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Theorem List for Intuitionistic Logic Explorer - 6901-7000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-inr 6901 Right injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.)
 |- inr 
 =  ( x  e. 
 _V  |->  <. 1o ,  x >. )
 
Theoremdjulclr 6902 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 6903 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 6904 Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)
 |-  ( C  e.  A  ->  (inl `  C )  e.  ( A B )
 )
 
Theoremdjurcl 6905 Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)
 |-  ( C  e.  B  ->  (inr `  C )  e.  ( A B )
 )
 
Theoremdjuf1olem 6906* Lemma for djulf1o 6911 and djurf1o 6912. (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 6907* Lemma for djulf1or 6909 and djurf1or 6910. For a version of this lemma with  F defined on  A and no restriction in the conclusion, see djuf1olem 6906. (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 6908 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 6909 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 6910 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 6911 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 6912 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 6913* Lemma for inlresf1 6914 and inrresf1 6915. (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 6914 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 6915 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 6916 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 6946 and djufun 6957) 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 6938). (Contributed by BJ and Jim Kingdon, 21-Jun-2022.)
 |-  ( ran  (inl  |`  A )  i^i  ran  (inr  |`  B ) )  =  (/)
 
Theoremdjuin 6917 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 6918 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 6919 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 6920 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 6921* 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 6922* 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 )
 ) )
 
2.6.35.3  Universal property of the disjoint union
 
Theoremdjuss 6923 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 6924 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 6925 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 6926 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 6927 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 6928 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 6929 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 6930 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 6931 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 6932* 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 6933* 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 6934* 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 6935* 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 6936 Syntax for the "case" construction.
 class case ( R ,  S )
 
Definitiondf-case 6937 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 6935. 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 6938 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 6939 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 6940 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 6941 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 6942 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 6943 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 6944 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 6945 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 ) )
 
2.6.35.4  Dominance and equinumerosity properties of disjoint union
 
Theoremdjudom 6946 Dominance law for disjoint union. (Contributed by Jim Kingdon, 25-Jul-2022.)
 |-  ( ( A  ~<_  B  /\  C 
 ~<_  D )  ->  ( A C )  ~<_  ( B D ) )
 
Theoremomp1eomlem 6947* Lemma for omp1eom 6948. (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 6948 Adding one to  om. (Contributed by Jim Kingdon, 10-Jul-2023.)
 |-  ( om 1o )  ~~  om
 
Theoremendjusym 6949 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 6950 Equinumerosity of a set and its image under left injection. (Contributed by Jim Kingdon, 30-Jul-2023.)
 |-  ( A  e.  V  ->  (inl " A )  ~~  A )
 
Theoremeninr 6951 Equinumerosity of a set and its image under right injection. (Contributed by Jim Kingdon, 30-Jul-2023.)
 |-  ( A  e.  V  ->  (inr " A )  ~~  A )
 
Theoremdifinfsnlem 6952* Lemma for difinfsn 6953. 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 6953* 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 6954* 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 ) )
 
2.6.35.5  Older definition temporarily kept for comparison, to be deleted
 
Syntaxcdjud 6955 Syntax for the domain-disjoint-union of two relations.
 class  ( R ⊔d  S )
 
Definitiondf-djud 6956 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 6935 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 6937, 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 6957 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 6958 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 6959 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 )
 )
 
2.6.35.6  Countable sets
 
Theorem0ct 6960 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 6961* Lemma for ctm 6962. 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 6962* 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 6963* Lemma for ctssdc 6966. 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 6964* A mapping from a decidable subset of the natural numbers onto a countable set. This is similar to one direction of ctssdc 6966 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 6965* Lemma for ctssdc 6966. 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 6966* 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 6992. (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 6967* Lemma for enumct 6968. 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 6968* 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 6969* A finite set is countable. (Contributed by Jim Kingdon, 17-Mar-2023.)
 |-  ( A  e.  Fin  ->  E. g  g : om -onto-> ( A 1o )
 )
 
Theoremomct 6970  om is countable. (Contributed by Jim Kingdon, 23-Dec-2023.)
 |- 
 E. f  f : om -onto-> ( om 1o )
 
Theoremctfoex 6971* A countable class is a set. (Contributed by Jim Kingdon, 25-Dec-2023.)
 |-  ( E. f  f : om -onto-> ( A 1o )  ->  A  e.  _V )
 
2.6.36  Omniscient sets
 
Syntaxcomni 6972 Extend class definition to include the class of omniscient sets.
 class Omni
 
Syntaxxnninf 6973 Set of nonincreasing sequences in 
2o  ^m  om.
 class
 
Definitiondf-omni 6974* 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 ) ) }
 
Definitiondf-nninf 6975* 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 9009 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 6295) 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 ) }
 
Theoremisomni 6976* 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 6977* 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 6978 Lemma for enomni 6979. One direction of the biconditional. (Contributed by Jim Kingdon, 13-Jul-2022.)
 |-  ( A  ~~  B  ->  ( A  e. Omni  ->  B  e. Omni ) )
 
Theoremenomni 6979 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 6295 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 6980 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 6981 Given excluded middle, every set is omniscient. Remark following Definition 3.1 of [Pierik], p. 14. This is one direction of the biconditional exmidomni 6982. (Contributed by Jim Kingdon, 29-Jun-2022.)
 |-  (EXMID 
 ->  A. x  x  e. Omni
 )
 
Theoremexmidomni 6982 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 6983 Excluded middle implies the Limited Principle of Omniscience (LPO). (Contributed by Jim Kingdon, 29-Mar-2023.)
 |-  (EXMID 
 ->  om  e. Omni )
 
Theoremfodjuomnilemdc 6984* Lemma for fodjuomni 6989. 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
 ) )
 
Theoremfodjuf 6985* Lemma for fodjuomni 6989 and fodjumkv 7002. Domain and range of  P. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.)
 |-  ( ph  ->  F : O -onto-> ( A B ) )   &    |-  P  =  ( y  e.  O  |->  if ( E. z  e.  A  ( F `  y )  =  (inl `  z ) ,  (/) ,  1o ) )   &    |-  ( ph  ->  O  e.  V )   =>    |-  ( ph  ->  P  e.  ( 2o  ^m  O ) )
 
Theoremfodjum 6986* Lemma for fodjuomni 6989 and fodjumkv 7002. A condition which shows that  A is inhabited. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.)
 |-  ( 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 )
 
Theoremfodju0 6987* Lemma for fodjuomni 6989 and fodjumkv 7002. A condition which shows that  A is empty. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.)
 |-  ( 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  =  (/) )
 
Theoremfodjuomnilemres 6988* Lemma for fodjuomni 6989. The final result with  P expressed as a local definition. (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  =  (/) ) )
 
Theoremfodjuomni 6989* 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  =  (/) ) )
 
Theoreminfnninf 6990 The point at infinity in ℕ (the constant sequence equal to  1o). (Contributed by Jim Kingdon, 14-Jul-2022.)
 |-  ( om  X.  { 1o } )  e.
 
Theoremnnnninf 6991* 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 6990. 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. )
 
Theoremctssexmid 6992* The decidability condition in ctssdc 6966 is needed. More specifically, ctssdc 6966 minus that condition, plus the Limited Principle of Omniscience (LPO), implies excluded middle. (Contributed by Jim Kingdon, 15-Aug-2023.)
 |-  ( ( y  C_  om 
 /\  E. f  f : y -onto-> x )  ->  E. f  f : om -onto-> ( x 1o ) )   &    |-  om  e. Omni   =>    |-  ( ph  \/  -.  ph )
 
2.6.37  Markov's principle
 
Syntaxcmarkov 6993 Extend class definition to include the class of Markov sets.
 class Markov
 
Definitiondf-markov 6994* A Markov set is one where if a predicate (here represented by a function  f) on that set does not hold (where hold means is equal to  1o) for all elements, then there exists an element where it fails (is equal to  (/)). Generalization of definition 2.5 of [Pierik], p. 9.

In particular,  om  e. Markov is known as Markov's Principle (MP). (Contributed by Jim Kingdon, 18-Mar-2023.)

 |- Markov  =  { y  |  A. f ( f : y --> 2o  ->  ( -. 
 A. x  e.  y  ( f `  x )  =  1o  ->  E. x  e.  y  ( f `  x )  =  (/) ) ) }
 
Theoremismkv 6995* The predicate of being Markov. (Contributed by Jim Kingdon, 18-Mar-2023.)
 |-  ( A  e.  V  ->  ( A  e. Markov  <->  A. f ( f : A --> 2o  ->  ( -.  A. x  e.  A  ( f `  x )  =  1o  ->  E. x  e.  A  ( f `  x )  =  (/) ) ) ) )
 
Theoremismkvmap 6996* The predicate of being Markov stated in terms of set exponentiation. (Contributed by Jim Kingdon, 18-Mar-2023.)
 |-  ( A  e.  V  ->  ( A  e. Markov  <->  A. f  e.  ( 2o  ^m  A ) ( -.  A. x  e.  A  ( f `  x )  =  1o  ->  E. x  e.  A  ( f `  x )  =  (/) ) ) )
 
Theoremismkvnex 6997* The predicate of being Markov stated in terms of double negation and comparison with  1o. (Contributed by Jim Kingdon, 29-Nov-2023.)
 |-  ( A  e.  V  ->  ( A  e. Markov  <->  A. f  e.  ( 2o  ^m  A ) ( -.  -.  E. x  e.  A  ( f `  x )  =  1o  ->  E. x  e.  A  ( f `  x )  =  1o )
 ) )
 
Theoremomnimkv 6998 An omniscient set is Markov. In particular, the case where  A is  om means that the Limited Principle of Omniscience (LPO) implies Markov's Principle (MP). (Contributed by Jim Kingdon, 18-Mar-2023.)
 |-  ( A  e. Omni  ->  A  e. Markov )
 
Theoremexmidmp 6999 Excluded middle implies Markov's Principle (MP). (Contributed by Jim Kingdon, 4-Apr-2023.)
 |-  (EXMID 
 ->  om  e. Markov )
 
Theoremmkvprop 7000* Markov's Principle expressed in terms of propositions (or more precisely, the  A  =  om case is Markov's Principle). (Contributed by Jim Kingdon, 19-Mar-2023.)
 |-  ( ( A  e. Markov  /\ 
 A. n  e.  A DECID  ph  /\  -.  A. n  e.  A  -.  ph )  ->  E. n  e.  A  ph )
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