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	ctfoex 7301*	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 7302	Set of nonincreasing sequences in 2o ↑𝑚 ω.
class ℕ∞
Definition	df-nninf 7303*	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 9449 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 6588) 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 7304	ℕ∞ is a set. (Contributed by Jim Kingdon, 10-Aug-2022.)
⊢ ℕ∞ ∈ V
Theorem	nninff 7305	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 7306	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 7307	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 4768 shows. (Contributed by Jim Kingdon, 14-Jul-2022.) Use maps-to notation. (Revised by BJ, 10-Aug-2024.)
⊢ (𝑖 ∈ ω ↦ 1o) ∈ ℕ∞
Theorem	infnninfOLD 7308	Obsolete version of infnninf 7307 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 7309*	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 7310. (Contributed by Jim Kingdon, 14-Jul-2022.)
⊢ (𝑁 ∈ ω → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) ∈ ℕ∞)
Theorem	nnnninf2 7310*	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 7311*	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 7312*	Mapping of a natural number to an element of ℕ∞. Similar to nnnninfeq 7311 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 7313*	Lemma for nninfisol 7316. 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 7314*	Lemma for nninfisol 7316. 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 7315*	Lemma for nninfisol 7316. 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 7316*	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 7363). (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 7317	Extend class definition to include the class of omniscient sets.
class Omni
Definition	df-omni 7318*	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 7319*	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 7320*	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 7321	Lemma for enomni 7322. One direction of the biconditional. (Contributed by Jim Kingdon, 13-Jul-2022.)
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Omni → 𝐵 ∈ Omni))
Theorem	enomni 7322	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 6588 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 7323	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 7324	Given excluded middle, every set is omniscient. Remark following Definition 3.1 of [Pierik], p. 14. This is one direction of the biconditional exmidomni 7325. (Contributed by Jim Kingdon, 29-Jun-2022.)
⊢ (EXMID → ∀𝑥 𝑥 ∈ Omni)
Theorem	exmidomni 7325	Excluded middle is equivalent to every set being omniscient. (Contributed by BJ and Jim Kingdon, 30-Jun-2022.)
⊢ (EXMID ↔ ∀𝑥 𝑥 ∈ Omni)
Theorem	exmidlpo 7326	Excluded middle implies the Limited Principle of Omniscience (LPO). (Contributed by Jim Kingdon, 29-Mar-2023.)
⊢ (EXMID → ω ∈ Omni)
Theorem	fodjuomnilemdc 7327*	Lemma for fodjuomni 7332. 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 7328*	Lemma for fodjuomni 7332 and fodjumkv 7343. 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 7329*	Lemma for fodjuomni 7332 and fodjumkv 7343. 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 7330*	Lemma for fodjuomni 7332 and fodjumkv 7343. 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 7331*	Lemma for fodjuomni 7332. 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 7332*	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 7333*	The decidability condition in ctssdc 7296 is needed. More specifically, ctssdc 7296 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 7334	Extend class definition to include the class of Markov sets.
class Markov
Definition	df-markov 7335*	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 7336*	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 7337*	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 7338*	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 7339	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 7340	Excluded middle implies Markov's Principle (MP). (Contributed by Jim Kingdon, 4-Apr-2023.)
⊢ (EXMID → ω ∈ Markov)
Theorem	mkvprop 7341*	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 7342*	Lemma for fodjumkv 7343. 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 7343*	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 7344	Lemma for enmkv 7345. One direction of the biconditional. (Contributed by Jim Kingdon, 25-Jun-2024.)
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Markov → 𝐵 ∈ Markov))
Theorem	enmkv 7345	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 6588 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 7346	Extend class definition to include the class of weakly omniscient sets.
class WOmni
Definition	df-womni 7347*	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 7348*	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 7349*	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 7350	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 7351	LPO implies WLPO. Easy corollary of the more general omniwomnimkv 7350. There is an analogue in terms of analytic omniscience principles at tridceq 16538. (Contributed by Jim Kingdon, 24-Jul-2024.)
⊢ (ω ∈ Omni → ω ∈ WOmni)
Theorem	enwomnilem 7352	Lemma for enwomni 7353. One direction of the biconditional. (Contributed by Jim Kingdon, 20-Jun-2024.)
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ WOmni → 𝐵 ∈ WOmni))
Theorem	enwomni 7353	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 6588 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 7354*	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 7355*	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 7356*	Lemma for nninfwlpor 7357. 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 7357*	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 7358*	Lemma for nninfwlpoim 7362. (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 7359*	Lemma for nninfwlpoim 7362. (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 7360*	Lemma for nninfwlpoim 7362. (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 7361*	Lemma for nninfinfwlpo 7363. (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 7362*	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 7363*	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 7316). (Contributed by Jim Kingdon, 25-Nov-2025.)
⊢ (∀𝑥 ∈ ℕ∞ DECID 𝑥 = (𝑖 ∈ ω ↦ 1o) ↔ ω ∈ WOmni)
Theorem	nninfwlpo 7364*	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 7365	Extend class definition to include the cardinal size function.
class card
Syntax	wacn 7366	The axiom of choice for limited-length sequences.
class AC 𝐴
Definition	df-card 7367*	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 7368*	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 7369*	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 7370	A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
⊢ ((𝐴 ∈ On ∧ 𝐴 ≈ 𝐵) → 𝐵 ∈ dom card)
Theorem	finnum 7371	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 7372	Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.)
⊢ (𝐴 ∈ On → 𝐴 ∈ dom card)
Theorem	cardval3ex 7373*	The value of (card‘𝐴). (Contributed by Jim Kingdon, 30-Aug-2021.)
⊢ (∃𝑥 ∈ On 𝑥 ≈ 𝐴 → (card‘𝐴) = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝐴})
Theorem	oncardval 7374*	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 7375	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 7376	The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.)
⊢ (card‘∅) = ∅
Theorem	ficardon 7377	The cardinal number of a finite set is an ordinal. (Contributed by Jim Kingdon, 1-Nov-2025.)
⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ On)
Theorem	carden2bex 7378*	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 7379	Theorem *54.43 of [WhiteheadRussell] p. 360. (Contributed by NM, 4-Apr-2007.)
⊢ ((𝐴 ≈ 1o ∧ 𝐵 ≈ 1o) → ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐴 ∪ 𝐵) ≈ 2o))
Theorem	pr2nelem 7380	Lemma for pr2ne 7381. (Contributed by FL, 17-Aug-2008.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ≈ 2o)
Theorem	pr2ne 7381	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 7382*	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 7383	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 7384	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 7385	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 7386	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 7387*	Lemma for exmidonfin 7388. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.)
⊢ 𝐴 = {{𝑥 ∈ {∅} ∣ 𝜑}, {𝑥 ∈ {∅} ∣ ¬ 𝜑}}    ⇒   ⊢ (ω = (On ∩ Fin) → DECID 𝜑)
Theorem	exmidonfin 7388	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 7047 and nnon 4703. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.)
⊢ (ω = (On ∩ Fin) → EXMID)
Theorem	en2eleq 7389	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 7390	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)
Theorem	dju1p1e2 7391	Disjoint union version of one plus one equals two. (Contributed by Jim Kingdon, 1-Jul-2022.)
⊢ (1o ⊔ 1o) ≈ 2o
Theorem	infpwfidom 7392	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 → 𝐴 ≼ (𝒫 𝐴 ∩ Fin))
Theorem	exmidfodomrlemeldju 7393	Lemma for exmidfodomr 7398. A variant of djur 7252. (Contributed by Jim Kingdon, 2-Jul-2022.)
⊢ (𝜑 → 𝐴 ⊆ 1o)    &   ⊢ (𝜑 → 𝐵 ∈ (𝐴 ⊔ 1o))    ⇒   ⊢ (𝜑 → (𝐵 = (inl‘∅) ∨ 𝐵 = (inr‘∅)))
Theorem	exmidfodomrlemreseldju 7394	Lemma for exmidfodomrlemrALT 7397. A variant of eldju 7251. (Contributed by Jim Kingdon, 9-Jul-2022.)
⊢ (𝜑 → 𝐴 ⊆ 1o)    &   ⊢ (𝜑 → 𝐵 ∈ (𝐴 ⊔ 1o))    ⇒   ⊢ (𝜑 → ((∅ ∈ 𝐴 ∧ 𝐵 = ((inl ↾ 𝐴)‘∅)) ∨ 𝐵 = ((inr ↾ 1o)‘∅)))
Theorem	exmidfodomrlemim 7395*	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 → ∀𝑥∀𝑦((∃𝑧 𝑧 ∈ 𝑦 ∧ 𝑦 ≼ 𝑥) → ∃𝑓 𝑓:𝑥–onto→𝑦))
Theorem	exmidfodomrlemr 7396*	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.)
⊢ (∀𝑥∀𝑦((∃𝑧 𝑧 ∈ 𝑦 ∧ 𝑦 ≼ 𝑥) → ∃𝑓 𝑓:𝑥–onto→𝑦) → EXMID)
Theorem	exmidfodomrlemrALT 7397*	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. An alternative proof of exmidfodomrlemr 7396. In particular, this proof uses eldju 7251 instead of djur 7252 and avoids djulclb 7238. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Jim Kingdon, 9-Jul-2022.)
⊢ (∀𝑥∀𝑦((∃𝑧 𝑧 ∈ 𝑦 ∧ 𝑦 ≼ 𝑥) → ∃𝑓 𝑓:𝑥–onto→𝑦) → EXMID)
Theorem	exmidfodomr 7398*	Excluded middle is equivalent to the existence of a mapping from any set onto any inhabited set that it dominates. (Contributed by Jim Kingdon, 1-Jul-2022.)
⊢ (EXMID ↔ ∀𝑥∀𝑦((∃𝑧 𝑧 ∈ 𝑦 ∧ 𝑦 ≼ 𝑥) → ∃𝑓 𝑓:𝑥–onto→𝑦))
Theorem	acnrcl 7399	Reverse closure for the choice set predicate. (Contributed by Mario Carneiro, 31-Aug-2015.)
⊢ (𝑋 ∈ AC 𝐴 → 𝐴 ∈ V)
Theorem	acneq 7400	Equality theorem for the choice set function. (Contributed by Mario Carneiro, 31-Aug-2015.)
⊢ (𝐴 = 𝐶 → AC 𝐴 = AC 𝐶)