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Theorem List for Intuitionistic Logic Explorer - 7001-7100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdjuexb 7001 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 ).

 
Syntaxcinl 7002 Extend class notation to include left injection of a disjoint union.
 class inl
 
Syntaxcinr 7003 Extend class notation to include right injection of a disjoint union.
 class inr
 
Definitiondf-inl 7004 Left injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.)
 |- inl 
 =  ( x  e. 
 _V  |->  <. (/) ,  x >. )
 
Definitiondf-inr 7005 Right injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.)
 |- inr 
 =  ( x  e. 
 _V  |->  <. 1o ,  x >. )
 
Theoremdjulclr 7006 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 )
 )
 
Theoremdjurclr 7007 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 )
 )
 
Theoremdjulcl 7008 Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)
 |-  ( C  e.  A  ->  (inl `  C )  e.  ( A B )
 )
 
Theoremdjurcl 7009 Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)
 |-  ( C  e.  B  ->  (inr `  C )  e.  ( A B )
 )
 
Theoremdjuf1olem 7010* Lemma for djulf1o 7015 and djurf1o 7016. (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 )
 
Theoremdjuf1olemr 7011* Lemma for djulf1or 7013 and djurf1or 7014. For a version of this lemma with  F defined on  A and no restriction in the conclusion, see djuf1olem 7010. (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 )
 
Theoremdjulclb 7012 Left biconditional closure of disjoint union. (Contributed by Jim Kingdon, 2-Jul-2022.)
 |-  ( C  e.  V  ->  ( C  e.  A  <->  (inl `  C )  e.  ( A B ) ) )
 
Theoremdjulf1or 7013 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 )
 
Theoremdjurf1or 7014 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 )
 
Theoremdjulf1o 7015 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 )
 
Theoremdjurf1o 7016 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 )
 
Theoreminresflem 7017* Lemma for inlresf1 7018 and inrresf1 7019. (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
 
Theoreminlresf1 7018 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 )
 
Theoreminrresf1 7019 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 )
 
Theoremdjuinr 7020 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 7050 and djufun 7061) 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 7042). (Contributed by BJ and Jim Kingdon, 21-Jun-2022.)
 |-  ( ran  (inl  |`  A )  i^i  ran  (inr  |`  B ) )  =  (/)
 
Theoremdjuin 7021 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 ) )  =  (/)
 
Theoreminl11 7022 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 ) )
 
Theoremdjuunr 7023 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 )
 
Theoremdjuun 7024 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 )
 
Theoremeldju 7025* 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 ) ) )
 
Theoremdjur 7026* 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
 
Theoremdjuss 7027 A disjoint union is a subset of a Cartesian product. (Contributed by AV, 25-Jun-2022.)
 |-  ( A B )  C_  ( { (/) ,  1o }  X.  ( A  u.  B ) )
 
Theoremeldju1st 7028 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 ) )
 
Theoremeldju2ndl 7029 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 )
 
Theoremeldju2ndr 7030 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 )
 
Theorem1stinl 7031 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 ) )  =  (/) )
 
Theorem2ndinl 7032 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 )
 
Theorem1stinr 7033 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 )
 
Theorem2ndinr 7034 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 )
 
Theoremdjune 7035 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 ) )
 
Theoremupdjudhf 7036* 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 )
 
Theoremupdjudhcoinlf 7037* 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 )
 
Theoremupdjudhcoinrg 7038* 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 )
 
Theoremupdjud 7039* 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 ) )
 
Syntaxcdjucase 7040 Syntax for the "case" construction.
 class case ( R ,  S )
 
Definitiondf-case 7041 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 7039. 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 ) )
 
Theoremcasefun 7042 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 ) )
 
Theoremcasedm 7043 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 )
 
Theoremcaserel 7044 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 ) )
 
Theoremcasef 7045 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 )
 
Theoremcaseinj 7046 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 ) )
 
Theoremcasef1 7047 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 )
 
Theoremcaseinl 7048 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 ) )
 
Theoremcaseinr 7049 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
 
Theoremdjudom 7050 Dominance law for disjoint union. (Contributed by Jim Kingdon, 25-Jul-2022.)
 |-  ( ( A  ~<_  B  /\  C 
 ~<_  D )  ->  ( A C )  ~<_  ( B D ) )
 
Theoremomp1eomlem 7051* Lemma for omp1eom 7052. (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 )
 
Theoremomp1eom 7052 Adding one to  om. (Contributed by Jim Kingdon, 10-Jul-2023.)
 |-  ( om 1o )  ~~  om
 
Theoremendjusym 7053 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 ) )
 
Theoremeninl 7054 Equinumerosity of a set and its image under left injection. (Contributed by Jim Kingdon, 30-Jul-2023.)
 |-  ( A  e.  V  ->  (inl " A )  ~~  A )
 
Theoremeninr 7055 Equinumerosity of a set and its image under right injection. (Contributed by Jim Kingdon, 30-Jul-2023.)
 |-  ( A  e.  V  ->  (inr " A )  ~~  A )
 
Theoremdifinfsnlem 7056* Lemma for difinfsn 7057. 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 } ) )
 
Theoremdifinfsn 7057* 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 }
 ) )
 
Theoremdifinfinf 7058* 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
 
Syntaxcdjud 7059 Syntax for the domain-disjoint-union of two relations.
 class  ( R ⊔d  S )
 
Definitiondf-djud 7060 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 7039 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 7041, 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 ) ) )
 
Theoremdjufun 7061 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 ) )
 
Theoremdjudm 7062 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 )
 
Theoremdjuinj 7063 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
 
Theorem0ct 7064 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 )
 
Theoremctmlemr 7065* Lemma for ctm 7066. 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 ) ) )
 
Theoremctm 7066* 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 ) )
 
Theoremctssdclemn0 7067* Lemma for ctssdc 7070. 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 ) )
 
Theoremctssdccl 7068* A mapping from a decidable subset of the natural numbers onto a countable set. This is similar to one direction of ctssdc 7070 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 ) )
 
Theoremctssdclemr 7069* Lemma for ctssdc 7070. 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 ) )
 
Theoremctssdc 7070* 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 7106. (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 )
 )
 
Theoremenumctlemm 7071* Lemma for enumct 7072. 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 )
 
Theoremenumct 7072* 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 ) )
 
Theoremfinct 7073* A finite set is countable. (Contributed by Jim Kingdon, 17-Mar-2023.)
 |-  ( A  e.  Fin  ->  E. g  g : om -onto-> ( A 1o )
 )
 
Theoremomct 7074  om is countable. (Contributed by Jim Kingdon, 23-Dec-2023.)
 |- 
 E. f  f : om -onto-> ( om 1o )
 
Theoremctfoex 7075* 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.

 
Syntaxxnninf 7076 Set of nonincreasing sequences in 
2o  ^m  om.
 class
 
Definitiondf-nninf 7077* 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 9170 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 6390) so we adopt it. (Contributed by Jim Kingdon, 14-Jul-2022.)
 |-  =  { f  e.  ( 2o  ^m  om )  | 
 A. i  e.  om  ( f `  suc  i )  C_  ( f `
  i ) }
 
Theoremnninfex 7078 is a set. (Contributed by Jim Kingdon, 10-Aug-2022.)
 |-  e.  _V
 
Theoremnninff 7079 An element of ℕ is a sequence of zeroes and ones. (Contributed by Jim Kingdon, 4-Aug-2022.)
 |-  ( A  e.  ->  A : om --> 2o )
 
Theoreminfnninf 7080 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 4646 shows. (Contributed by Jim Kingdon, 14-Jul-2022.) Use maps-to notation. (Revised by BJ, 10-Aug-2024.)
 |-  ( i  e.  om  |->  1o )  e.
 
TheoreminfnninfOLD 7081 Obsolete version of infnninf 7080 as of 10-Aug-2024. (Contributed by Jim Kingdon, 14-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( om  X.  { 1o } )  e.
 
Theoremnnnninf 7082* Elements of ℕ corresponding to natural numbers. The natural number  N corresponds to a sequence of  N ones followed by zeroes. This can be strengthened to include infinity, see nnnninf2 7083. (Contributed by Jim Kingdon, 14-Jul-2022.)
 |-  ( N  e.  om  ->  ( i  e.  om  |->  if ( i  e.  N ,  1o ,  (/) ) )  e. )
 
Theoremnnnninf2 7083* Canonical embedding of  suc  om into ℕ. (Contributed by BJ, 10-Aug-2024.)
 |-  ( N  e.  suc  om 
 ->  ( i  e.  om  |->  if ( i  e.  N ,  1o ,  (/) ) )  e. )
 
Theoremnnnninfeq 7084* Mapping of a natural number to an element of ℕ. (Contributed by Jim Kingdon, 4-Aug-2022.)
 |-  ( ph  ->  P  e. )   &    |-  ( ph  ->  N  e.  om )   &    |-  ( ph  ->  A. x  e.  N  ( P `  x )  =  1o )   &    |-  ( ph  ->  ( P `  N )  =  (/) )   =>    |-  ( ph  ->  P  =  ( i  e. 
 om  |->  if ( i  e.  N ,  1o ,  (/) ) ) )
 
Theoremnnnninfeq2 7085* Mapping of a natural number to an element of ℕ. Similar to nnnninfeq 7084 but if we have information about a single  1o digit, that gives information about all previous digits. (Contributed by Jim Kingdon, 4-Aug-2022.)
 |-  ( ph  ->  P  e. )   &    |-  ( ph  ->  N  e.  om )   &    |-  ( ph  ->  ( P `  U. N )  =  1o )   &    |-  ( ph  ->  ( P `  N )  =  (/) )   =>    |-  ( ph  ->  P  =  ( i  e. 
 om  |->  if ( i  e.  N ,  1o ,  (/) ) ) )
 
Theoremnninfisollem0 7086* Lemma for nninfisol 7089. The case where  N is zero. (Contributed by Jim Kingdon, 13-Sep-2024.)
 |-  ( ph  ->  X  e. )   &    |-  ( ph  ->  ( X `  N )  =  (/) )   &    |-  ( ph  ->  N  e.  om )   &    |-  ( ph  ->  N  =  (/) )   =>    |-  ( ph  -> DECID  ( i  e.  om  |->  if ( i  e.  N ,  1o ,  (/) ) )  =  X )
 
Theoremnninfisollemne 7087* Lemma for nninfisol 7089. A case where  N is a successor and  N and  X are not equal. (Contributed by Jim Kingdon, 13-Sep-2024.)
 |-  ( ph  ->  X  e. )   &    |-  ( ph  ->  ( X `  N )  =  (/) )   &    |-  ( ph  ->  N  e.  om )   &    |-  ( ph  ->  N  =/=  (/) )   &    |-  ( ph  ->  ( X `  U. N )  =  (/) )   =>    |-  ( ph  -> DECID  ( i  e.  om  |->  if (
 i  e.  N ,  1o ,  (/) ) )  =  X )
 
Theoremnninfisollemeq 7088* Lemma for nninfisol 7089. The case where  N is a successor and  N and  X are equal. (Contributed by Jim Kingdon, 13-Sep-2024.)
 |-  ( ph  ->  X  e. )   &    |-  ( ph  ->  ( X `  N )  =  (/) )   &    |-  ( ph  ->  N  e.  om )   &    |-  ( ph  ->  N  =/=  (/) )   &    |-  ( ph  ->  ( X `  U. N )  =  1o )   =>    |-  ( ph  -> DECID 
 ( i  e.  om  |->  if ( i  e.  N ,  1o ,  (/) ) )  =  X )
 
Theoremnninfisol 7089* Finite elements of ℕ are isolated. That is, given a natural number and any element of ℕ, it is decidable whether the natural number (when converted to an element of ℕ) is equal to the given element of ℕ. Stated in an online post by Martin Escardo. One way to understand this theorem is that you do not need to look at an unbounded number of elements of the sequence  X to decide whether it is equal to  N (in fact, you only need to look at two elements and  N tells you where to look). (Contributed by BJ and Jim Kingdon, 12-Sep-2024.)
 |-  ( ( N  e.  om 
 /\  X  e. )  -> DECID  ( i  e.  om  |->  if ( i  e.  N ,  1o ,  (/) ) )  =  X )
 
2.6.38  Omniscient sets
 
Syntaxcomni 7090 Extend class definition to include the class of omniscient sets.
 class Omni
 
Definitiondf-omni 7091* 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 Limited 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 ) ) }
 
Theoremisomni 7092* 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 ) ) ) )
 
Theoremisomnimap 7093* 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 ) ) )
 
Theoremenomnilem 7094 Lemma for enomni 7095. One direction of the biconditional. (Contributed by Jim Kingdon, 13-Jul-2022.)
 |-  ( A  ~~  B  ->  ( A  e. Omni  ->  B  e. Omni ) )
 
Theoremenomni 7095 Omniscience is invariant with respect to equinumerosity. For example, this means that we can express the Limited 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 6390 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 ) )
 
Theoremfinomni 7096 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 )
 
Theoremexmidomniim 7097 Given excluded middle, every set is omniscient. Remark following Definition 3.1 of [Pierik], p. 14. This is one direction of the biconditional exmidomni 7098. (Contributed by Jim Kingdon, 29-Jun-2022.)
 |-  (EXMID 
 ->  A. x  x  e. Omni
 )
 
Theoremexmidomni 7098 Excluded middle is equivalent to every set being omniscient. (Contributed by BJ and Jim Kingdon, 30-Jun-2022.)
 |-  (EXMID  <->  A. x  x  e. Omni )
 
Theoremexmidlpo 7099 Excluded middle implies the Limited Principle of Omniscience (LPO). (Contributed by Jim Kingdon, 29-Mar-2023.)
 |-  (EXMID 
 ->  om  e. Omni )
 
Theoremfodjuomnilemdc 7100* Lemma for fodjuomni 7105. 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
 ) )
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