P. 75 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7401-7500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	djuun 7401	The disjoint union of two classes is the union of the images of those two classes under right and left injection. (Contributed by Jim Kingdon, 22-Jun-2022.) (Proof shortened by BJ, 9-Jul-2023.)
⊢ ((inl “ 𝐴) ∪ (inr “ 𝐵)) = (𝐴 ⊔ 𝐵)
Theorem	eldju 7402*	Element of a disjoint union. (Contributed by BJ and Jim Kingdon, 23-Jun-2022.)
⊢ (𝐶 ∈ (𝐴 ⊔ 𝐵) ↔ (∃𝑥 ∈ 𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥)))
Theorem	djur 7403*	A member of a disjoint union can be mapped from one of the classes which produced it. (Contributed by Jim Kingdon, 23-Jun-2022.) Upgrade implication to biconditional and shorten proof. (Revised by BJ, 14-Jul-2023.)
⊢ (𝐶 ∈ (𝐴 ⊔ 𝐵) ↔ (∃𝑥 ∈ 𝐴 𝐶 = (inl‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = (inr‘𝑥)))
2.6.39.3  Universal property of the disjoint union
Theorem	djuss 7404	A disjoint union is a subset of a Cartesian product. (Contributed by AV, 25-Jun-2022.)
⊢ (𝐴 ⊔ 𝐵) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵))
Theorem	eldju1st 7405	The first component of an element of a disjoint union is either ∅ or 1o. (Contributed by AV, 26-Jun-2022.)
⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) → ((1st ‘𝑋) = ∅ ∨ (1st ‘𝑋) = 1o))
Theorem	eldju2ndl 7406	The second component of an element of a disjoint union is an element of the left class of the disjoint union if its first component is the empty set. (Contributed by AV, 26-Jun-2022.)
⊢ ((𝑋 ∈ (𝐴 ⊔ 𝐵) ∧ (1st ‘𝑋) = ∅) → (2nd ‘𝑋) ∈ 𝐴)
Theorem	eldju2ndr 7407	The second component of an element of a disjoint union is an element of the right class of the disjoint union if its first component is not the empty set. (Contributed by AV, 26-Jun-2022.)
⊢ ((𝑋 ∈ (𝐴 ⊔ 𝐵) ∧ (1st ‘𝑋) ≠ ∅) → (2nd ‘𝑋) ∈ 𝐵)
Theorem	1stinl 7408	The first component of the value of a left injection is the empty set. (Contributed by AV, 27-Jun-2022.)
⊢ (𝑋 ∈ 𝑉 → (1st ‘(inl‘𝑋)) = ∅)
Theorem	2ndinl 7409	The second component of the value of a left injection is its argument. (Contributed by AV, 27-Jun-2022.)
⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inl‘𝑋)) = 𝑋)
Theorem	1stinr 7410	The first component of the value of a right injection is 1o. (Contributed by AV, 27-Jun-2022.)
⊢ (𝑋 ∈ 𝑉 → (1st ‘(inr‘𝑋)) = 1o)
Theorem	2ndinr 7411	The second component of the value of a right injection is its argument. (Contributed by AV, 27-Jun-2022.)
⊢ (𝑋 ∈ 𝑉 → (2nd ‘(inr‘𝑋)) = 𝑋)
Theorem	djune 7412	Left and right injection never produce equal values. (Contributed by Jim Kingdon, 2-Jul-2022.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (inl‘𝐴) ≠ (inr‘𝐵))
Theorem	updjudhf 7413*	The mapping of an element of the disjoint union to the value of the corresponding function is a function. (Contributed by AV, 26-Jun-2022.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐶)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝐶)    &   ⊢ 𝐻 = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))    ⇒   ⊢ (𝜑 → 𝐻:(𝐴 ⊔ 𝐵)⟶𝐶)
Theorem	updjudhcoinlf 7414*	The composition of the mapping of an element of the disjoint union to the value of the corresponding function and the left injection equals the first function. (Contributed by AV, 27-Jun-2022.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐶)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝐶)    &   ⊢ 𝐻 = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))    ⇒   ⊢ (𝜑 → (𝐻 ∘ (inl ↾ 𝐴)) = 𝐹)
Theorem	updjudhcoinrg 7415*	The composition of the mapping of an element of the disjoint union to the value of the corresponding function and the right injection equals the second function. (Contributed by AV, 27-Jun-2022.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐶)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝐶)    &   ⊢ 𝐻 = (𝑥 ∈ (𝐴 ⊔ 𝐵) ↦ if((1st ‘𝑥) = ∅, (𝐹‘(2nd ‘𝑥)), (𝐺‘(2nd ‘𝑥))))    ⇒   ⊢ (𝜑 → (𝐻 ∘ (inr ↾ 𝐵)) = 𝐺)
Theorem	updjud 7416*	Universal property of the disjoint union. (Proposed by BJ, 25-Jun-2022.) (Contributed by AV, 28-Jun-2022.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐶)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ∃!ℎ(ℎ:(𝐴 ⊔ 𝐵)⟶𝐶 ∧ (ℎ ∘ (inl ↾ 𝐴)) = 𝐹 ∧ (ℎ ∘ (inr ↾ 𝐵)) = 𝐺))
Syntax	cdjucase 7417	Syntax for the "case" construction.
class case(𝑅, 𝑆)
Definition	df-case 7418	The "case" construction: if 𝐹:𝐴⟶𝑋 and 𝐺:𝐵⟶𝑋 are functions, then case(𝐹, 𝐺):(𝐴 ⊔ 𝐵)⟶𝑋 is the natural function obtained by a definition by cases, hence the name. It is the unique function whose existence is asserted by the universal property of disjoint unions updjud 7416. The definition is adapted to make sense also for binary relations (where the universal property also holds). (Contributed by MC and BJ, 10-Jul-2022.)
⊢ case(𝑅, 𝑆) = ((𝑅 ∘ ◡inl) ∪ (𝑆 ∘ ◡inr))
Theorem	casefun 7419	The "case" construction of two functions is a function. (Contributed by BJ, 10-Jul-2022.)
⊢ (𝜑 → Fun 𝐹)    &   ⊢ (𝜑 → Fun 𝐺)    ⇒   ⊢ (𝜑 → Fun case(𝐹, 𝐺))
Theorem	casedm 7420	The domain of the "case" construction is the disjoint union of the domains. TODO (although less important): ⊢ ran case(𝐹, 𝐺) = (ran 𝐹 ∪ ran 𝐺). (Contributed by BJ, 10-Jul-2022.)
⊢ dom case(𝐹, 𝐺) = (dom 𝐹 ⊔ dom 𝐺)
Theorem	caserel 7421	The "case" construction of two relations is a relation, with bounds on its domain and codomain. Typically, the "case" construction is used when both relations have a common codomain. (Contributed by BJ, 10-Jul-2022.)
⊢ case(𝑅, 𝑆) ⊆ ((dom 𝑅 ⊔ dom 𝑆) × (ran 𝑅 ∪ ran 𝑆))
Theorem	casef 7422	The "case" construction of two functions is a function on the disjoint union of their domains. (Contributed by BJ, 10-Jul-2022.)
⊢ (𝜑 → 𝐹:𝐴⟶𝑋)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝑋)    ⇒   ⊢ (𝜑 → case(𝐹, 𝐺):(𝐴 ⊔ 𝐵)⟶𝑋)
Theorem	caseinj 7423	The "case" construction of two injective relations with disjoint ranges is an injective relation. (Contributed by BJ, 10-Jul-2022.)
⊢ (𝜑 → Fun ◡𝑅)    &   ⊢ (𝜑 → Fun ◡𝑆)    &   ⊢ (𝜑 → (ran 𝑅 ∩ ran 𝑆) = ∅)    ⇒   ⊢ (𝜑 → Fun ◡case(𝑅, 𝑆))
Theorem	casef1 7424	The "case" construction of two injective functions with disjoint ranges is an injective function. (Contributed by BJ, 10-Jul-2022.)
⊢ (𝜑 → 𝐹:𝐴–1-1→𝑋)    &   ⊢ (𝜑 → 𝐺:𝐵–1-1→𝑋)    &   ⊢ (𝜑 → (ran 𝐹 ∩ ran 𝐺) = ∅)    ⇒   ⊢ (𝜑 → case(𝐹, 𝐺):(𝐴 ⊔ 𝐵)–1-1→𝑋)
Theorem	caseinl 7425	Applying the "case" construction to a left injection. (Contributed by Jim Kingdon, 15-Mar-2023.)
⊢ (𝜑 → 𝐹 Fn 𝐵)    &   ⊢ (𝜑 → Fun 𝐺)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (case(𝐹, 𝐺)‘(inl‘𝐴)) = (𝐹‘𝐴))
Theorem	caseinr 7426	Applying the "case" construction to a right injection. (Contributed by Jim Kingdon, 12-Jul-2023.)
⊢ (𝜑 → Fun 𝐹)    &   ⊢ (𝜑 → 𝐺 Fn 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (case(𝐹, 𝐺)‘(inr‘𝐴)) = (𝐺‘𝐴))
2.6.39.4  Dominance and equinumerosity properties of disjoint union
Theorem	djudom 7427	Dominance law for disjoint union. (Contributed by Jim Kingdon, 25-Jul-2022.)
⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) → (𝐴 ⊔ 𝐶) ≼ (𝐵 ⊔ 𝐷))
Theorem	omp1eomlem 7428*	Lemma for omp1eom 7429. (Contributed by Jim Kingdon, 11-Jul-2023.)
⊢ 𝐹 = (𝑥 ∈ ω ↦ if(𝑥 = ∅, (inr‘𝑥), (inl‘∪ 𝑥)))    &   ⊢ 𝑆 = (𝑥 ∈ ω ↦ suc 𝑥)    &   ⊢ 𝐺 = case(𝑆, ( I ↾ 1o))    ⇒   ⊢ 𝐹:ω–1-1-onto→(ω ⊔ 1o)
Theorem	omp1eom 7429	Adding one to ω. (Contributed by Jim Kingdon, 10-Jul-2023.)
⊢ (ω ⊔ 1o) ≈ ω
Theorem	endjusym 7430	Reversing right and left operands of a disjoint union produces an equinumerous result. (Contributed by Jim Kingdon, 10-Jul-2023.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ≈ (𝐵 ⊔ 𝐴))
Theorem	eninl 7431	Equinumerosity of a set and its image under left injection. (Contributed by Jim Kingdon, 30-Jul-2023.)
⊢ (𝐴 ∈ 𝑉 → (inl “ 𝐴) ≈ 𝐴)
Theorem	eninr 7432	Equinumerosity of a set and its image under right injection. (Contributed by Jim Kingdon, 30-Jul-2023.)
⊢ (𝐴 ∈ 𝑉 → (inr “ 𝐴) ≈ 𝐴)
Theorem	difinfsnlem 7433*	Lemma for difinfsn 7434. The case where we need to swap 𝐵 and (inr‘∅) in building the mapping 𝐺. (Contributed by Jim Kingdon, 9-Aug-2023.)
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐹:(ω ⊔ 1o)–1-1→𝐴)    &   ⊢ (𝜑 → (𝐹‘(inr‘∅)) ≠ 𝐵)    &   ⊢ 𝐺 = (𝑛 ∈ ω ↦ if((𝐹‘(inl‘𝑛)) = 𝐵, (𝐹‘(inr‘∅)), (𝐹‘(inl‘𝑛))))    ⇒   ⊢ (𝜑 → 𝐺:ω–1-1→(𝐴 ∖ {𝐵}))
Theorem	difinfsn 7434*	An infinite set minus one element is infinite. We require that the set has decidable equality. (Contributed by Jim Kingdon, 8-Aug-2023.)
⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) → ω ≼ (𝐴 ∖ {𝐵}))
Theorem	difinfinf 7435*	An infinite set minus a finite subset is infinite. We require that the set has decidable equality. (Contributed by Jim Kingdon, 8-Aug-2023.)
⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ∈ Fin)) → ω ≼ (𝐴 ∖ 𝐵))
2.6.39.5  Older definition temporarily kept for comparison, to be deleted
Syntax	cdjud 7436	Syntax for the domain-disjoint-union of two relations.
class (𝑅 ⊔d 𝑆)
Definition	df-djud 7437	The "domain-disjoint-union" of two relations: if 𝑅 ⊆ (𝐴 × 𝑋) and 𝑆 ⊆ (𝐵 × 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 7416 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 7418, but they are explicitly written for clarity. (Contributed by MC and BJ, 10-Jul-2022.)
⊢ (𝑅 ⊔d 𝑆) = ((𝑅 ∘ ◡(inl ↾ dom 𝑅)) ∪ (𝑆 ∘ ◡(inr ↾ dom 𝑆)))
Theorem	djufun 7438	The "domain-disjoint-union" of two functions is a function. (Contributed by BJ, 10-Jul-2022.)
⊢ (𝜑 → Fun 𝐹)    &   ⊢ (𝜑 → Fun 𝐺)    ⇒   ⊢ (𝜑 → Fun (𝐹 ⊔d 𝐺))
Theorem	djudm 7439	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)
Theorem	djuinj 7440	The "domain-disjoint-union" of two injective relations with disjoint ranges is an injective relation. (Contributed by BJ, 10-Jul-2022.)
⊢ (𝜑 → Fun ◡𝑅)    &   ⊢ (𝜑 → Fun ◡𝑆)    &   ⊢ (𝜑 → (ran 𝑅 ∩ ran 𝑆) = ∅)    ⇒   ⊢ (𝜑 → Fun ◡(𝑅 ⊔d 𝑆))
2.6.39.6  Countable sets
Theorem	0ct 7441	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.)
⊢ ∃𝑓 𝑓:ω–onto→(∅ ⊔ 1o)
Theorem	ctmlemr 7442*	Lemma for ctm 7443. 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 7443*	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 7444*	Lemma for ctssdc 7447. 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 7445*	A mapping from a decidable subset of the natural numbers onto a countable set. This is similar to one direction of ctssdc 7447 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 7446*	Lemma for ctssdc 7447. 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 7447*	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 7484. (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 7448*	Lemma for enumct 7449. 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 7449*	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 Definition on page 14:3 of [BauerSwan], p. 3. (Contributed by Jim Kingdon, 13-Mar-2023.)
⊢ (∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛–onto→𝐴 → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
Theorem	finct 7450*	A finite set is countable. (Contributed by Jim Kingdon, 17-Mar-2023.)
⊢ (𝐴 ∈ Fin → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
Theorem	omct 7451	ω is countable. (Contributed by Jim Kingdon, 23-Dec-2023.)
⊢ ∃𝑓 𝑓:ω–onto→(ω ⊔ 1o)
Theorem	ctfoex 7452*	A countable class is a set. (Contributed by Jim Kingdon, 25-Dec-2023.)
⊢ (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) → 𝐴 ∈ V)
2.6.40  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 7453	Set of nonincreasing sequences in 2o ↑𝑚 ω.
class ℕ∞
Definition	df-nninf 7454*	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 9614 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 6696) 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 7455	ℕ∞ is a set. (Contributed by Jim Kingdon, 10-Aug-2022.)
⊢ ℕ∞ ∈ V
Theorem	nninff 7456	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 7457	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 7458	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 4820 shows. (Contributed by Jim Kingdon, 14-Jul-2022.) Use maps-to notation. (Revised by BJ, 10-Aug-2024.)
⊢ (𝑖 ∈ ω ↦ 1o) ∈ ℕ∞
Theorem	infnninfOLD 7459	Obsolete version of infnninf 7458 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 7460*	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 7461. (Contributed by Jim Kingdon, 14-Jul-2022.)
⊢ (𝑁 ∈ ω → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) ∈ ℕ∞)
Theorem	nnnninf2 7461*	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 7462*	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 7463*	Mapping of a natural number to an element of ℕ∞. Similar to nnnninfeq 7462 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 7464*	Lemma for nninfisol 7467. 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 7465*	Lemma for nninfisol 7467. 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 7466*	Lemma for nninfisol 7467. 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 7467*	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 7514). (Contributed by BJ and Jim Kingdon, 12-Sep-2024.)
⊢ ((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
2.6.41  Omniscient sets
Syntax	comni 7468	Extend class definition to include the class of omniscient sets.
class Omni
Definition	df-omni 7469*	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 7470*	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 7471*	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 7472	Lemma for enomni 7473. One direction of the biconditional. (Contributed by Jim Kingdon, 13-Jul-2022.)
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Omni → 𝐵 ∈ Omni))
Theorem	enomni 7473	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 6696 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 7474	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 7475	Given excluded middle, every set is omniscient. Remark following Definition 3.1 of [Pierik], p. 14. This is one direction of the biconditional exmidomni 7476. (Contributed by Jim Kingdon, 29-Jun-2022.)
⊢ (EXMID → ∀𝑥 𝑥 ∈ Omni)
Theorem	exmidomni 7476	Excluded middle is equivalent to every set being omniscient. (Contributed by BJ and Jim Kingdon, 30-Jun-2022.)
⊢ (EXMID ↔ ∀𝑥 𝑥 ∈ Omni)
Theorem	exmidlpo 7477	Excluded middle implies the Limited Principle of Omniscience (LPO). (Contributed by Jim Kingdon, 29-Mar-2023.)
⊢ (EXMID → ω ∈ Omni)
Theorem	fodjuomnilemdc 7478*	Lemma for fodjuomni 7483. 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 7479*	Lemma for fodjuomni 7483 and fodjumkv 7494. 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 7480*	Lemma for fodjuomni 7483 and fodjumkv 7494. 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 7481*	Lemma for fodjuomni 7483 and fodjumkv 7494. 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 7482*	Lemma for fodjuomni 7483. 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 7483*	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 7484*	The decidability condition in ctssdc 7447 is needed. More specifically, ctssdc 7447 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)
2.6.42  Markov's principle
Syntax	cmarkov 7485	Extend class definition to include the class of Markov sets.
class Markov
Definition	df-markov 7486*	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 7487*	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 7488*	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 7489*	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 7490	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 7491	Excluded middle implies Markov's Principle (MP). (Contributed by Jim Kingdon, 4-Apr-2023.)
⊢ (EXMID → ω ∈ Markov)
Theorem	mkvprop 7492*	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 7493*	Lemma for fodjumkv 7494. 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 7494*	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 7495	Lemma for enmkv 7496. One direction of the biconditional. (Contributed by Jim Kingdon, 25-Jun-2024.)
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Markov → 𝐵 ∈ Markov))
Theorem	enmkv 7496	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 6696 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))
2.6.43  Weakly omniscient sets
Syntax	cwomni 7497	Extend class definition to include the class of weakly omniscient sets.
class WOmni
Definition	df-womni 7498*	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 7499*	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 7500*	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))