection. (Contributed by Jim Kingdon, 30-Jul-2023.)
|- ( A e. V -> (inr " A ) ~~ A )
Theorem	difinfsnlem 7174*	Lemma for difinfsn 7175. The case where we need to swap B and (inr ` (/) ) in building the mapping G. (Contributed by Jim Kingdon, 9-Aug-2023.)
|- ( ph -> A. x e. A A. y e. A DECID x = y )   &    |- ( ph -> B e. A )   &    |- ( ph -> F : ( om ⊔ 1o ) -1-1-> A )   &    |- ( ph -> ( F ` (inr ` (/) ) ) =/= B )   &    |- G = ( n e. om |-> if ( ( F ` (inl ` n ) ) = B , ( F ` (inr ` (/) ) ) , ( F ` (inl ` n ) ) ) )   =>    |- ( ph -> G : om -1-1-> ( A \ { B } ) )
Theorem	difinfsn 7175*	An infinite set minus one element is infinite. We require that the set has decidable equality. (Contributed by Jim Kingdon, 8-Aug-2023.)
|- ( ( A. x e. A A. y e. A DECID x = y /\ om ~<_ A /\ B e. A ) -> om ~<_ ( A \ { B } ) )
Theorem	difinfinf 7176*	An infinite set minus a finite subset is infinite. We require that the set has decidable equality. (Contributed by Jim Kingdon, 8-Aug-2023.)
|- ( ( ( A. x e. A A. y e. A DECID x = y /\ om ~<_ A ) /\ ( B C_ A /\ B e. Fin ) ) -> om ~<_ ( A \ B ) )
2.6.36.5  Older definition temporarily kept for comparison, to be deleted
Syntax	cdjud 7177	Syntax for the domain-disjoint-union of two relations.
class ( R ⊔d S )
Definition	df-djud 7178	The "domain-disjoint-union" of two relations: if R C_ ( A X. X ) and S C_ ( B X. X ) are two binary relations, then ( R ⊔d S ) is the binary relation from ( A ⊔ B ) to X having the universal property of disjoint unions (see updjud 7157 in the case of functions). Remark: the restrictions to dom R (resp. dom S) are not necessary since extra stuff would be thrown away in the post-composition with R (resp. S), as in df-case 7159, but they are explicitly written for clarity. (Contributed by MC and BJ, 10-Jul-2022.)
|- ( R ⊔d S ) = ( ( R o. `' (inl |` dom R ) ) u. ( S o. `' (inr |` dom S ) ) )
Theorem	djufun 7179	The "domain-disjoint-union" of two functions is a function. (Contributed by BJ, 10-Jul-2022.)
|- ( ph -> Fun F )   &    |- ( ph -> Fun G )   =>    |- ( ph -> Fun ( F ⊔d G ) )
Theorem	djudm 7180	The domain of the "domain-disjoint-union" is the disjoint union of the domains. Remark: its range is the (standard) union of the ranges. (Contributed by BJ, 10-Jul-2022.)
|- dom ( F ⊔d G ) = ( dom F ⊔ dom G )
Theorem	djuinj 7181	The "domain-disjoint-union" of two injective relations with disjoint ranges is an injective relation. (Contributed by BJ, 10-Jul-2022.)
|- ( ph -> Fun `' R )   &    |- ( ph -> Fun `' S )   &    |- ( ph -> ( ran R i^i ran S ) = (/) )   =>    |- ( ph -> Fun `' ( R ⊔d S ) )
2.6.36.6  Countable sets
Theorem	0ct 7182	The empty set is countable. Remark of [BauerSwan], p. 14:3 which also has the definition of countable used here. (Contributed by Jim Kingdon, 13-Mar-2023.)
|- E. f f : om -onto-> ( (/) ⊔ 1o )
Theorem	ctmlemr 7183*	Lemma for ctm 7184. One of the directions of the biconditional. (Contributed by Jim Kingdon, 16-Mar-2023.)
|- ( E. x x e. A -> ( E. f f : om -onto-> A -> E. f f : om -onto-> ( A ⊔ 1o ) ) )
Theorem	ctm 7184*	Two equivalent definitions of countable for an inhabited set. Remark of [BauerSwan], p. 14:3. (Contributed by Jim Kingdon, 13-Mar-2023.)
|- ( E. x x e. A -> ( E. f f : om -onto-> ( A ⊔ 1o ) <-> E. f f : om -onto-> A ) )
Theorem	ctssdclemn0 7185*	Lemma for ctssdc 7188. The -. (/) e. S case. (Contributed by Jim Kingdon, 16-Aug-2023.)
|- ( ph -> S C_ om )   &    |- ( ph -> A. n e. om DECID n e. S )   &    |- ( ph -> F : S -onto-> A )   &    |- ( ph -> -. (/) e. S )   =>    |- ( ph -> E. g g : om -onto-> ( A ⊔ 1o ) )
Theorem	ctssdccl 7186*	A mapping from a decidable subset of the natural numbers onto a countable set. This is similar to one direction of ctssdc 7188 but expressed in terms of classes rather than E.. (Contributed by Jim Kingdon, 30-Oct-2023.)
|- ( ph -> F : om -onto-> ( A ⊔ 1o ) )   &    |- S = { x e. om | ( F ` x ) e. (inl " A ) }   &    |- G = ( `'inl o. F )   =>    |- ( ph -> ( S C_ om /\ G : S -onto-> A /\ A. n e. om DECID n e. S ) )
Theorem	ctssdclemr 7187*	Lemma for ctssdc 7188. Showing that our usual definition of countable implies the alternate one. (Contributed by Jim Kingdon, 16-Aug-2023.)
|- ( E. f f : om -onto-> ( A ⊔ 1o ) -> E. s ( s C_ om /\ E. f f : s -onto-> A /\ A. n e. om DECID n e. s ) )
Theorem	ctssdc 7188*	A set is countable iff there is a surjection from a decidable subset of the natural numbers onto it. The decidability condition is needed as shown at ctssexmid 7225. (Contributed by Jim Kingdon, 15-Aug-2023.)
|- ( E. s ( s C_ om /\ E. f f : s -onto-> A /\ A. n e. om DECID n e. s ) <-> E. f f : om -onto-> ( A ⊔ 1o ) )
Theorem	enumctlemm 7189*	Lemma for enumct 7190. The case where N is greater than zero. (Contributed by Jim Kingdon, 13-Mar-2023.)
|- ( ph -> F : N -onto-> A )   &    |- ( ph -> N e. om )   &    |- ( ph -> (/) e. N )   &    |- G = ( k e. om |-> if ( k e. N , ( F ` k ) , ( F ` (/) ) ) )   =>    |- ( ph -> G : om -onto-> A )
Theorem	enumct 7190*	A finitely enumerable set is countable. Lemma 8.1.14 of [AczelRathjen], p. 73 (except that our definition of countable does not require the set to be inhabited). "Finitely enumerable" is defined as E. n e. om E. f f : n -onto-> A per Definition 8.1.4 of [AczelRathjen], p. 71 and "countable" is defined as E. g g : om -onto-> ( A ⊔ 1o ) per [BauerSwan], p. 14:3. (Contributed by Jim Kingdon, 13-Mar-2023.)
|- ( E. n e. om E. f f : n -onto-> A -> E. g g : om -onto-> ( A ⊔ 1o ) )
Theorem	finct 7191*	A finite set is countable. (Contributed by Jim Kingdon, 17-Mar-2023.)
|- ( A e. Fin -> E. g g : om -onto-> ( A ⊔ 1o ) )
Theorem	omct 7192	om is countable. (Contributed by Jim Kingdon, 23-Dec-2023.)
|- E. f f : om -onto-> ( om ⊔ 1o )
Theorem	ctfoex 7193*	A countable class is a set. (Contributed by Jim Kingdon, 25-Dec-2023.)
|- ( E. f f : om -onto-> ( A ⊔ 1o ) -> A e. _V )
2.6.37  The one-point compactification of the natural numbers This section introduces the one-point compactification of the set of natural numbers, introduced by Escardo as the set of nonincreasing sequences on om with values in 2o. The topological results justifying its name will be proved later.
Syntax	xnninf 7194	Set of nonincreasing sequences in 2o ^m om.
class ℕ∞
Definition	df-nninf 7195*	Define the set of nonincreasing sequences in 2o ^m om. Definition in Section 3.1 of [Pierik], p. 15. If we assumed excluded middle, this would be essentially the same as NN0* as defined at df-xnn0 9330 but in its absence the relationship between the two is more complicated. This definition would function much the same whether we used om or NN0, but the former allows us to take advantage of 2o = { (/) , 1o } (df2o3 6497) so we adopt it. (Contributed by Jim Kingdon, 14-Jul-2022.)
|- ℕ∞ = { f e. ( 2o ^m om ) | A. i e. om ( f ` suc i ) C_ ( f ` i ) }
Theorem	nninfex 7196	ℕ∞ is a set. (Contributed by Jim Kingdon, 10-Aug-2022.)
|- ℕ∞ e. _V
Theorem	nninff 7197	An element of ℕ∞ is a sequence of zeroes and ones. (Contributed by Jim Kingdon, 4-Aug-2022.)
|- ( A e. ℕ∞ -> A : om --> 2o )
Theorem	nninfninc 7198	All values beyond a zero in an ℕ∞ sequence are zero. This is another way of stating that elements of ℕ∞ are nonincreasing. (Contributed by Jim Kingdon, 12-Jul-2025.)
|- ( ph -> A e. ℕ∞ )   &    |- ( ph -> X e. om )   &    |- ( ph -> Y e. om )   &    |- ( ph -> X C_ Y )   &    |- ( ph -> ( A ` X ) = (/) )   =>    |- ( ph -> ( A ` Y ) = (/) )
Theorem	infnninf 7199	The point at infinity in ℕ∞ is the constant sequence equal to 1o. Note that with our encoding of functions, that constant function can also be expressed as ( om X. { 1o } ), as fconstmpt 4711 shows. (Contributed by Jim Kingdon, 14-Jul-2022.) Use maps-to notation. (Revised by BJ, 10-Aug-2024.)
|- ( i e. om |-> 1o ) e. ℕ∞
Theorem	infnninfOLD 7200	Obsolete version of infnninf 7199 as of 10-Aug-2024. (Contributed by Jim Kingdon, 14-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( om X. { 1o } ) e. ℕ∞