P. 74 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7301-7400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ctmlemr 7301*	Lemma for ctm 7302. One of the directions of the biconditional. (Contributed by Jim Kingdon, 16-Mar-2023.)
⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∃𝑓 𝑓:ω–onto→𝐴 → ∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o)))
Theorem	ctm 7302*	Two equivalent definitions of countable for an inhabited set. Remark of [BauerSwan], p. 14:3. (Contributed by Jim Kingdon, 13-Mar-2023.)
⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) ↔ ∃𝑓 𝑓:ω–onto→𝐴))
Theorem	ctssdclemn0 7303*	Lemma for ctssdc 7306. The ¬ ∅ ∈ 𝑆 case. (Contributed by Jim Kingdon, 16-Aug-2023.)
⊢ (𝜑 → 𝑆 ⊆ ω)    &   ⊢ (𝜑 → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹:𝑆–onto→𝐴)    &   ⊢ (𝜑 → ¬ ∅ ∈ 𝑆)    ⇒   ⊢ (𝜑 → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
Theorem	ctssdccl 7304*	A mapping from a decidable subset of the natural numbers onto a countable set. This is similar to one direction of ctssdc 7306 but expressed in terms of classes rather than ∃. (Contributed by Jim Kingdon, 30-Oct-2023.)
⊢ (𝜑 → 𝐹:ω–onto→(𝐴 ⊔ 1o))    &   ⊢ 𝑆 = {𝑥 ∈ ω ∣ (𝐹‘𝑥) ∈ (inl “ 𝐴)}    &   ⊢ 𝐺 = (◡inl ∘ 𝐹)    ⇒   ⊢ (𝜑 → (𝑆 ⊆ ω ∧ 𝐺:𝑆–onto→𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑆))
Theorem	ctssdclemr 7305*	Lemma for ctssdc 7306. Showing that our usual definition of countable implies the alternate one. (Contributed by Jim Kingdon, 16-Aug-2023.)
⊢ (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) → ∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑠))
Theorem	ctssdc 7306*	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 7343. (Contributed by Jim Kingdon, 15-Aug-2023.)
⊢ (∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠–onto→𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑠) ↔ ∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o))
Theorem	enumctlemm 7307*	Lemma for enumct 7308. The case where 𝑁 is greater than zero. (Contributed by Jim Kingdon, 13-Mar-2023.)
⊢ (𝜑 → 𝐹:𝑁–onto→𝐴)    &   ⊢ (𝜑 → 𝑁 ∈ ω)    &   ⊢ (𝜑 → ∅ ∈ 𝑁)    &   ⊢ 𝐺 = (𝑘 ∈ ω ↦ if(𝑘 ∈ 𝑁, (𝐹‘𝑘), (𝐹‘∅)))    ⇒   ⊢ (𝜑 → 𝐺:ω–onto→𝐴)
Theorem	enumct 7308*	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 ∃𝑛 ∈ ω∃𝑓𝑓:𝑛–onto→𝐴 per Definition 8.1.4 of [AczelRathjen], p. 71 and "countable" is defined as ∃𝑔𝑔:ω–onto→(𝐴 ⊔ 1o) per [BauerSwan], p. 14:3. (Contributed by Jim Kingdon, 13-Mar-2023.)
⊢ (∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛–onto→𝐴 → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
Theorem	finct 7309*	A finite set is countable. (Contributed by Jim Kingdon, 17-Mar-2023.)
⊢ (𝐴 ∈ Fin → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
Theorem	omct 7310	ω is countable. (Contributed by Jim Kingdon, 23-Dec-2023.)
⊢ ∃𝑓 𝑓:ω–onto→(ω ⊔ 1o)
Theorem	ctfoex 7311*	A countable class is a set. (Contributed by Jim Kingdon, 25-Dec-2023.)
⊢ (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) → 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 ω with values in 2o. The topological results justifying its name will be proved later.
Syntax	xnninf 7312	Set of nonincreasing sequences in 2o ↑𝑚 ω.
class ℕ∞
Definition	df-nninf 7313*	Define the set of nonincreasing sequences in 2o ↑𝑚 ω. Definition in Section 3.1 of [Pierik], p. 15. If we assumed excluded middle, this would be essentially the same as ℕ0* as defined at df-xnn0 9459 but in its absence the relationship between the two is more complicated. This definition would function much the same whether we used ω or ℕ0, but the former allows us to take advantage of 2o = {∅, 1o} (df2o3 6592) so we adopt it. (Contributed by Jim Kingdon, 14-Jul-2022.)
⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖)}
Theorem	nninfex 7314	ℕ∞ is a set. (Contributed by Jim Kingdon, 10-Aug-2022.)
⊢ ℕ∞ ∈ V
Theorem	nninff 7315	An element of ℕ∞ is a sequence of zeroes and ones. (Contributed by Jim Kingdon, 4-Aug-2022.)
⊢ (𝐴 ∈ ℕ∞ → 𝐴:ω⟶2o)
Theorem	nninfninc 7316	All values beyond a zero in an ℕ∞ sequence are zero. This is another way of stating that elements of ℕ∞ are nonincreasing. (Contributed by Jim Kingdon, 12-Jul-2025.)
⊢ (𝜑 → 𝐴 ∈ ℕ∞)    &   ⊢ (𝜑 → 𝑋 ∈ ω)    &   ⊢ (𝜑 → 𝑌 ∈ ω)    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑌)    &   ⊢ (𝜑 → (𝐴‘𝑋) = ∅)    ⇒   ⊢ (𝜑 → (𝐴‘𝑌) = ∅)
Theorem	infnninf 7317	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 (ω × {1o}), as fconstmpt 4771 shows. (Contributed by Jim Kingdon, 14-Jul-2022.) Use maps-to notation. (Revised by BJ, 10-Aug-2024.)
⊢ (𝑖 ∈ ω ↦ 1o) ∈ ℕ∞
Theorem	infnninfOLD 7318	Obsolete version of infnninf 7317 as of 10-Aug-2024. (Contributed by Jim Kingdon, 14-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (ω × {1o}) ∈ ℕ∞
Theorem	nnnninf 7319*	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 7320. (Contributed by Jim Kingdon, 14-Jul-2022.)
⊢ (𝑁 ∈ ω → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) ∈ ℕ∞)
Theorem	nnnninf2 7320*	Canonical embedding of suc ω into ℕ∞. (Contributed by BJ, 10-Aug-2024.)
⊢ (𝑁 ∈ suc ω → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) ∈ ℕ∞)
Theorem	nnnninfeq 7321*	Mapping of a natural number to an element of ℕ∞. (Contributed by Jim Kingdon, 4-Aug-2022.)
⊢ (𝜑 → 𝑃 ∈ ℕ∞)    &   ⊢ (𝜑 → 𝑁 ∈ ω)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝑁 (𝑃‘𝑥) = 1o)    &   ⊢ (𝜑 → (𝑃‘𝑁) = ∅)    ⇒   ⊢ (𝜑 → 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)))
Theorem	nnnninfeq2 7322*	Mapping of a natural number to an element of ℕ∞. Similar to nnnninfeq 7321 but if we have information about a single 1o digit, that gives information about all previous digits. (Contributed by Jim Kingdon, 4-Aug-2022.)
⊢ (𝜑 → 𝑃 ∈ ℕ∞)    &   ⊢ (𝜑 → 𝑁 ∈ ω)    &   ⊢ (𝜑 → (𝑃‘∪ 𝑁) = 1o)    &   ⊢ (𝜑 → (𝑃‘𝑁) = ∅)    ⇒   ⊢ (𝜑 → 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)))
Theorem	nninfisollem0 7323*	Lemma for nninfisol 7326. The case where 𝑁 is zero. (Contributed by Jim Kingdon, 13-Sep-2024.)
⊢ (𝜑 → 𝑋 ∈ ℕ∞)    &   ⊢ (𝜑 → (𝑋‘𝑁) = ∅)    &   ⊢ (𝜑 → 𝑁 ∈ ω)    &   ⊢ (𝜑 → 𝑁 = ∅)    ⇒   ⊢ (𝜑 → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
Theorem	nninfisollemne 7324*	Lemma for nninfisol 7326. A case where 𝑁 is a successor and 𝑁 and 𝑋 are not equal. (Contributed by Jim Kingdon, 13-Sep-2024.)
⊢ (𝜑 → 𝑋 ∈ ℕ∞)    &   ⊢ (𝜑 → (𝑋‘𝑁) = ∅)    &   ⊢ (𝜑 → 𝑁 ∈ ω)    &   ⊢ (𝜑 → 𝑁 ≠ ∅)    &   ⊢ (𝜑 → (𝑋‘∪ 𝑁) = ∅)    ⇒   ⊢ (𝜑 → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
Theorem	nninfisollemeq 7325*	Lemma for nninfisol 7326. The case where 𝑁 is a successor and 𝑁 and 𝑋 are equal. (Contributed by Jim Kingdon, 13-Sep-2024.)
⊢ (𝜑 → 𝑋 ∈ ℕ∞)    &   ⊢ (𝜑 → (𝑋‘𝑁) = ∅)    &   ⊢ (𝜑 → 𝑁 ∈ ω)    &   ⊢ (𝜑 → 𝑁 ≠ ∅)    &   ⊢ (𝜑 → (𝑋‘∪ 𝑁) = 1o)    ⇒   ⊢ (𝜑 → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
Theorem	nninfisol 7326*	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). By contrast, the point at infinity being isolated is equivalent to the Weak Limited Principle of Omniscience (WLPO) (nninfinfwlpo 7373). (Contributed by BJ and Jim Kingdon, 12-Sep-2024.)
⊢ ((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
2.6.38  Omniscient sets
Syntax	comni 7327	Extend class definition to include the class of omniscient sets.
class Omni
Definition	df-omni 7328*	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, ω ∈ Omni is known as the Limited Principle of Omniscience (LPO). (Contributed by Jim Kingdon, 28-Jun-2022.)
⊢ Omni = {𝑦 ∣ ∀𝑓(𝑓:𝑦⟶2o → (∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o))}
Theorem	isomni 7329*	The predicate of being omniscient. (Contributed by Jim Kingdon, 28-Jun-2022.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Omni ↔ ∀𝑓(𝑓:𝐴⟶2o → (∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))))
Theorem	isomnimap 7330*	The predicate of being omniscient stated in terms of set exponentiation. (Contributed by Jim Kingdon, 13-Jul-2022.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Omni ↔ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅ ∨ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)))
Theorem	enomnilem 7331	Lemma for enomni 7332. One direction of the biconditional. (Contributed by Jim Kingdon, 13-Jul-2022.)
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Omni → 𝐵 ∈ Omni))
Theorem	enomni 7332	Omniscience is invariant with respect to equinumerosity. For example, this means that we can express the Limited Principle of Omniscience as either ω ∈ Omni or ℕ0 ∈ Omni. The former is a better match to conventional notation in the sense that df2o3 6592 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 ∈ Omni ↔ 𝐵 ∈ Omni))
Theorem	finomni 7333	A finite set is omniscient. Remark right after Definition 3.1 of [Pierik], p. 14. (Contributed by Jim Kingdon, 28-Jun-2022.)
⊢ (𝐴 ∈ Fin → 𝐴 ∈ Omni)
Theorem	exmidomniim 7334	Given excluded middle, every set is omniscient. Remark following Definition 3.1 of [Pierik], p. 14. This is one direction of the biconditional exmidomni 7335. (Contributed by Jim Kingdon, 29-Jun-2022.)
⊢ (EXMID → ∀𝑥 𝑥 ∈ Omni)
Theorem	exmidomni 7335	Excluded middle is equivalent to every set being omniscient. (Contributed by BJ and Jim Kingdon, 30-Jun-2022.)
⊢ (EXMID ↔ ∀𝑥 𝑥 ∈ Omni)
Theorem	exmidlpo 7336	Excluded middle implies the Limited Principle of Omniscience (LPO). (Contributed by Jim Kingdon, 29-Mar-2023.)
⊢ (EXMID → ω ∈ Omni)
Theorem	fodjuomnilemdc 7337*	Lemma for fodjuomni 7342. Decidability of a condition we use in various lemmas. (Contributed by Jim Kingdon, 27-Jul-2022.)
⊢ (𝜑 → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑂) → DECID ∃𝑧 ∈ 𝐴 (𝐹‘𝑋) = (inl‘𝑧))
Theorem	fodjuf 7338*	Lemma for fodjuomni 7342 and fodjumkv 7353. Domain and range of 𝑃. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.)
⊢ (𝜑 → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵))    &   ⊢ 𝑃 = (𝑦 ∈ 𝑂 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o))    &   ⊢ (𝜑 → 𝑂 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝑃 ∈ (2o ↑𝑚 𝑂))
Theorem	fodjum 7339*	Lemma for fodjuomni 7342 and fodjumkv 7353. A condition which shows that 𝐴 is inhabited. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.)
⊢ (𝜑 → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵))    &   ⊢ 𝑃 = (𝑦 ∈ 𝑂 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o))    &   ⊢ (𝜑 → ∃𝑤 ∈ 𝑂 (𝑃‘𝑤) = ∅)    ⇒   ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
Theorem	fodju0 7340*	Lemma for fodjuomni 7342 and fodjumkv 7353. A condition which shows that 𝐴 is empty. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.)
⊢ (𝜑 → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵))    &   ⊢ 𝑃 = (𝑦 ∈ 𝑂 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o))    &   ⊢ (𝜑 → ∀𝑤 ∈ 𝑂 (𝑃‘𝑤) = 1o)    ⇒   ⊢ (𝜑 → 𝐴 = ∅)
Theorem	fodjuomnilemres 7341*	Lemma for fodjuomni 7342. The final result with 𝑃 expressed as a local definition. (Contributed by Jim Kingdon, 29-Jul-2022.)
⊢ (𝜑 → 𝑂 ∈ Omni)    &   ⊢ (𝜑 → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵))    &   ⊢ 𝑃 = (𝑦 ∈ 𝑂 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o))    ⇒   ⊢ (𝜑 → (∃𝑥 𝑥 ∈ 𝐴 ∨ 𝐴 = ∅))
Theorem	fodjuomni 7342*	A condition which ensures 𝐴 is either inhabited or empty. Lemma 3.2 of [PradicBrown2022], p. 4. (Contributed by Jim Kingdon, 27-Jul-2022.)
⊢ (𝜑 → 𝑂 ∈ Omni)    &   ⊢ (𝜑 → 𝐹:𝑂–onto→(𝐴 ⊔ 𝐵))    ⇒   ⊢ (𝜑 → (∃𝑥 𝑥 ∈ 𝐴 ∨ 𝐴 = ∅))
Theorem	ctssexmid 7343*	The decidability condition in ctssdc 7306 is needed. More specifically, ctssdc 7306 minus that condition, plus the Limited Principle of Omniscience (LPO), implies excluded middle. (Contributed by Jim Kingdon, 15-Aug-2023.)
⊢ ((𝑦 ⊆ ω ∧ ∃𝑓 𝑓:𝑦–onto→𝑥) → ∃𝑓 𝑓:ω–onto→(𝑥 ⊔ 1o))    &   ⊢ ω ∈ Omni    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
2.6.39  Markov's principle
Syntax	cmarkov 7344	Extend class definition to include the class of Markov sets.
class Markov
Definition	df-markov 7345*	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, ω ∈ Markov is known as Markov's Principle (MP). (Contributed by Jim Kingdon, 18-Mar-2023.)
⊢ Markov = {𝑦 ∣ ∀𝑓(𝑓:𝑦⟶2o → (¬ ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝑦 (𝑓‘𝑥) = ∅))}
Theorem	ismkv 7346*	The predicate of being Markov. (Contributed by Jim Kingdon, 18-Mar-2023.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Markov ↔ ∀𝑓(𝑓:𝐴⟶2o → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))))
Theorem	ismkvmap 7347*	The predicate of being Markov stated in terms of set exponentiation. (Contributed by Jim Kingdon, 18-Mar-2023.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Markov ↔ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅)))
Theorem	ismkvnex 7348*	The predicate of being Markov stated in terms of double negation and comparison with 1o. (Contributed by Jim Kingdon, 29-Nov-2023.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Markov ↔ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ¬ ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)))
Theorem	omnimkv 7349	An omniscient set is Markov. In particular, the case where 𝐴 is ω means that the Limited Principle of Omniscience (LPO) implies Markov's Principle (MP). (Contributed by Jim Kingdon, 18-Mar-2023.)
⊢ (𝐴 ∈ Omni → 𝐴 ∈ Markov)
Theorem	exmidmp 7350	Excluded middle implies Markov's Principle (MP). (Contributed by Jim Kingdon, 4-Apr-2023.)
⊢ (EXMID → ω ∈ Markov)
Theorem	mkvprop 7351*	Markov's Principle expressed in terms of propositions (or more precisely, the 𝐴 = ω case is Markov's Principle). (Contributed by Jim Kingdon, 19-Mar-2023.)
⊢ ((𝐴 ∈ Markov ∧ ∀𝑛 ∈ 𝐴 DECID 𝜑 ∧ ¬ ∀𝑛 ∈ 𝐴 ¬ 𝜑) → ∃𝑛 ∈ 𝐴 𝜑)
Theorem	fodjumkvlemres 7352*	Lemma for fodjumkv 7353. The final result with 𝑃 expressed as a local definition. (Contributed by Jim Kingdon, 25-Mar-2023.)
⊢ (𝜑 → 𝑀 ∈ Markov)    &   ⊢ (𝜑 → 𝐹:𝑀–onto→(𝐴 ⊔ 𝐵))    &   ⊢ 𝑃 = (𝑦 ∈ 𝑀 ↦ if(∃𝑧 ∈ 𝐴 (𝐹‘𝑦) = (inl‘𝑧), ∅, 1o))    ⇒   ⊢ (𝜑 → (𝐴 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝐴))
Theorem	fodjumkv 7353*	A condition which ensures that a nonempty set is inhabited. (Contributed by Jim Kingdon, 25-Mar-2023.)
⊢ (𝜑 → 𝑀 ∈ Markov)    &   ⊢ (𝜑 → 𝐹:𝑀–onto→(𝐴 ⊔ 𝐵))    ⇒   ⊢ (𝜑 → (𝐴 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝐴))
Theorem	enmkvlem 7354	Lemma for enmkv 7355. One direction of the biconditional. (Contributed by Jim Kingdon, 25-Jun-2024.)
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Markov → 𝐵 ∈ Markov))
Theorem	enmkv 7355	Being Markov is invariant with respect to equinumerosity. For example, this means that we can express the Markov's Principle as either ω ∈ Markov or ℕ0 ∈ Markov. The former is a better match to conventional notation in the sense that df2o3 6592 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 ∈ Markov ↔ 𝐵 ∈ Markov))
2.6.40  Weakly omniscient sets
Syntax	cwomni 7356	Extend class definition to include the class of weakly omniscient sets.
class WOmni
Definition	df-womni 7357*	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, ω ∈ 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 ∣ ∀𝑓(𝑓:𝑦⟶2o → DECID ∀𝑥 ∈ 𝑦 (𝑓‘𝑥) = 1o)}
Theorem	iswomni 7358*	The predicate of being weakly omniscient. (Contributed by Jim Kingdon, 9-Jun-2024.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓(𝑓:𝐴⟶2o → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)))
Theorem	iswomnimap 7359*	The predicate of being weakly omniscient stated in terms of set exponentiation. (Contributed by Jim Kingdon, 9-Jun-2024.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓 ∈ (2o ↑𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))
Theorem	omniwomnimkv 7360	A set is omniscient if and only if it is weakly omniscient and Markov. The case 𝐴 = ω 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))
Theorem	lpowlpo 7361	LPO implies WLPO. Easy corollary of the more general omniwomnimkv 7360. There is an analogue in terms of analytic omniscience principles at tridceq 16610. (Contributed by Jim Kingdon, 24-Jul-2024.)
⊢ (ω ∈ Omni → ω ∈ WOmni)
Theorem	enwomnilem 7362	Lemma for enwomni 7363. One direction of the biconditional. (Contributed by Jim Kingdon, 20-Jun-2024.)
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ WOmni → 𝐵 ∈ WOmni))
Theorem	enwomni 7363	Weak omniscience is invariant with respect to equinumerosity. For example, this means that we can express the Weak Limited Principle of Omniscience as either ω ∈ WOmni or ℕ0 ∈ WOmni. The former is a better match to conventional notation in the sense that df2o3 6592 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 ∈ WOmni ↔ 𝐵 ∈ WOmni))
Theorem	nninfdcinf 7364*	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.)
⊢ (𝜑 → ω ∈ WOmni)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ∞)    ⇒   ⊢ (𝜑 → DECID 𝑁 = (𝑖 ∈ ω ↦ 1o))
Theorem	nninfwlporlemd 7365*	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.)
⊢ (𝜑 → 𝑋:ω⟶2o)    &   ⊢ (𝜑 → 𝑌:ω⟶2o)    &   ⊢ 𝐷 = (𝑖 ∈ ω ↦ if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅))    ⇒   ⊢ (𝜑 → (𝑋 = 𝑌 ↔ 𝐷 = (𝑖 ∈ ω ↦ 1o)))
Theorem	nninfwlporlem 7366*	Lemma for nninfwlpor 7367. The result. (Contributed by Jim Kingdon, 7-Dec-2024.)
⊢ (𝜑 → 𝑋:ω⟶2o)    &   ⊢ (𝜑 → 𝑌:ω⟶2o)    &   ⊢ 𝐷 = (𝑖 ∈ ω ↦ if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅))    &   ⊢ (𝜑 → ω ∈ WOmni)    ⇒   ⊢ (𝜑 → DECID 𝑋 = 𝑌)
Theorem	nninfwlpor 7367*	The Weak Limited Principle of Omniscience (WLPO) implies that equality for ℕ∞ is decidable. (Contributed by Jim Kingdon, 7-Dec-2024.)
⊢ (ω ∈ WOmni → ∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ DECID 𝑥 = 𝑦)
Theorem	nninfwlpoimlemg 7368*	Lemma for nninfwlpoim 7372. (Contributed by Jim Kingdon, 8-Dec-2024.)
⊢ (𝜑 → 𝐹:ω⟶2o)    &   ⊢ 𝐺 = (𝑖 ∈ ω ↦ if(∃𝑥 ∈ suc 𝑖(𝐹‘𝑥) = ∅, ∅, 1o))    ⇒   ⊢ (𝜑 → 𝐺 ∈ ℕ∞)
Theorem	nninfwlpoimlemginf 7369*	Lemma for nninfwlpoim 7372. (Contributed by Jim Kingdon, 8-Dec-2024.)
⊢ (𝜑 → 𝐹:ω⟶2o)    &   ⊢ 𝐺 = (𝑖 ∈ ω ↦ if(∃𝑥 ∈ suc 𝑖(𝐹‘𝑥) = ∅, ∅, 1o))    ⇒   ⊢ (𝜑 → (𝐺 = (𝑖 ∈ ω ↦ 1o) ↔ ∀𝑛 ∈ ω (𝐹‘𝑛) = 1o))
Theorem	nninfwlpoimlemdc 7370*	Lemma for nninfwlpoim 7372. (Contributed by Jim Kingdon, 8-Dec-2024.)
⊢ (𝜑 → 𝐹:ω⟶2o)    &   ⊢ 𝐺 = (𝑖 ∈ ω ↦ if(∃𝑥 ∈ suc 𝑖(𝐹‘𝑥) = ∅, ∅, 1o))    &   ⊢ (𝜑 → ∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ DECID 𝑥 = 𝑦)    ⇒   ⊢ (𝜑 → DECID ∀𝑛 ∈ ω (𝐹‘𝑛) = 1o)
Theorem	nninfinfwlpolem 7371*	Lemma for nninfinfwlpo 7373. (Contributed by Jim Kingdon, 8-Dec-2024.)
⊢ (𝜑 → 𝐹:ω⟶2o)    &   ⊢ 𝐺 = (𝑖 ∈ ω ↦ if(∃𝑥 ∈ suc 𝑖(𝐹‘𝑥) = ∅, ∅, 1o))    &   ⊢ (𝜑 → ∀𝑥 ∈ ℕ∞ DECID 𝑥 = (𝑖 ∈ ω ↦ 1o))    ⇒   ⊢ (𝜑 → DECID ∀𝑛 ∈ ω (𝐹‘𝑛) = 1o)
Theorem	nninfwlpoim 7372*	Decidable equality for ℕ∞ implies the Weak Limited Principle of Omniscience (WLPO). (Contributed by Jim Kingdon, 9-Dec-2024.)
⊢ (∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ DECID 𝑥 = 𝑦 → ω ∈ WOmni)
Theorem	nninfinfwlpo 7373*	The point at infinity in ℕ∞ being isolated is equivalent to the Weak Limited Principle of Omniscience (WLPO). By isolated, we mean that the equality of that point with every other element of ℕ∞ is decidable. From an online post by Martin Escardo. By contrast, elements of ℕ∞ corresponding to natural numbers are isolated (nninfisol 7326). (Contributed by Jim Kingdon, 25-Nov-2025.)
⊢ (∀𝑥 ∈ ℕ∞ DECID 𝑥 = (𝑖 ∈ ω ↦ 1o) ↔ ω ∈ WOmni)
Theorem	nninfwlpo 7374*	Decidability of equality for ℕ∞ is equivalent to the Weak Limited Principle of Omniscience (WLPO). (Contributed by Jim Kingdon, 3-Dec-2024.)
⊢ (∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ DECID 𝑥 = 𝑦 ↔ ω ∈ WOmni)
2.6.41  Cardinal numbers
Syntax	ccrd 7375	Extend class definition to include the cardinal size function.
class card
Syntax	wacn 7376	The axiom of choice for limited-length sequences.
class AC 𝐴
Definition	df-card 7377*	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 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥})
Definition	df-acnm 7378*	Define a local and length-limited version of the axiom of choice. The definition of the predicate 𝑋 ∈ AC 𝐴 is that for all families of inhabited subsets of 𝑋 indexed on 𝐴 (i.e. functions 𝐴⟶{𝑧 ∈ 𝒫 𝑋 ∣ ∃𝑗𝑗 ∈ 𝑧}), there is a function which selects an element from each set in the family. (Contributed by Mario Carneiro, 31-Aug-2015.) Change nonempty to inhabited. (Revised by Jim Kingdon, 22-Nov-2025.)
⊢ AC 𝐴 = {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑥 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦))}
Theorem	cardcl 7379*	The cardinality of a well-orderable set is an ordinal. (Contributed by Jim Kingdon, 30-Aug-2021.)
⊢ (∃𝑦 ∈ On 𝑦 ≈ 𝐴 → (card‘𝐴) ∈ On)
Theorem	isnumi 7380	A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → 𝐵 ∈ dom card)
Theorem	finnum 7381	Every finite set is numerable. (Contributed by Mario Carneiro, 4-Feb-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
⊢ (𝐴 ∈ Fin → 𝐴 ∈ dom card)
Theorem	onenon 7382	Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
⊢ (𝐴 ∈ On → 𝐴 ∈ dom card)
Theorem	cardval3ex 7383*	The value of (card‘𝐴). (Contributed by Jim Kingdon, 30-Aug-2021.)
⊢ (∃𝑥 ∈ On 𝑥 ≈ 𝐴 → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
Theorem	oncardval 7384*	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 → (card‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐴})
Theorem	cardonle 7385	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 → (card‘𝐴) ⊆ 𝐴)
Theorem	card0 7386	The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.)
⊢ (card‘∅) = ∅
Theorem	ficardon 7387	The cardinal number of a finite set is an ordinal. (Contributed by Jim Kingdon, 1-Nov-2025.)
⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ On)
Theorem	carden2bex 7388*	If two numerable sets are equinumerous, then they have equal cardinalities. (Contributed by Jim Kingdon, 30-Aug-2021.)
⊢ ((𝐴 ≈ 𝐵 ∧ ∃𝑥 ∈ On 𝑥 ≈ 𝐴) → (card‘𝐴) = (card‘𝐵))
Theorem	pm54.43 7389	Theorem *54.43 of [WhiteheadRussell] p. 360. (Contributed by NM, 4-Apr-2007.)
⊢ ((𝐴 ≈ 1o ∧ 𝐵 ≈ 1o) → ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐴 ∪ 𝐵) ≈ 2o))
Theorem	pr2nelem 7390	Lemma for pr2ne 7391. (Contributed by FL, 17-Aug-2008.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ≈ 2o)
Theorem	pr2ne 7391	If an unordered pair has two elements they are different. (Contributed by FL, 14-Feb-2010.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ({𝐴, 𝐵} ≈ 2o ↔ 𝐴 ≠ 𝐵))
Theorem	en2prde 7392*	A set of size two is an unordered pair of two different elements. (Contributed by Alexander van der Vekens, 8-Dec-2017.) (Revised by Jim Kingdon, 11-Jan-2026.)
⊢ (𝑉 ≈ 2o → ∃𝑎∃𝑏(𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}))
Theorem	pr1or2 7393	An unordered pair, with decidable equality for the specified elements, has either one or two elements. (Contributed by Jim Kingdon, 7-Jan-2026.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ DECID 𝐴 = 𝐵) → ({𝐴, 𝐵} ≈ 1o ∨ {𝐴, 𝐵} ≈ 2o))
Theorem	pr2cv1 7394	If an unordered pair is equinumerous to ordinal two, then a part is a set. (Contributed by RP, 21-Oct-2023.)
⊢ ({𝐴, 𝐵} ≈ 2o → 𝐴 ∈ V)
Theorem	pr2cv2 7395	If an unordered pair is equinumerous to ordinal two, then a part is a set. (Contributed by RP, 21-Oct-2023.)
⊢ ({𝐴, 𝐵} ≈ 2o → 𝐵 ∈ V)
Theorem	pr2cv 7396	If an unordered pair is equinumerous to ordinal two, then both parts are sets. (Contributed by RP, 8-Oct-2023.)
⊢ ({𝐴, 𝐵} ≈ 2o → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Theorem	exmidonfinlem 7397*	Lemma for exmidonfin 7398. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.)
⊢ 𝐴 = {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}}    ⇒   ⊢ (ω = (On ∩ Fin) → DECID 𝜑)
Theorem	exmidonfin 7398	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 7054 and nnon 4706. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.)
⊢ (ω = (On ∩ Fin) → EXMID)
Theorem	en2eleq 7399	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 ≈ 2o) → 𝑃 = {𝑋, ∪ (𝑃 ∖ {𝑋})})
Theorem	en2other2 7400	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 ≈ 2o) → ∪ (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}) = 𝑋)