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Theorem List for Intuitionistic Logic Explorer - 7101-7200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-djud 7101 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 7080 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 7082, 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 7102 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 7103 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 7104 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 7105 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 7106* Lemma for ctm 7107. 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 7107* 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 7108* Lemma for ctssdc 7111. 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 7109* A mapping from a decidable subset of the natural numbers onto a countable set. This is similar to one direction of ctssdc 7111 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 7110* Lemma for ctssdc 7111. 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 7111* 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 7147. (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 7112* Lemma for enumct 7113. 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 7113* 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 7114* A finite set is countable. (Contributed by Jim Kingdon, 17-Mar-2023.)
 |-  ( A  e.  Fin  ->  E. g  g : om -onto-> ( A 1o )
 )
 
Theoremomct 7115  om is countable. (Contributed by Jim Kingdon, 23-Dec-2023.)
 |- 
 E. f  f : om -onto-> ( om 1o )
 
Theoremctfoex 7116* 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 7117 Set of nonincreasing sequences in 
2o  ^m  om.
 class
 
Definitiondf-nninf 7118* 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 9238 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 6430) 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 7119 is a set. (Contributed by Jim Kingdon, 10-Aug-2022.)
 |-  e.  _V
 
Theoremnninff 7120 An element of ℕ is a sequence of zeroes and ones. (Contributed by Jim Kingdon, 4-Aug-2022.)
 |-  ( A  e.  ->  A : om --> 2o )
 
Theoreminfnninf 7121 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 4673 shows. (Contributed by Jim Kingdon, 14-Jul-2022.) Use maps-to notation. (Revised by BJ, 10-Aug-2024.)
 |-  ( i  e.  om  |->  1o )  e.
 
TheoreminfnninfOLD 7122 Obsolete version of infnninf 7121 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 7123* 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 7124. (Contributed by Jim Kingdon, 14-Jul-2022.)
 |-  ( N  e.  om  ->  ( i  e.  om  |->  if ( i  e.  N ,  1o ,  (/) ) )  e. )
 
Theoremnnnninf2 7124* 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 7125* 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 7126* Mapping of a natural number to an element of ℕ. Similar to nnnninfeq 7125 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 7127* Lemma for nninfisol 7130. 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 7128* Lemma for nninfisol 7130. 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 7129* Lemma for nninfisol 7130. 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 7130* 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 7131 Extend class definition to include the class of omniscient sets.
 class Omni
 
Definitiondf-omni 7132* 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 7133* 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 7134* 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 7135 Lemma for enomni 7136. One direction of the biconditional. (Contributed by Jim Kingdon, 13-Jul-2022.)
 |-  ( A  ~~  B  ->  ( A  e. Omni  ->  B  e. Omni ) )
 
Theoremenomni 7136 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 6430 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 7137 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 7138 Given excluded middle, every set is omniscient. Remark following Definition 3.1 of [Pierik], p. 14. This is one direction of the biconditional exmidomni 7139. (Contributed by Jim Kingdon, 29-Jun-2022.)
 |-  (EXMID 
 ->  A. x  x  e. Omni
 )
 
Theoremexmidomni 7139 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 7140 Excluded middle implies the Limited Principle of Omniscience (LPO). (Contributed by Jim Kingdon, 29-Mar-2023.)
 |-  (EXMID 
 ->  om  e. Omni )
 
Theoremfodjuomnilemdc 7141* Lemma for fodjuomni 7146. 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 7142* Lemma for fodjuomni 7146 and fodjumkv 7157. 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 7143* Lemma for fodjuomni 7146 and fodjumkv 7157. 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 7144* Lemma for fodjuomni 7146 and fodjumkv 7157. 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 7145* Lemma for fodjuomni 7146. 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 7146* 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  =  (/) ) )
 
Theoremctssexmid 7147* The decidability condition in ctssdc 7111 is needed. More specifically, ctssdc 7111 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 )
 
2.6.39  Markov's principle
 
Syntaxcmarkov 7148 Extend class definition to include the class of Markov sets.
 class Markov
 
Definitiondf-markov 7149* 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 7150* 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 7151* 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 7152* 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 7153 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 7154 Excluded middle implies Markov's Principle (MP). (Contributed by Jim Kingdon, 4-Apr-2023.)
 |-  (EXMID 
 ->  om  e. Markov )
 
Theoremmkvprop 7155* 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 )
 
Theoremfodjumkvlemres 7156* Lemma for fodjumkv 7157. The final result with  P expressed as a local definition. (Contributed by Jim Kingdon, 25-Mar-2023.)
 |-  ( ph  ->  M  e. Markov )   &    |-  ( ph  ->  F : M -onto-> ( A B ) )   &    |-  P  =  ( y  e.  M  |->  if ( E. z  e.  A  ( F `  y )  =  (inl `  z ) ,  (/) ,  1o ) )   =>    |-  ( ph  ->  ( A  =/=  (/)  ->  E. x  x  e.  A )
 )
 
Theoremfodjumkv 7157* A condition which ensures that a nonempty set is inhabited. (Contributed by Jim Kingdon, 25-Mar-2023.)
 |-  ( ph  ->  M  e. Markov )   &    |-  ( ph  ->  F : M -onto-> ( A B ) )   =>    |-  ( ph  ->  ( A  =/=  (/)  ->  E. x  x  e.  A )
 )
 
Theoremenmkvlem 7158 Lemma for enmkv 7159. One direction of the biconditional. (Contributed by Jim Kingdon, 25-Jun-2024.)
 |-  ( A  ~~  B  ->  ( A  e. Markov  ->  B  e. Markov ) )
 
Theoremenmkv 7159 Being Markov is invariant with respect to equinumerosity. For example, this means that we can express the Markov's Principle as either  om  e. Markov or  NN0  e. Markov. The former is a better match to conventional notation in the sense that df2o3 6430 says that  2o  =  { (/)
,  1o } whereas the corresponding relationship does not exist between  2 and  { 0 ,  1 }. (Contributed by Jim Kingdon, 24-Jun-2024.)
 |-  ( A  ~~  B  ->  ( A  e. Markov  <->  B  e. Markov ) )
 
2.6.40  Weakly omniscient sets
 
Syntaxcwomni 7160 Extend class definition to include the class of weakly omniscient sets.
 class WOmni
 
Definitiondf-womni 7161* A weakly 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 not. Generalization of definition 2.4 of [Pierik], p. 9.

In particular,  om  e. WOmni is known as the Weak Limited Principle of Omniscience (WLPO).

The term WLPO is common in the literature; there appears to be no widespread term for what we are calling a weakly omniscient set. (Contributed by Jim Kingdon, 9-Jun-2024.)

 |- WOmni  =  { y  |  A. f ( f : y --> 2o  -> DECID  A. x  e.  y  ( f `  x )  =  1o ) }
 
Theoremiswomni 7162* The predicate of being weakly omniscient. (Contributed by Jim Kingdon, 9-Jun-2024.)
 |-  ( A  e.  V  ->  ( A  e. WOmni  <->  A. f ( f : A --> 2o  -> DECID  A. x  e.  A  ( f `  x )  =  1o ) ) )
 
Theoremiswomnimap 7163* The predicate of being weakly omniscient stated in terms of set exponentiation. (Contributed by Jim Kingdon, 9-Jun-2024.)
 |-  ( A  e.  V  ->  ( A  e. WOmni  <->  A. f  e.  ( 2o  ^m  A )DECID  A. x  e.  A  ( f `  x )  =  1o ) )
 
Theoremomniwomnimkv 7164 A set is omniscient if and only if it is weakly omniscient and Markov. The case  A  =  om says that LPO  <-> WLPO  /\ MP which is a remark following Definition 2.5 of [Pierik], p. 9. (Contributed by Jim Kingdon, 9-Jun-2024.)
 |-  ( A  e. Omni  <->  ( A  e. WOmni  /\  A  e. Markov ) )
 
Theoremlpowlpo 7165 LPO implies WLPO. Easy corollary of the more general omniwomnimkv 7164. There is an analogue in terms of analytic omniscience principles at tridceq 14686. (Contributed by Jim Kingdon, 24-Jul-2024.)
 |-  ( om  e. Omni  ->  om  e. WOmni )
 
Theoremenwomnilem 7166 Lemma for enwomni 7167. One direction of the biconditional. (Contributed by Jim Kingdon, 20-Jun-2024.)
 |-  ( A  ~~  B  ->  ( A  e. WOmni  ->  B  e. WOmni ) )
 
Theoremenwomni 7167 Weak omniscience is invariant with respect to equinumerosity. For example, this means that we can express the Weak Limited Principle of Omniscience as either  om  e. WOmni or  NN0  e. WOmni. The former is a better match to conventional notation in the sense that df2o3 6430 says that  2o  =  { (/)
,  1o } whereas the corresponding relationship does not exist between  2 and  { 0 ,  1 }. (Contributed by Jim Kingdon, 20-Jun-2024.)
 |-  ( A  ~~  B  ->  ( A  e. WOmni  <->  B  e. WOmni ) )
 
Theoremnninfdcinf 7168* The Weak Limited Principle of Omniscience (WLPO) implies that it is decidable whether an element of ℕ equals the point at infinity. (Contributed by Jim Kingdon, 3-Dec-2024.)
 |-  ( ph  ->  om  e. WOmni )   &    |-  ( ph  ->  N  e. )   =>    |-  ( ph  -> DECID  N  =  ( i  e.  om  |->  1o ) )
 
Theoremnninfwlporlemd 7169* Given two countably infinite sequences of zeroes and ones, they are equal if and only if a sequence formed by pointwise comparing them is all ones. (Contributed by Jim Kingdon, 6-Dec-2024.)
 |-  ( ph  ->  X : om --> 2o )   &    |-  ( ph  ->  Y : om --> 2o )   &    |-  D  =  ( i  e.  om  |->  if ( ( X `  i )  =  ( Y `  i ) ,  1o ,  (/) ) )   =>    |-  ( ph  ->  ( X  =  Y  <->  D  =  (
 i  e.  om  |->  1o ) ) )
 
Theoremnninfwlporlem 7170* Lemma for nninfwlpor 7171. The result. (Contributed by Jim Kingdon, 7-Dec-2024.)
 |-  ( ph  ->  X : om --> 2o )   &    |-  ( ph  ->  Y : om --> 2o )   &    |-  D  =  ( i  e.  om  |->  if ( ( X `  i )  =  ( Y `  i ) ,  1o ,  (/) ) )   &    |-  ( ph  ->  om  e. WOmni )   =>    |-  ( ph  -> DECID  X  =  Y )
 
Theoremnninfwlpor 7171* The Weak Limited Principle of Omniscience (WLPO) implies that equality for ℕ is decidable. (Contributed by Jim Kingdon, 7-Dec-2024.)
 |-  ( om  e. WOmni  ->  A. x  e.  A. y  e. DECID  x  =  y )
 
Theoremnninfwlpoimlemg 7172* Lemma for nninfwlpoim 7175. (Contributed by Jim Kingdon, 8-Dec-2024.)
 |-  ( ph  ->  F : om --> 2o )   &    |-  G  =  ( i  e.  om  |->  if ( E. x  e. 
 suc  i ( F `
  x )  =  (/) ,  (/) ,  1o )
 )   =>    |-  ( ph  ->  G  e. )
 
Theoremnninfwlpoimlemginf 7173* Lemma for nninfwlpoim 7175. (Contributed by Jim Kingdon, 8-Dec-2024.)
 |-  ( ph  ->  F : om --> 2o )   &    |-  G  =  ( i  e.  om  |->  if ( E. x  e. 
 suc  i ( F `
  x )  =  (/) ,  (/) ,  1o )
 )   =>    |-  ( ph  ->  ( G  =  ( i  e.  om  |->  1o )  <->  A. n  e.  om  ( F `  n )  =  1o ) )
 
Theoremnninfwlpoimlemdc 7174* Lemma for nninfwlpoim 7175. (Contributed by Jim Kingdon, 8-Dec-2024.)
 |-  ( ph  ->  F : om --> 2o )   &    |-  G  =  ( i  e.  om  |->  if ( E. x  e. 
 suc  i ( F `
  x )  =  (/) ,  (/) ,  1o )
 )   &    |-  ( ph  ->  A. x  e.  A. y  e. DECID  x  =  y )   =>    |-  ( ph  -> DECID  A. n  e.  om  ( F `  n )  =  1o )
 
Theoremnninfwlpoim 7175* Decidable equality for ℕ implies the Weak Limited Principle of Omniscience (WLPO). (Contributed by Jim Kingdon, 9-Dec-2024.)
 |-  ( A. x  e.  A. y  e. DECID  x  =  y  ->  om  e. WOmni )
 
Theoremnninfwlpo 7176* Decidability of equality for ℕ is equivalent to the Weak Limited Principle of Omniscience (WLPO). (Contributed by Jim Kingdon, 3-Dec-2024.)
 |-  ( A. x  e.  A. y  e. DECID  x  =  y  <->  om  e. WOmni )
 
2.6.41  Cardinal numbers
 
Syntaxccrd 7177 Extend class definition to include the cardinal size function.
 class  card
 
Definitiondf-card 7178* 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 } )
 
Theoremcardcl 7179* 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 )
 
Theoremisnumi 7180 A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
 |-  ( ( A  e.  On  /\  A  ~~  B )  ->  B  e.  dom  card
 )
 
Theoremfinnum 7181 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 )
 
Theoremonenon 7182 Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
 |-  ( A  e.  On  ->  A  e.  dom  card )
 
Theoremcardval3ex 7183* 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 } )
 
Theoremoncardval 7184* 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 } )
 
Theoremcardonle 7185 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 )
 
Theoremcard0 7186 The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.)
 |-  ( card `  (/) )  =  (/)
 
Theoremcarden2bex 7187* 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 ) )
 
Theorempm54.43 7188 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 ) )
 
Theorempr2nelem 7189 Lemma for pr2ne 7190. (Contributed by FL, 17-Aug-2008.)
 |-  ( ( A  e.  C  /\  B  e.  D  /\  A  =/=  B ) 
 ->  { A ,  B }  ~~  2o )
 
Theorempr2ne 7190 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 ) )
 
Theoremexmidonfinlem 7191* Lemma for exmidonfin 7192. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.)
 |-  A  =  { { x  e.  { (/) }  |  ph
 } ,  { x  e.  { (/) }  |  -.  ph
 } }   =>    |-  ( om  =  ( On  i^i  Fin )  -> DECID  ph )
 
Theoremexmidonfin 7192 If a finite ordinal is a natural number, excluded middle follows. That excluded middle implies that a finite ordinal is a natural number is proved in the Metamath Proof Explorer. That a natural number is a finite ordinal is shown at nnfi 6871 and nnon 4609. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.)
 |-  ( om  =  ( On  i^i  Fin )  -> EXMID )
 
Theoremen2eleq 7193 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 }
 ) } )
 
Theoremen2other2 7194 Taking the other element twice in a pair gets back to the original element. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  ( ( X  e.  P  /\  P  ~~  2o )  ->  U. ( P  \  { U. ( P  \  { X } ) }
 )  =  X )
 
Theoremdju1p1e2 7195 Disjoint union version of one plus one equals two. (Contributed by Jim Kingdon, 1-Jul-2022.)
 |-  ( 1o 1o )  ~~  2o
 
Theoreminfpwfidom 7196 The collection of finite subsets of a set dominates the set. (We use the weaker sethood assumption 
( ~P A  i^i  Fin )  e.  _V because this theorem also implies that  A is a set if  ~P A  i^i  Fin is.) (Contributed by Mario Carneiro, 17-May-2015.)
 |-  ( ( ~P A  i^i  Fin )  e.  _V  ->  A  ~<_  ( ~P A  i^i  Fin ) )
 
Theoremexmidfodomrlemeldju 7197 Lemma for exmidfodomr 7202. A variant of djur 7067. (Contributed by Jim Kingdon, 2-Jul-2022.)
 |-  ( ph  ->  A  C_ 
 1o )   &    |-  ( ph  ->  B  e.  ( A 1o )
 )   =>    |-  ( ph  ->  ( B  =  (inl `  (/) )  \/  B  =  (inr `  (/) ) ) )
 
Theoremexmidfodomrlemreseldju 7198 Lemma for exmidfodomrlemrALT 7201. A variant of eldju 7066. (Contributed by Jim Kingdon, 9-Jul-2022.)
 |-  ( ph  ->  A  C_ 
 1o )   &    |-  ( ph  ->  B  e.  ( A 1o )
 )   =>    |-  ( ph  ->  (
 ( (/)  e.  A  /\  B  =  ( (inl  |`  A ) `  (/) ) )  \/  B  =  ( (inr  |`  1o ) `  (/) ) ) )
 
Theoremexmidfodomrlemim 7199* Excluded middle implies the existence of a mapping from any set onto any inhabited set that it dominates. Proposition 1.1 of [PradicBrown2022], p. 2. (Contributed by Jim Kingdon, 1-Jul-2022.)
 |-  (EXMID 
 ->  A. x A. y
 ( ( E. z  z  e.  y  /\  y 
 ~<_  x )  ->  E. f  f : x -onto-> y ) )
 
Theoremexmidfodomrlemr 7200* The existence of a mapping from any set onto any inhabited set that it dominates implies excluded middle. Proposition 1.2 of [PradicBrown2022], p. 2. (Contributed by Jim Kingdon, 1-Jul-2022.)
 |-  ( A. x A. y ( ( E. z  z  e.  y  /\  y  ~<_  x )  ->  E. f  f : x -onto-> y )  -> EXMID )
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