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Theorem List for Intuitionistic Logic Explorer - 7101-7200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
2.6.36.5  Older definition temporarily kept for comparison, to be deleted
 
Syntaxcdjud 7101 Syntax for the domain-disjoint-union of two relations.
class (𝑅 βŠ”d 𝑆)
 
Definitiondf-djud 7102 The "domain-disjoint-union" of two relations: if 𝑅 βŠ† (𝐴 Γ— 𝑋) and 𝑆 βŠ† (𝐡 Γ— 𝑋) are two binary relations, then (𝑅 βŠ”d 𝑆) is the binary relation from (𝐴 βŠ” 𝐡) to 𝑋 having the universal property of disjoint unions (see updjud 7081 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 7083, but they are explicitly written for clarity. (Contributed by MC and BJ, 10-Jul-2022.)

(𝑅 βŠ”d 𝑆) = ((𝑅 ∘ β—‘(inl β†Ύ dom 𝑅)) βˆͺ (𝑆 ∘ β—‘(inr β†Ύ dom 𝑆)))
 
Theoremdjufun 7103 The "domain-disjoint-union" of two functions is a function. (Contributed by BJ, 10-Jul-2022.)
(πœ‘ β†’ Fun 𝐹)    &   (πœ‘ β†’ Fun 𝐺)    β‡’   (πœ‘ β†’ Fun (𝐹 βŠ”d 𝐺))
 
Theoremdjudm 7104 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 βŠ”d 𝐺) = (dom 𝐹 βŠ” dom 𝐺)
 
Theoremdjuinj 7105 The "domain-disjoint-union" of two injective relations with disjoint ranges is an injective relation. (Contributed by BJ, 10-Jul-2022.)
(πœ‘ β†’ Fun ◑𝑅)    &   (πœ‘ β†’ Fun ◑𝑆)    &   (πœ‘ β†’ (ran 𝑅 ∩ ran 𝑆) = βˆ…)    β‡’   (πœ‘ β†’ Fun β—‘(𝑅 βŠ”d 𝑆))
 
2.6.36.6  Countable sets
 
Theorem0ct 7106 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.)
βˆƒπ‘“ 𝑓:ω–ontoβ†’(βˆ… βŠ” 1o)
 
Theoremctmlemr 7107* Lemma for ctm 7108. One of the directions of the biconditional. (Contributed by Jim Kingdon, 16-Mar-2023.)
(βˆƒπ‘₯ π‘₯ ∈ 𝐴 β†’ (βˆƒπ‘“ 𝑓:ω–onto→𝐴 β†’ βˆƒπ‘“ 𝑓:ω–ontoβ†’(𝐴 βŠ” 1o)))
 
Theoremctm 7108* Two equivalent definitions of countable for an inhabited set. Remark of [BauerSwan], p. 14:3. (Contributed by Jim Kingdon, 13-Mar-2023.)
(βˆƒπ‘₯ π‘₯ ∈ 𝐴 β†’ (βˆƒπ‘“ 𝑓:ω–ontoβ†’(𝐴 βŠ” 1o) ↔ βˆƒπ‘“ 𝑓:ω–onto→𝐴))
 
Theoremctssdclemn0 7109* Lemma for ctssdc 7112. The Β¬ βˆ… ∈ 𝑆 case. (Contributed by Jim Kingdon, 16-Aug-2023.)
(πœ‘ β†’ 𝑆 βŠ† Ο‰)    &   (πœ‘ β†’ βˆ€π‘› ∈ Ο‰ DECID 𝑛 ∈ 𝑆)    &   (πœ‘ β†’ 𝐹:𝑆–onto→𝐴)    &   (πœ‘ β†’ Β¬ βˆ… ∈ 𝑆)    β‡’   (πœ‘ β†’ βˆƒπ‘” 𝑔:ω–ontoβ†’(𝐴 βŠ” 1o))
 
Theoremctssdccl 7110* A mapping from a decidable subset of the natural numbers onto a countable set. This is similar to one direction of ctssdc 7112 but expressed in terms of classes rather than βˆƒ. (Contributed by Jim Kingdon, 30-Oct-2023.)
(πœ‘ β†’ 𝐹:ω–ontoβ†’(𝐴 βŠ” 1o))    &   π‘† = {π‘₯ ∈ Ο‰ ∣ (πΉβ€˜π‘₯) ∈ (inl β€œ 𝐴)}    &   πΊ = (β—‘inl ∘ 𝐹)    β‡’   (πœ‘ β†’ (𝑆 βŠ† Ο‰ ∧ 𝐺:𝑆–onto→𝐴 ∧ βˆ€π‘› ∈ Ο‰ DECID 𝑛 ∈ 𝑆))
 
Theoremctssdclemr 7111* Lemma for ctssdc 7112. Showing that our usual definition of countable implies the alternate one. (Contributed by Jim Kingdon, 16-Aug-2023.)
(βˆƒπ‘“ 𝑓:ω–ontoβ†’(𝐴 βŠ” 1o) β†’ βˆƒπ‘ (𝑠 βŠ† Ο‰ ∧ βˆƒπ‘“ 𝑓:𝑠–onto→𝐴 ∧ βˆ€π‘› ∈ Ο‰ DECID 𝑛 ∈ 𝑠))
 
Theoremctssdc 7112* 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 7148. (Contributed by Jim Kingdon, 15-Aug-2023.)
(βˆƒπ‘ (𝑠 βŠ† Ο‰ ∧ βˆƒπ‘“ 𝑓:𝑠–onto→𝐴 ∧ βˆ€π‘› ∈ Ο‰ DECID 𝑛 ∈ 𝑠) ↔ βˆƒπ‘“ 𝑓:ω–ontoβ†’(𝐴 βŠ” 1o))
 
Theoremenumctlemm 7113* Lemma for enumct 7114. The case where 𝑁 is greater than zero. (Contributed by Jim Kingdon, 13-Mar-2023.)
(πœ‘ β†’ 𝐹:𝑁–onto→𝐴)    &   (πœ‘ β†’ 𝑁 ∈ Ο‰)    &   (πœ‘ β†’ βˆ… ∈ 𝑁)    &   πΊ = (π‘˜ ∈ Ο‰ ↦ if(π‘˜ ∈ 𝑁, (πΉβ€˜π‘˜), (πΉβ€˜βˆ…)))    β‡’   (πœ‘ β†’ 𝐺:ω–onto→𝐴)
 
Theoremenumct 7114* 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 βˆƒπ‘› ∈ Ο‰βˆƒπ‘“π‘“:𝑛–onto→𝐴 per Definition 8.1.4 of [AczelRathjen], p. 71 and "countable" is defined as βˆƒπ‘”π‘”:ω–ontoβ†’(𝐴 βŠ” 1o) per [BauerSwan], p. 14:3. (Contributed by Jim Kingdon, 13-Mar-2023.)
(βˆƒπ‘› ∈ Ο‰ βˆƒπ‘“ 𝑓:𝑛–onto→𝐴 β†’ βˆƒπ‘” 𝑔:ω–ontoβ†’(𝐴 βŠ” 1o))
 
Theoremfinct 7115* A finite set is countable. (Contributed by Jim Kingdon, 17-Mar-2023.)
(𝐴 ∈ Fin β†’ βˆƒπ‘” 𝑔:ω–ontoβ†’(𝐴 βŠ” 1o))
 
Theoremomct 7116 Ο‰ is countable. (Contributed by Jim Kingdon, 23-Dec-2023.)
βˆƒπ‘“ 𝑓:ω–ontoβ†’(Ο‰ βŠ” 1o)
 
Theoremctfoex 7117* A countable class is a set. (Contributed by Jim Kingdon, 25-Dec-2023.)
(βˆƒπ‘“ 𝑓:ω–ontoβ†’(𝐴 βŠ” 1o) β†’ 𝐴 ∈ 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 Ο‰ with values in 2o. The topological results justifying its name will be proved later.

 
Syntaxxnninf 7118 Set of nonincreasing sequences in 2o β†‘π‘š Ο‰.
class β„•βˆž
 
Definitiondf-nninf 7119* Define the set of nonincreasing sequences in 2o β†‘π‘š Ο‰. Definition in Section 3.1 of [Pierik], p. 15. If we assumed excluded middle, this would be essentially the same as β„•0* as defined at df-xnn0 9240 but in its absence the relationship between the two is more complicated. This definition would function much the same whether we used Ο‰ or β„•0, but the former allows us to take advantage of 2o = {βˆ…, 1o} (df2o3 6431) so we adopt it. (Contributed by Jim Kingdon, 14-Jul-2022.)
β„•βˆž = {𝑓 ∈ (2o β†‘π‘š Ο‰) ∣ βˆ€π‘– ∈ Ο‰ (π‘“β€˜suc 𝑖) βŠ† (π‘“β€˜π‘–)}
 
Theoremnninfex 7120 β„•βˆž is a set. (Contributed by Jim Kingdon, 10-Aug-2022.)
β„•βˆž ∈ V
 
Theoremnninff 7121 An element of β„•βˆž is a sequence of zeroes and ones. (Contributed by Jim Kingdon, 4-Aug-2022.)
(𝐴 ∈ β„•βˆž β†’ 𝐴:Ο‰βŸΆ2o)
 
Theoreminfnninf 7122 The point at infinity in β„•βˆž is the constant sequence equal to 1o. Note that with our encoding of functions, that constant function can also be expressed as (Ο‰ Γ— {1o}), as fconstmpt 4674 shows. (Contributed by Jim Kingdon, 14-Jul-2022.) Use maps-to notation. (Revised by BJ, 10-Aug-2024.)
(𝑖 ∈ Ο‰ ↦ 1o) ∈ β„•βˆž
 
TheoreminfnninfOLD 7123 Obsolete version of infnninf 7122 as of 10-Aug-2024. (Contributed by Jim Kingdon, 14-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
(Ο‰ Γ— {1o}) ∈ β„•βˆž
 
Theoremnnnninf 7124* Elements of β„•βˆž 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 7125. (Contributed by Jim Kingdon, 14-Jul-2022.)
(𝑁 ∈ Ο‰ β†’ (𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑁, 1o, βˆ…)) ∈ β„•βˆž)
 
Theoremnnnninf2 7125* Canonical embedding of suc Ο‰ into β„•βˆž. (Contributed by BJ, 10-Aug-2024.)
(𝑁 ∈ suc Ο‰ β†’ (𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑁, 1o, βˆ…)) ∈ β„•βˆž)
 
Theoremnnnninfeq 7126* Mapping of a natural number to an element of β„•βˆž. (Contributed by Jim Kingdon, 4-Aug-2022.)
(πœ‘ β†’ 𝑃 ∈ β„•βˆž)    &   (πœ‘ β†’ 𝑁 ∈ Ο‰)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑁 (π‘ƒβ€˜π‘₯) = 1o)    &   (πœ‘ β†’ (π‘ƒβ€˜π‘) = βˆ…)    β‡’   (πœ‘ β†’ 𝑃 = (𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑁, 1o, βˆ…)))
 
Theoremnnnninfeq2 7127* Mapping of a natural number to an element of β„•βˆž. Similar to nnnninfeq 7126 but if we have information about a single 1o digit, that gives information about all previous digits. (Contributed by Jim Kingdon, 4-Aug-2022.)
(πœ‘ β†’ 𝑃 ∈ β„•βˆž)    &   (πœ‘ β†’ 𝑁 ∈ Ο‰)    &   (πœ‘ β†’ (π‘ƒβ€˜βˆͺ 𝑁) = 1o)    &   (πœ‘ β†’ (π‘ƒβ€˜π‘) = βˆ…)    β‡’   (πœ‘ β†’ 𝑃 = (𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑁, 1o, βˆ…)))
 
Theoremnninfisollem0 7128* Lemma for nninfisol 7131. The case where 𝑁 is zero. (Contributed by Jim Kingdon, 13-Sep-2024.)
(πœ‘ β†’ 𝑋 ∈ β„•βˆž)    &   (πœ‘ β†’ (π‘‹β€˜π‘) = βˆ…)    &   (πœ‘ β†’ 𝑁 ∈ Ο‰)    &   (πœ‘ β†’ 𝑁 = βˆ…)    β‡’   (πœ‘ β†’ DECID (𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑁, 1o, βˆ…)) = 𝑋)
 
Theoremnninfisollemne 7129* Lemma for nninfisol 7131. A case where 𝑁 is a successor and 𝑁 and 𝑋 are not equal. (Contributed by Jim Kingdon, 13-Sep-2024.)
(πœ‘ β†’ 𝑋 ∈ β„•βˆž)    &   (πœ‘ β†’ (π‘‹β€˜π‘) = βˆ…)    &   (πœ‘ β†’ 𝑁 ∈ Ο‰)    &   (πœ‘ β†’ 𝑁 β‰  βˆ…)    &   (πœ‘ β†’ (π‘‹β€˜βˆͺ 𝑁) = βˆ…)    β‡’   (πœ‘ β†’ DECID (𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑁, 1o, βˆ…)) = 𝑋)
 
Theoremnninfisollemeq 7130* Lemma for nninfisol 7131. The case where 𝑁 is a successor and 𝑁 and 𝑋 are equal. (Contributed by Jim Kingdon, 13-Sep-2024.)
(πœ‘ β†’ 𝑋 ∈ β„•βˆž)    &   (πœ‘ β†’ (π‘‹β€˜π‘) = βˆ…)    &   (πœ‘ β†’ 𝑁 ∈ Ο‰)    &   (πœ‘ β†’ 𝑁 β‰  βˆ…)    &   (πœ‘ β†’ (π‘‹β€˜βˆͺ 𝑁) = 1o)    β‡’   (πœ‘ β†’ DECID (𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑁, 1o, βˆ…)) = 𝑋)
 
Theoremnninfisol 7131* Finite elements of β„•βˆž are isolated. That is, given a natural number and any element of β„•βˆž, it is decidable whether the natural number (when converted to an element of β„•βˆž) is equal to the given element of β„•βˆž. 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 ∈ Ο‰ ∧ 𝑋 ∈ β„•βˆž) β†’ DECID (𝑖 ∈ Ο‰ ↦ if(𝑖 ∈ 𝑁, 1o, βˆ…)) = 𝑋)
 
2.6.38  Omniscient sets
 
Syntaxcomni 7132 Extend class definition to include the class of omniscient sets.
class Omni
 
Definitiondf-omni 7133* 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 βˆ…) for some element. Definition 3.1 of [Pierik], p. 14.

In particular, Ο‰ ∈ Omni is known as the Limited Principle of Omniscience (LPO). (Contributed by Jim Kingdon, 28-Jun-2022.)

Omni = {𝑦 ∣ βˆ€π‘“(𝑓:π‘¦βŸΆ2o β†’ (βˆƒπ‘₯ ∈ 𝑦 (π‘“β€˜π‘₯) = βˆ… ∨ βˆ€π‘₯ ∈ 𝑦 (π‘“β€˜π‘₯) = 1o))}
 
Theoremisomni 7134* The predicate of being omniscient. (Contributed by Jim Kingdon, 28-Jun-2022.)
(𝐴 ∈ 𝑉 β†’ (𝐴 ∈ Omni ↔ βˆ€π‘“(𝑓:𝐴⟢2o β†’ (βˆƒπ‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) = βˆ… ∨ βˆ€π‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) = 1o))))
 
Theoremisomnimap 7135* The predicate of being omniscient stated in terms of set exponentiation. (Contributed by Jim Kingdon, 13-Jul-2022.)
(𝐴 ∈ 𝑉 β†’ (𝐴 ∈ Omni ↔ βˆ€π‘“ ∈ (2o β†‘π‘š 𝐴)(βˆƒπ‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) = βˆ… ∨ βˆ€π‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) = 1o)))
 
Theoremenomnilem 7136 Lemma for enomni 7137. One direction of the biconditional. (Contributed by Jim Kingdon, 13-Jul-2022.)
(𝐴 β‰ˆ 𝐡 β†’ (𝐴 ∈ Omni β†’ 𝐡 ∈ Omni))
 
Theoremenomni 7137 Omniscience is invariant with respect to equinumerosity. For example, this means that we can express the Limited Principle of Omniscience as either Ο‰ ∈ Omni or β„•0 ∈ Omni. The former is a better match to conventional notation in the sense that df2o3 6431 says that 2o = {βˆ…, 1o} whereas the corresponding relationship does not exist between 2 and {0, 1}. (Contributed by Jim Kingdon, 13-Jul-2022.)
(𝐴 β‰ˆ 𝐡 β†’ (𝐴 ∈ Omni ↔ 𝐡 ∈ Omni))
 
Theoremfinomni 7138 A finite set is omniscient. Remark right after Definition 3.1 of [Pierik], p. 14. (Contributed by Jim Kingdon, 28-Jun-2022.)
(𝐴 ∈ Fin β†’ 𝐴 ∈ Omni)
 
Theoremexmidomniim 7139 Given excluded middle, every set is omniscient. Remark following Definition 3.1 of [Pierik], p. 14. This is one direction of the biconditional exmidomni 7140. (Contributed by Jim Kingdon, 29-Jun-2022.)
(EXMID β†’ βˆ€π‘₯ π‘₯ ∈ Omni)
 
Theoremexmidomni 7140 Excluded middle is equivalent to every set being omniscient. (Contributed by BJ and Jim Kingdon, 30-Jun-2022.)
(EXMID ↔ βˆ€π‘₯ π‘₯ ∈ Omni)
 
Theoremexmidlpo 7141 Excluded middle implies the Limited Principle of Omniscience (LPO). (Contributed by Jim Kingdon, 29-Mar-2023.)
(EXMID β†’ Ο‰ ∈ Omni)
 
Theoremfodjuomnilemdc 7142* Lemma for fodjuomni 7147. Decidability of a condition we use in various lemmas. (Contributed by Jim Kingdon, 27-Jul-2022.)
(πœ‘ β†’ 𝐹:𝑂–ontoβ†’(𝐴 βŠ” 𝐡))    β‡’   ((πœ‘ ∧ 𝑋 ∈ 𝑂) β†’ DECID βˆƒπ‘§ ∈ 𝐴 (πΉβ€˜π‘‹) = (inlβ€˜π‘§))
 
Theoremfodjuf 7143* Lemma for fodjuomni 7147 and fodjumkv 7158. Domain and range of 𝑃. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.)
(πœ‘ β†’ 𝐹:𝑂–ontoβ†’(𝐴 βŠ” 𝐡))    &   π‘ƒ = (𝑦 ∈ 𝑂 ↦ if(βˆƒπ‘§ ∈ 𝐴 (πΉβ€˜π‘¦) = (inlβ€˜π‘§), βˆ…, 1o))    &   (πœ‘ β†’ 𝑂 ∈ 𝑉)    β‡’   (πœ‘ β†’ 𝑃 ∈ (2o β†‘π‘š 𝑂))
 
Theoremfodjum 7144* Lemma for fodjuomni 7147 and fodjumkv 7158. A condition which shows that 𝐴 is inhabited. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.)
(πœ‘ β†’ 𝐹:𝑂–ontoβ†’(𝐴 βŠ” 𝐡))    &   π‘ƒ = (𝑦 ∈ 𝑂 ↦ if(βˆƒπ‘§ ∈ 𝐴 (πΉβ€˜π‘¦) = (inlβ€˜π‘§), βˆ…, 1o))    &   (πœ‘ β†’ βˆƒπ‘€ ∈ 𝑂 (π‘ƒβ€˜π‘€) = βˆ…)    β‡’   (πœ‘ β†’ βˆƒπ‘₯ π‘₯ ∈ 𝐴)
 
Theoremfodju0 7145* Lemma for fodjuomni 7147 and fodjumkv 7158. A condition which shows that 𝐴 is empty. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.)
(πœ‘ β†’ 𝐹:𝑂–ontoβ†’(𝐴 βŠ” 𝐡))    &   π‘ƒ = (𝑦 ∈ 𝑂 ↦ if(βˆƒπ‘§ ∈ 𝐴 (πΉβ€˜π‘¦) = (inlβ€˜π‘§), βˆ…, 1o))    &   (πœ‘ β†’ βˆ€π‘€ ∈ 𝑂 (π‘ƒβ€˜π‘€) = 1o)    β‡’   (πœ‘ β†’ 𝐴 = βˆ…)
 
Theoremfodjuomnilemres 7146* Lemma for fodjuomni 7147. The final result with 𝑃 expressed as a local definition. (Contributed by Jim Kingdon, 29-Jul-2022.)
(πœ‘ β†’ 𝑂 ∈ Omni)    &   (πœ‘ β†’ 𝐹:𝑂–ontoβ†’(𝐴 βŠ” 𝐡))    &   π‘ƒ = (𝑦 ∈ 𝑂 ↦ if(βˆƒπ‘§ ∈ 𝐴 (πΉβ€˜π‘¦) = (inlβ€˜π‘§), βˆ…, 1o))    β‡’   (πœ‘ β†’ (βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∨ 𝐴 = βˆ…))
 
Theoremfodjuomni 7147* A condition which ensures 𝐴 is either inhabited or empty. Lemma 3.2 of [PradicBrown2022], p. 4. (Contributed by Jim Kingdon, 27-Jul-2022.)
(πœ‘ β†’ 𝑂 ∈ Omni)    &   (πœ‘ β†’ 𝐹:𝑂–ontoβ†’(𝐴 βŠ” 𝐡))    β‡’   (πœ‘ β†’ (βˆƒπ‘₯ π‘₯ ∈ 𝐴 ∨ 𝐴 = βˆ…))
 
Theoremctssexmid 7148* The decidability condition in ctssdc 7112 is needed. More specifically, ctssdc 7112 minus that condition, plus the Limited Principle of Omniscience (LPO), implies excluded middle. (Contributed by Jim Kingdon, 15-Aug-2023.)
((𝑦 βŠ† Ο‰ ∧ βˆƒπ‘“ 𝑓:𝑦–ontoβ†’π‘₯) β†’ βˆƒπ‘“ 𝑓:ω–ontoβ†’(π‘₯ βŠ” 1o))    &   Ο‰ ∈ Omni    β‡’   (πœ‘ ∨ Β¬ πœ‘)
 
2.6.39  Markov's principle
 
Syntaxcmarkov 7149 Extend class definition to include the class of Markov sets.
class Markov
 
Definitiondf-markov 7150* 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 βˆ…). Generalization of definition 2.5 of [Pierik], p. 9.

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

Markov = {𝑦 ∣ βˆ€π‘“(𝑓:π‘¦βŸΆ2o β†’ (Β¬ βˆ€π‘₯ ∈ 𝑦 (π‘“β€˜π‘₯) = 1o β†’ βˆƒπ‘₯ ∈ 𝑦 (π‘“β€˜π‘₯) = βˆ…))}
 
Theoremismkv 7151* The predicate of being Markov. (Contributed by Jim Kingdon, 18-Mar-2023.)
(𝐴 ∈ 𝑉 β†’ (𝐴 ∈ Markov ↔ βˆ€π‘“(𝑓:𝐴⟢2o β†’ (Β¬ βˆ€π‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) = 1o β†’ βˆƒπ‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) = βˆ…))))
 
Theoremismkvmap 7152* The predicate of being Markov stated in terms of set exponentiation. (Contributed by Jim Kingdon, 18-Mar-2023.)
(𝐴 ∈ 𝑉 β†’ (𝐴 ∈ Markov ↔ βˆ€π‘“ ∈ (2o β†‘π‘š 𝐴)(Β¬ βˆ€π‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) = 1o β†’ βˆƒπ‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) = βˆ…)))
 
Theoremismkvnex 7153* The predicate of being Markov stated in terms of double negation and comparison with 1o. (Contributed by Jim Kingdon, 29-Nov-2023.)
(𝐴 ∈ 𝑉 β†’ (𝐴 ∈ Markov ↔ βˆ€π‘“ ∈ (2o β†‘π‘š 𝐴)(Β¬ Β¬ βˆƒπ‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) = 1o β†’ βˆƒπ‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) = 1o)))
 
Theoremomnimkv 7154 An omniscient set is Markov. In particular, the case where 𝐴 is Ο‰ means that the Limited Principle of Omniscience (LPO) implies Markov's Principle (MP). (Contributed by Jim Kingdon, 18-Mar-2023.)
(𝐴 ∈ Omni β†’ 𝐴 ∈ Markov)
 
Theoremexmidmp 7155 Excluded middle implies Markov's Principle (MP). (Contributed by Jim Kingdon, 4-Apr-2023.)
(EXMID β†’ Ο‰ ∈ Markov)
 
Theoremmkvprop 7156* Markov's Principle expressed in terms of propositions (or more precisely, the 𝐴 = Ο‰ case is Markov's Principle). (Contributed by Jim Kingdon, 19-Mar-2023.)
((𝐴 ∈ Markov ∧ βˆ€π‘› ∈ 𝐴 DECID πœ‘ ∧ Β¬ βˆ€π‘› ∈ 𝐴 Β¬ πœ‘) β†’ βˆƒπ‘› ∈ 𝐴 πœ‘)
 
Theoremfodjumkvlemres 7157* Lemma for fodjumkv 7158. The final result with 𝑃 expressed as a local definition. (Contributed by Jim Kingdon, 25-Mar-2023.)
(πœ‘ β†’ 𝑀 ∈ Markov)    &   (πœ‘ β†’ 𝐹:𝑀–ontoβ†’(𝐴 βŠ” 𝐡))    &   π‘ƒ = (𝑦 ∈ 𝑀 ↦ if(βˆƒπ‘§ ∈ 𝐴 (πΉβ€˜π‘¦) = (inlβ€˜π‘§), βˆ…, 1o))    β‡’   (πœ‘ β†’ (𝐴 β‰  βˆ… β†’ βˆƒπ‘₯ π‘₯ ∈ 𝐴))
 
Theoremfodjumkv 7158* A condition which ensures that a nonempty set is inhabited. (Contributed by Jim Kingdon, 25-Mar-2023.)
(πœ‘ β†’ 𝑀 ∈ Markov)    &   (πœ‘ β†’ 𝐹:𝑀–ontoβ†’(𝐴 βŠ” 𝐡))    β‡’   (πœ‘ β†’ (𝐴 β‰  βˆ… β†’ βˆƒπ‘₯ π‘₯ ∈ 𝐴))
 
Theoremenmkvlem 7159 Lemma for enmkv 7160. One direction of the biconditional. (Contributed by Jim Kingdon, 25-Jun-2024.)
(𝐴 β‰ˆ 𝐡 β†’ (𝐴 ∈ Markov β†’ 𝐡 ∈ Markov))
 
Theoremenmkv 7160 Being Markov is invariant with respect to equinumerosity. For example, this means that we can express the Markov's Principle as either Ο‰ ∈ Markov or β„•0 ∈ Markov. The former is a better match to conventional notation in the sense that df2o3 6431 says that 2o = {βˆ…, 1o} whereas the corresponding relationship does not exist between 2 and {0, 1}. (Contributed by Jim Kingdon, 24-Jun-2024.)
(𝐴 β‰ˆ 𝐡 β†’ (𝐴 ∈ Markov ↔ 𝐡 ∈ Markov))
 
2.6.40  Weakly omniscient sets
 
Syntaxcwomni 7161 Extend class definition to include the class of weakly omniscient sets.
class WOmni
 
Definitiondf-womni 7162* 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, Ο‰ ∈ 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 ∣ βˆ€π‘“(𝑓:π‘¦βŸΆ2o β†’ DECID βˆ€π‘₯ ∈ 𝑦 (π‘“β€˜π‘₯) = 1o)}
 
Theoremiswomni 7163* The predicate of being weakly omniscient. (Contributed by Jim Kingdon, 9-Jun-2024.)
(𝐴 ∈ 𝑉 β†’ (𝐴 ∈ WOmni ↔ βˆ€π‘“(𝑓:𝐴⟢2o β†’ DECID βˆ€π‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) = 1o)))
 
Theoremiswomnimap 7164* The predicate of being weakly omniscient stated in terms of set exponentiation. (Contributed by Jim Kingdon, 9-Jun-2024.)
(𝐴 ∈ 𝑉 β†’ (𝐴 ∈ WOmni ↔ βˆ€π‘“ ∈ (2o β†‘π‘š 𝐴)DECID βˆ€π‘₯ ∈ 𝐴 (π‘“β€˜π‘₯) = 1o))
 
Theoremomniwomnimkv 7165 A set is omniscient if and only if it is weakly omniscient and Markov. The case 𝐴 = Ο‰ 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 ∈ Omni ↔ (𝐴 ∈ WOmni ∧ 𝐴 ∈ Markov))
 
Theoremlpowlpo 7166 LPO implies WLPO. Easy corollary of the more general omniwomnimkv 7165. There is an analogue in terms of analytic omniscience principles at tridceq 14807. (Contributed by Jim Kingdon, 24-Jul-2024.)
(Ο‰ ∈ Omni β†’ Ο‰ ∈ WOmni)
 
Theoremenwomnilem 7167 Lemma for enwomni 7168. One direction of the biconditional. (Contributed by Jim Kingdon, 20-Jun-2024.)
(𝐴 β‰ˆ 𝐡 β†’ (𝐴 ∈ WOmni β†’ 𝐡 ∈ WOmni))
 
Theoremenwomni 7168 Weak omniscience is invariant with respect to equinumerosity. For example, this means that we can express the Weak Limited Principle of Omniscience as either Ο‰ ∈ WOmni or β„•0 ∈ WOmni. The former is a better match to conventional notation in the sense that df2o3 6431 says that 2o = {βˆ…, 1o} whereas the corresponding relationship does not exist between 2 and {0, 1}. (Contributed by Jim Kingdon, 20-Jun-2024.)
(𝐴 β‰ˆ 𝐡 β†’ (𝐴 ∈ WOmni ↔ 𝐡 ∈ WOmni))
 
Theoremnninfdcinf 7169* The Weak Limited Principle of Omniscience (WLPO) implies that it is decidable whether an element of β„•βˆž equals the point at infinity. (Contributed by Jim Kingdon, 3-Dec-2024.)
(πœ‘ β†’ Ο‰ ∈ WOmni)    &   (πœ‘ β†’ 𝑁 ∈ β„•βˆž)    β‡’   (πœ‘ β†’ DECID 𝑁 = (𝑖 ∈ Ο‰ ↦ 1o))
 
Theoremnninfwlporlemd 7170* 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.)
(πœ‘ β†’ 𝑋:Ο‰βŸΆ2o)    &   (πœ‘ β†’ π‘Œ:Ο‰βŸΆ2o)    &   π· = (𝑖 ∈ Ο‰ ↦ if((π‘‹β€˜π‘–) = (π‘Œβ€˜π‘–), 1o, βˆ…))    β‡’   (πœ‘ β†’ (𝑋 = π‘Œ ↔ 𝐷 = (𝑖 ∈ Ο‰ ↦ 1o)))
 
Theoremnninfwlporlem 7171* Lemma for nninfwlpor 7172. The result. (Contributed by Jim Kingdon, 7-Dec-2024.)
(πœ‘ β†’ 𝑋:Ο‰βŸΆ2o)    &   (πœ‘ β†’ π‘Œ:Ο‰βŸΆ2o)    &   π· = (𝑖 ∈ Ο‰ ↦ if((π‘‹β€˜π‘–) = (π‘Œβ€˜π‘–), 1o, βˆ…))    &   (πœ‘ β†’ Ο‰ ∈ WOmni)    β‡’   (πœ‘ β†’ DECID 𝑋 = π‘Œ)
 
Theoremnninfwlpor 7172* The Weak Limited Principle of Omniscience (WLPO) implies that equality for β„•βˆž is decidable. (Contributed by Jim Kingdon, 7-Dec-2024.)
(Ο‰ ∈ WOmni β†’ βˆ€π‘₯ ∈ β„•βˆž βˆ€π‘¦ ∈ β„•βˆž DECID π‘₯ = 𝑦)
 
Theoremnninfwlpoimlemg 7173* Lemma for nninfwlpoim 7176. (Contributed by Jim Kingdon, 8-Dec-2024.)
(πœ‘ β†’ 𝐹:Ο‰βŸΆ2o)    &   πΊ = (𝑖 ∈ Ο‰ ↦ if(βˆƒπ‘₯ ∈ suc 𝑖(πΉβ€˜π‘₯) = βˆ…, βˆ…, 1o))    β‡’   (πœ‘ β†’ 𝐺 ∈ β„•βˆž)
 
Theoremnninfwlpoimlemginf 7174* Lemma for nninfwlpoim 7176. (Contributed by Jim Kingdon, 8-Dec-2024.)
(πœ‘ β†’ 𝐹:Ο‰βŸΆ2o)    &   πΊ = (𝑖 ∈ Ο‰ ↦ if(βˆƒπ‘₯ ∈ suc 𝑖(πΉβ€˜π‘₯) = βˆ…, βˆ…, 1o))    β‡’   (πœ‘ β†’ (𝐺 = (𝑖 ∈ Ο‰ ↦ 1o) ↔ βˆ€π‘› ∈ Ο‰ (πΉβ€˜π‘›) = 1o))
 
Theoremnninfwlpoimlemdc 7175* Lemma for nninfwlpoim 7176. (Contributed by Jim Kingdon, 8-Dec-2024.)
(πœ‘ β†’ 𝐹:Ο‰βŸΆ2o)    &   πΊ = (𝑖 ∈ Ο‰ ↦ if(βˆƒπ‘₯ ∈ suc 𝑖(πΉβ€˜π‘₯) = βˆ…, βˆ…, 1o))    &   (πœ‘ β†’ βˆ€π‘₯ ∈ β„•βˆž βˆ€π‘¦ ∈ β„•βˆž DECID π‘₯ = 𝑦)    β‡’   (πœ‘ β†’ DECID βˆ€π‘› ∈ Ο‰ (πΉβ€˜π‘›) = 1o)
 
Theoremnninfwlpoim 7176* Decidable equality for β„•βˆž implies the Weak Limited Principle of Omniscience (WLPO). (Contributed by Jim Kingdon, 9-Dec-2024.)
(βˆ€π‘₯ ∈ β„•βˆž βˆ€π‘¦ ∈ β„•βˆž DECID π‘₯ = 𝑦 β†’ Ο‰ ∈ WOmni)
 
Theoremnninfwlpo 7177* Decidability of equality for β„•βˆž is equivalent to the Weak Limited Principle of Omniscience (WLPO). (Contributed by Jim Kingdon, 3-Dec-2024.)
(βˆ€π‘₯ ∈ β„•βˆž βˆ€π‘¦ ∈ β„•βˆž DECID π‘₯ = 𝑦 ↔ Ο‰ ∈ WOmni)
 
2.6.41  Cardinal numbers
 
Syntaxccrd 7178 Extend class definition to include the cardinal size function.
class card
 
Definitiondf-card 7179* 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 = (π‘₯ ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ π‘₯})
 
Theoremcardcl 7180* The cardinality of a well-orderable set is an ordinal. (Contributed by Jim Kingdon, 30-Aug-2021.)
(βˆƒπ‘¦ ∈ On 𝑦 β‰ˆ 𝐴 β†’ (cardβ€˜π΄) ∈ On)
 
Theoremisnumi 7181 A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
((𝐴 ∈ On ∧ 𝐴 β‰ˆ 𝐡) β†’ 𝐡 ∈ dom card)
 
Theoremfinnum 7182 Every finite set is numerable. (Contributed by Mario Carneiro, 4-Feb-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
(𝐴 ∈ Fin β†’ 𝐴 ∈ dom card)
 
Theoremonenon 7183 Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
(𝐴 ∈ On β†’ 𝐴 ∈ dom card)
 
Theoremcardval3ex 7184* The value of (cardβ€˜π΄). (Contributed by Jim Kingdon, 30-Aug-2021.)
(βˆƒπ‘₯ ∈ On π‘₯ β‰ˆ 𝐴 β†’ (cardβ€˜π΄) = ∩ {𝑦 ∈ On ∣ 𝑦 β‰ˆ 𝐴})
 
Theoremoncardval 7185* 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 ∈ On β†’ (cardβ€˜π΄) = ∩ {π‘₯ ∈ On ∣ π‘₯ β‰ˆ 𝐴})
 
Theoremcardonle 7186 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 ∈ On β†’ (cardβ€˜π΄) βŠ† 𝐴)
 
Theoremcard0 7187 The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.)
(cardβ€˜βˆ…) = βˆ…
 
Theoremcarden2bex 7188* If two numerable sets are equinumerous, then they have equal cardinalities. (Contributed by Jim Kingdon, 30-Aug-2021.)
((𝐴 β‰ˆ 𝐡 ∧ βˆƒπ‘₯ ∈ On π‘₯ β‰ˆ 𝐴) β†’ (cardβ€˜π΄) = (cardβ€˜π΅))
 
Theorempm54.43 7189 Theorem *54.43 of [WhiteheadRussell] p. 360. (Contributed by NM, 4-Apr-2007.)
((𝐴 β‰ˆ 1o ∧ 𝐡 β‰ˆ 1o) β†’ ((𝐴 ∩ 𝐡) = βˆ… ↔ (𝐴 βˆͺ 𝐡) β‰ˆ 2o))
 
Theorempr2nelem 7190 Lemma for pr2ne 7191. (Contributed by FL, 17-Aug-2008.)
((𝐴 ∈ 𝐢 ∧ 𝐡 ∈ 𝐷 ∧ 𝐴 β‰  𝐡) β†’ {𝐴, 𝐡} β‰ˆ 2o)
 
Theorempr2ne 7191 If an unordered pair has two elements they are different. (Contributed by FL, 14-Feb-2010.)
((𝐴 ∈ 𝐢 ∧ 𝐡 ∈ 𝐷) β†’ ({𝐴, 𝐡} β‰ˆ 2o ↔ 𝐴 β‰  𝐡))
 
Theoremexmidonfinlem 7192* Lemma for exmidonfin 7193. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.)
𝐴 = {{π‘₯ ∈ {βˆ…} ∣ πœ‘}, {π‘₯ ∈ {βˆ…} ∣ Β¬ πœ‘}}    β‡’   (Ο‰ = (On ∩ Fin) β†’ DECID πœ‘)
 
Theoremexmidonfin 7193 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 6872 and nnon 4610. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.)
(Ο‰ = (On ∩ Fin) β†’ EXMID)
 
Theoremen2eleq 7194 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 ∈ 𝑃 ∧ 𝑃 β‰ˆ 2o) β†’ 𝑃 = {𝑋, βˆͺ (𝑃 βˆ– {𝑋})})
 
Theoremen2other2 7195 Taking the other element twice in a pair gets back to the original element. (Contributed by Stefan O'Rear, 22-Aug-2015.)
((𝑋 ∈ 𝑃 ∧ 𝑃 β‰ˆ 2o) β†’ βˆͺ (𝑃 βˆ– {βˆͺ (𝑃 βˆ– {𝑋})}) = 𝑋)
 
Theoremdju1p1e2 7196 Disjoint union version of one plus one equals two. (Contributed by Jim Kingdon, 1-Jul-2022.)
(1o βŠ” 1o) β‰ˆ 2o
 
Theoreminfpwfidom 7197 The collection of finite subsets of a set dominates the set. (We use the weaker sethood assumption (𝒫 𝐴 ∩ Fin) ∈ V because this theorem also implies that 𝐴 is a set if 𝒫 𝐴 ∩ Fin is.) (Contributed by Mario Carneiro, 17-May-2015.)
((𝒫 𝐴 ∩ Fin) ∈ V β†’ 𝐴 β‰Ό (𝒫 𝐴 ∩ Fin))
 
Theoremexmidfodomrlemeldju 7198 Lemma for exmidfodomr 7203. A variant of djur 7068. (Contributed by Jim Kingdon, 2-Jul-2022.)
(πœ‘ β†’ 𝐴 βŠ† 1o)    &   (πœ‘ β†’ 𝐡 ∈ (𝐴 βŠ” 1o))    β‡’   (πœ‘ β†’ (𝐡 = (inlβ€˜βˆ…) ∨ 𝐡 = (inrβ€˜βˆ…)))
 
Theoremexmidfodomrlemreseldju 7199 Lemma for exmidfodomrlemrALT 7202. A variant of eldju 7067. (Contributed by Jim Kingdon, 9-Jul-2022.)
(πœ‘ β†’ 𝐴 βŠ† 1o)    &   (πœ‘ β†’ 𝐡 ∈ (𝐴 βŠ” 1o))    β‡’   (πœ‘ β†’ ((βˆ… ∈ 𝐴 ∧ 𝐡 = ((inl β†Ύ 𝐴)β€˜βˆ…)) ∨ 𝐡 = ((inr β†Ύ 1o)β€˜βˆ…)))
 
Theoremexmidfodomrlemim 7200* 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 β†’ βˆ€π‘₯βˆ€π‘¦((βˆƒπ‘§ 𝑧 ∈ 𝑦 ∧ 𝑦 β‰Ό π‘₯) β†’ βˆƒπ‘“ 𝑓:π‘₯–onto→𝑦))
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