P. 68 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6701-6800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	infminti 6701*	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 6702*	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 6703*	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 6704*	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 6705	The infimum regarding an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
|- inf ( B , (/) , R ) = (/)
Theorem	infisoti 6706*	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 ) ) )
2.6.32  Ordinal isomorphism
Theorem	ordiso2 6707	Generalize ordiso 6708 to proper classes. (Contributed by Mario Carneiro, 24-Jun-2015.)
|- ( ( F Isom _E , _E ( A , B ) /\ Ord A /\ Ord B ) -> A = B )
Theorem	ordiso 6708*	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 ) ) )
2.6.33  Disjoint union
2.6.33.1  Disjoint union
Syntax	cdju 6709	Extend class notation to include disjoint union of two classes.
class ( A ⊔ B )
Definition	df-dju 6710	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 6711	Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
|- ( ( A = B /\ C = D ) -> ( A ⊔ C ) = ( B ⊔ D ) )
Theorem	djueq1 6712	Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
|- ( A = B -> ( A ⊔ C ) = ( B ⊔ C ) )
Theorem	djueq2 6713	Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
|- ( A = B -> ( C ⊔ A ) = ( C ⊔ B ) )
Theorem	nfdju 6714	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 6715	The disjoint union of sets is a set. (Contributed by AV, 28-Jun-2022.)
|- ( ( A e. V /\ B e. W ) -> ( A ⊔ B ) e. _V )
2.6.33.2  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 6716	Extend class notation to include left injection of a disjoint union.
class inl
Syntax	cinr 6717	Extend class notation to include right injection of a disjoint union.
class inr
Definition	df-inl 6718	Left injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.)
|- inl = ( x e. _V |-> <. (/) , x >. )
Definition	df-inr 6719	Right injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.)
|- inr = ( x e. _V |-> <. 1o , x >. )
Theorem	djulclr 6720	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 6721	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 6722	Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)
|- ( C e. A -> (inl ` C ) e. ( A ⊔ B ) )
Theorem	djurcl 6723	Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)
|- ( C e. B -> (inr ` C ) e. ( A ⊔ B ) )
Theorem	djuf1olem 6724*	Lemma for djulf1o 6729 and djurf1o 6730. (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 6725*	Lemma for djulf1or 6727 and djurf1or 6728. Remark: maybe a version of this lemma with F defined on A and no restriction in the conclusion would be more usable. (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 6726	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 6727	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 6728	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 6729	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 6730	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 6731*	Lemma for inlresf1 6732 and inrresf1 6733. (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 6732	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 6733	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 6734	The ranges of any left and right injections are disjoint. Remark: the restrictions seem not necessary, but the proof is not much longer than the proof of |- ( ran inl i^i ran inr ) = (/) (which is easily recovered from it, as in the proof of casefun 6755). (Contributed by BJ and Jim Kingdon, 21-Jun-2022.)
|- ( ran (inl |` A ) i^i ran (inr |` B ) ) = (/)
Theorem	djuin 6735	The images of any classes under right and left injection produce disjoint sets. (Contributed by Jim Kingdon, 21-Jun-2022.)
|- ( (inl " A ) i^i (inr " B ) ) = (/)
Theorem	djur 6736*	A member of a disjoint union can be mapped from one of the classes which produced it. (Contributed by Jim Kingdon, 23-Jun-2022.)
|- ( C e. ( A ⊔ B ) -> ( E. x e. A C = (inl ` x ) \/ E. x e. B C = (inr ` x ) ) )
Theorem	djuunr 6737	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	eldju 6738*	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	djuun 6739	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.)
|- ( (inl " A ) u. (inr " B ) ) = ( A ⊔ B )
2.6.33.3  Universal property of the disjoint union
Theorem	djuss 6740	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 6741	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 6742	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 6743	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 6744	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 6745	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 6746	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 6747	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 6748	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 6749*	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 6750*	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 6751*	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 6752*	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 6753	Syntax for the "case" construction.
class case ( R , S )
Definition	df-case 6754	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. 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 6755	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 6756	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 6757	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 6758	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 6759	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 6760	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 )
2.6.33.4  Older definition temporarily kept for comparison, to be deleted
Syntax	cdjud 6761	Syntax for the domain-disjoint-union of two relations.
class ( R ⊔d S )
Definition	df-djud 6762	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. 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), 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 6763	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 6764	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 6765	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 ) )
Theorem	djudom 6766	Dominance law for disjoint union. (Contributed by Jim Kingdon, 25-Jul-2022.)
|- ( ( A ~<_ B /\ C ~<_ D ) -> ( A ⊔ C ) ~<_ ( B ⊔ D ) )
2.6.34  Omniscient sets
Syntax	comni 6767	Extend class definition to include the class of omniscient sets.
class Omni
Syntax	xnninf 6768	Set of nonincreasing sequences in 2o ^m om.
class ℕ∞
Definition	df-omni 6769*	An omniscient set is one where we can decide whether a predicate (here represented by a function f) holds (is equal to 1o) for all elements or fails to hold (is equal to (/)) for some element. Definition 3.1 of [Pierik], p. 14. In particular, om e. Omni is known as the Lesser Principle of Omniscience (LPO). (Contributed by Jim Kingdon, 28-Jun-2022.)
|- Omni = { y | A. f ( f : y --> 2o -> ( E. x e. y ( f ` x ) = (/) \/ A. x e. y ( f ` x ) = 1o ) ) }
Definition	df-nninf 6770*	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 8707 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 6177) 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	isomni 6771*	The predicate of being omniscient. (Contributed by Jim Kingdon, 28-Jun-2022.)
|- ( A e. V -> ( A e. Omni <-> A. f ( f : A --> 2o -> ( E. x e. A ( f ` x ) = (/) \/ A. x e. A ( f ` x ) = 1o ) ) ) )
Theorem	isomnimap 6772*	The predicate of being omniscient stated in terms of set exponentiation. (Contributed by Jim Kingdon, 13-Jul-2022.)
|- ( A e. V -> ( A e. Omni <-> A. f e. ( 2o ^m A ) ( E. x e. A ( f ` x ) = (/) \/ A. x e. A ( f ` x ) = 1o ) ) )
Theorem	enomnilem 6773	Lemma for enomni 6774. One direction of the biconditional. (Contributed by Jim Kingdon, 13-Jul-2022.)
|- ( A ~~ B -> ( A e. Omni -> B e. Omni ) )
Theorem	enomni 6774	Omniscience is invariant with respect to equinumerosity. For example, this means that we can express the Lesser Principle of Omniscience as either om e. Omni or NN0 e. Omni. The former is a better match to conventional notation in the sense that df2o3 6177 says that 2o = { (/) , 1o } whereas the corresponding relationship does not exist between 2 and { 0 , 1 }. (Contributed by Jim Kingdon, 13-Jul-2022.)
|- ( A ~~ B -> ( A e. Omni <-> B e. Omni ) )
Theorem	finomni 6775	A finite set is omniscient. Remark right after Definition 3.1 of [Pierik], p. 14. (Contributed by Jim Kingdon, 28-Jun-2022.)
|- ( A e. Fin -> A e. Omni )
Theorem	exmidomniim 6776	Given excluded middle, every set is omniscient. Remark following Definition 3.1 of [Pierik], p. 14. This is one direction of the biconditional exmidomni 6777. (Contributed by Jim Kingdon, 29-Jun-2022.)
|- (EXMID -> A. x x e. Omni )
Theorem	exmidomni 6777	Excluded middle is equivalent to every set being omniscient. (Contributed by BJ and Jim Kingdon, 30-Jun-2022.)
|- (EXMID <-> A. x x e. Omni )
Theorem	fodjuomnilemdc 6778*	Lemma for fodjuomni 6783. Decidability of a condition we use in various lemmas. (Contributed by Jim Kingdon, 27-Jul-2022.)
|- ( ph -> F : O -onto-> ( A ⊔ B ) )   =>    |- ( ( ph /\ X e. O ) -> DECID E. z e. A ( F ` X ) = (inl ` z ) )
Theorem	fodjuomnilemf 6779*	Lemma for fodjuomni 6783. Domain and range of P. (Contributed by Jim Kingdon, 27-Jul-2022.)
|- ( ph -> O e. Omni )   &    |- ( ph -> F : O -onto-> ( A ⊔ B ) )   &    |- P = ( y e. O |-> if ( E. z e. A ( F ` y ) = (inl ` z ) , (/) , 1o ) )   =>    |- ( ph -> P e. ( 2o ^m O ) )
Theorem	fodjuomnilemm 6780*	Lemma for fodjuomni 6783. The case where A is inhabited. (Contributed by Jim Kingdon, 27-Jul-2022.)
|- ( ph -> O e. Omni )   &    |- ( ph -> F : O -onto-> ( A ⊔ B ) )   &    |- P = ( y e. O |-> if ( E. z e. A ( F ` y ) = (inl ` z ) , (/) , 1o ) )   &    |- ( ph -> E. w e. O ( P ` w ) = (/) )   =>    |- ( ph -> E. x x e. A )
Theorem	fodjuomnilem0 6781*	Lemma for fodjuomni 6783. The case where A is empty. (Contributed by Jim Kingdon, 27-Jul-2022.)
|- ( ph -> O e. Omni )   &    |- ( ph -> F : O -onto-> ( A ⊔ B ) )   &    |- P = ( y e. O |-> if ( E. z e. A ( F ` y ) = (inl ` z ) , (/) , 1o ) )   &    |- ( ph -> A. w e. O ( P ` w ) = 1o )   =>    |- ( ph -> A = (/) )
Theorem	fodjuomnilemres 6782*	Lemma for fodjuomni 6783. The final result with P broken out into a hypothesis. (Contributed by Jim Kingdon, 29-Jul-2022.)
|- ( ph -> O e. Omni )   &    |- ( ph -> F : O -onto-> ( A ⊔ B ) )   &    |- P = ( y e. O |-> if ( E. z e. A ( F ` y ) = (inl ` z ) , (/) , 1o ) )   =>    |- ( ph -> ( E. x x e. A \/ A = (/) ) )
Theorem	fodjuomni 6783*	A condition which ensures A is either inhabited or empty. Lemma 3.2 of [PradicBrown2022], p. 4. (Contributed by Jim Kingdon, 27-Jul-2022.)
|- ( ph -> O e. Omni )   &    |- ( ph -> F : O -onto-> ( A ⊔ B ) )   =>    |- ( ph -> ( E. x x e. A \/ A = (/) ) )
Theorem	infnninf 6784	The point at infinity in ℕ∞ (the constant sequence equal to 1o). (Contributed by Jim Kingdon, 14-Jul-2022.)
|- ( om X. { 1o } ) e. ℕ∞
Theorem	nnnninf 6785*	Elements of ℕ∞ corresponding to natural numbers. The natural number N corresponds to a sequence of N ones followed by zeroes. Contrast to a sequence which is all ones as seen at infnninf 6784. Remark/TODO: the theorem still holds if N = om, that is, the antecedent could be weakened to N e. suc om. (Contributed by Jim Kingdon, 14-Jul-2022.)
|- ( N e. om -> ( i e. om |-> if ( i e. N , 1o , (/) ) ) e. ℕ∞ )
2.6.35  Cardinal numbers
Syntax	ccrd 6786	Extend class definition to include the cardinal size function.
class card
Definition	df-card 6787*	Define the cardinal number function. The cardinal number of a set is the least ordinal number equinumerous to it. In other words, it is the "size" of the set. Definition of [Enderton] p. 197. Our notation is from Enderton. Other textbooks often use a double bar over the set to express this function. (Contributed by NM, 21-Oct-2003.)
|- card = ( x e. _V |-> |^| { y e. On | y ~~ x } )
Theorem	cardcl 6788*	The cardinality of a well-orderable set is an ordinal. (Contributed by Jim Kingdon, 30-Aug-2021.)
|- ( E. y e. On y ~~ A -> ( card ` A ) e. On )
Theorem	isnumi 6789	A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
|- ( ( A e. On /\ A ~~ B ) -> B e. dom card )
Theorem	finnum 6790	Every finite set is numerable. (Contributed by Mario Carneiro, 4-Feb-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
|- ( A e. Fin -> A e. dom card )
Theorem	onenon 6791	Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
|- ( A e. On -> A e. dom card )
Theorem	cardval3ex 6792*	The value of ( card ` A ). (Contributed by Jim Kingdon, 30-Aug-2021.)
|- ( E. x e. On x ~~ A -> ( card ` A ) = |^| { y e. On | y ~~ A } )
Theorem	oncardval 6793*	The value of the cardinal number function with an ordinal number as its argument. (Contributed by NM, 24-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)
|- ( A e. On -> ( card ` A ) = |^| { x e. On | x ~~ A } )
Theorem	cardonle 6794	The cardinal of an ordinal number is less than or equal to the ordinal number. Proposition 10.6(3) of [TakeutiZaring] p. 85. (Contributed by NM, 22-Oct-2003.)
|- ( A e. On -> ( card ` A ) C_ A )
Theorem	card0 6795	The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.)
|- ( card ` (/) ) = (/)
Theorem	carden2bex 6796*	If two numerable sets are equinumerous, then they have equal cardinalities. (Contributed by Jim Kingdon, 30-Aug-2021.)
|- ( ( A ~~ B /\ E. x e. On x ~~ A ) -> ( card ` A ) = ( card ` B ) )
Theorem	pm54.43 6797	Theorem *54.43 of [WhiteheadRussell] p. 360. (Contributed by NM, 4-Apr-2007.)
|- ( ( A ~~ 1o /\ B ~~ 1o ) -> ( ( A i^i B ) = (/) <-> ( A u. B ) ~~ 2o ) )
Theorem	pr2nelem 6798	Lemma for pr2ne 6799. (Contributed by FL, 17-Aug-2008.)
|- ( ( A e. C /\ B e. D /\ A =/= B ) -> { A , B } ~~ 2o )
Theorem	pr2ne 6799	If an unordered pair has two elements they are different. (Contributed by FL, 14-Feb-2010.)
|- ( ( A e. C /\ B e. D ) -> ( { A , B } ~~ 2o <-> A =/= B ) )
Theorem	en2eleq 6800	Express a set of pair cardinality as the unordered pair of a given element and the other element. (Contributed by Stefan O'Rear, 22-Aug-2015.)
|- ( ( X e. P /\ P ~~ 2o ) -> P = { X , U. ( P \ { X } ) } )