P. 71 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 7001-7100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	inflbti 7001*	An infimum is a lower bound. See also infclti 7000 and infglbti 7002. (Contributed by Jim Kingdon, 18-Dec-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))    ⇒   ⊢ (𝜑 → (𝐶 ∈ 𝐵 → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅)))
Theorem	infglbti 7002*	An infimum is the greatest lower bound. See also infclti 7000 and inflbti 7001. (Contributed by Jim Kingdon, 18-Dec-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))    ⇒   ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ inf(𝐵, 𝐴, 𝑅)𝑅𝐶) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝐶))
Theorem	infnlbti 7003*	A lower bound is not greater than the infimum. (Contributed by Jim Kingdon, 18-Dec-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))    ⇒   ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝐶) → ¬ inf(𝐵, 𝐴, 𝑅)𝑅𝐶))
Theorem	infminti 7004*	The smallest element of a set is its infimum. Note that the converse is not true; the infimum might not be an element of the set considered. (Contributed by Jim Kingdon, 18-Dec-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝑦𝑅𝐶)    ⇒   ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)
Theorem	infmoti 7005*	Any class 𝐵 has at most one infimum in 𝐴 (where 𝑅 is interpreted as 'less than'). (Contributed by Jim Kingdon, 18-Dec-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    ⇒   ⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))
Theorem	infeuti 7006*	An infimum is unique. (Contributed by Jim Kingdon, 19-Dec-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))    ⇒   ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))
Theorem	infsnti 7007*	The infimum of a singleton. (Contributed by Jim Kingdon, 19-Dec-2021.)
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    ⇒   ⊢ (𝜑 → inf({𝐵}, 𝐴, 𝑅) = 𝐵)
Theorem	inf00 7008	The infimum regarding an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
⊢ inf(𝐵, ∅, 𝑅) = ∅
Theorem	infisoti 7009*	Image of an infimum under an isomorphism. (Contributed by Jim Kingdon, 19-Dec-2021.)
⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐶 𝑧𝑅𝑦)))    &   ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    ⇒   ⊢ (𝜑 → inf((𝐹 “ 𝐶), 𝐵, 𝑆) = (𝐹‘inf(𝐶, 𝐴, 𝑅)))
Theorem	supex2g 7010	Existence of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝐴 ∈ 𝐶 → sup(𝐵, 𝐴, 𝑅) ∈ V)
Theorem	infex2g 7011	Existence of infimum. (Contributed by Jim Kingdon, 1-Oct-2024.)
⊢ (𝐴 ∈ 𝐶 → inf(𝐵, 𝐴, 𝑅) ∈ V)
2.6.35  Ordinal isomorphism
Theorem	ordiso2 7012	Generalize ordiso 7013 to proper classes. (Contributed by Mario Carneiro, 24-Jun-2015.)
⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → 𝐴 = 𝐵)
Theorem	ordiso 7013*	Order-isomorphic ordinal numbers are equal. (Contributed by Jeff Hankins, 16-Oct-2009.) (Proof shortened by Mario Carneiro, 24-Jun-2015.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 = 𝐵 ↔ ∃𝑓 𝑓 Isom E , E (𝐴, 𝐵)))
2.6.36  Disjoint union
2.6.36.1  Disjoint union
Syntax	cdju 7014	Extend class notation to include disjoint union of two classes.
class (𝐴 ⊔ 𝐵)
Definition	df-dju 7015	Disjoint union of two classes. This is a way of creating a class which contains elements corresponding to each element of 𝐴 or 𝐵, tagging each one with whether it came from 𝐴 or 𝐵. (Contributed by Jim Kingdon, 20-Jun-2022.)
⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
Theorem	djueq12 7016	Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⊔ 𝐶) = (𝐵 ⊔ 𝐷))
Theorem	djueq1 7017	Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
⊢ (𝐴 = 𝐵 → (𝐴 ⊔ 𝐶) = (𝐵 ⊔ 𝐶))
Theorem	djueq2 7018	Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
⊢ (𝐴 = 𝐵 → (𝐶 ⊔ 𝐴) = (𝐶 ⊔ 𝐵))
Theorem	nfdju 7019	Bound-variable hypothesis builder for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥(𝐴 ⊔ 𝐵)
Theorem	djuex 7020	The disjoint union of sets is a set. See also the more precise djuss 7047. (Contributed by AV, 28-Jun-2022.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ∈ V)
Theorem	djuexb 7021	The disjoint union of two classes is a set iff both classes are sets. (Contributed by Jim Kingdon, 6-Sep-2023.)
⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ⊔ 𝐵) ∈ V)
2.6.36.2  Left and right injections of a disjoint union In this section, we define the left and right injections of a disjoint union and prove their main properties. These injections are restrictions of the "template" functions inl and inr, which appear in most applications in the form (inl ↾ 𝐴) and (inr ↾ 𝐵).
Syntax	cinl 7022	Extend class notation to include left injection of a disjoint union.
class inl
Syntax	cinr 7023	Extend class notation to include right injection of a disjoint union.
class inr
Definition	df-inl 7024	Left injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.)
⊢ inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
Definition	df-inr 7025	Right injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.)
⊢ inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
Theorem	djulclr 7026	Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) (Revised by BJ, 6-Jul-2022.)
⊢ (𝐶 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝐶) ∈ (𝐴 ⊔ 𝐵))
Theorem	djurclr 7027	Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) (Revised by BJ, 6-Jul-2022.)
⊢ (𝐶 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝐶) ∈ (𝐴 ⊔ 𝐵))
Theorem	djulcl 7028	Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)
⊢ (𝐶 ∈ 𝐴 → (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵))
Theorem	djurcl 7029	Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)
⊢ (𝐶 ∈ 𝐵 → (inr‘𝐶) ∈ (𝐴 ⊔ 𝐵))
Theorem	djuf1olem 7030*	Lemma for djulf1o 7035 and djurf1o 7036. (Contributed by BJ and Jim Kingdon, 4-Jul-2022.)
⊢ 𝑋 ∈ V    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ ⟨𝑋, 𝑥⟩)    ⇒   ⊢ 𝐹:𝐴–1-1-onto→({𝑋} × 𝐴)
Theorem	djuf1olemr 7031*	Lemma for djulf1or 7033 and djurf1or 7034. For a version of this lemma with 𝐹 defined on 𝐴 and no restriction in the conclusion, see djuf1olem 7030. (Contributed by BJ and Jim Kingdon, 4-Jul-2022.)
⊢ 𝑋 ∈ V    &   ⊢ 𝐹 = (𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩)    ⇒   ⊢ (𝐹 ↾ 𝐴):𝐴–1-1-onto→({𝑋} × 𝐴)
Theorem	djulclb 7032	Left biconditional closure of disjoint union. (Contributed by Jim Kingdon, 2-Jul-2022.)
⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ 𝐴 ↔ (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵)))
Theorem	djulf1or 7033	The left injection function on all sets is one to one and onto. (Contributed by BJ and Jim Kingdon, 22-Jun-2022.)
⊢ (inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴)
Theorem	djurf1or 7034	The right injection function on all sets is one to one and onto. (Contributed by BJ and Jim Kingdon, 22-Jun-2022.)
⊢ (inr ↾ 𝐴):𝐴–1-1-onto→({1o} × 𝐴)
Theorem	djulf1o 7035	The left injection function on all sets is one to one and onto. (Contributed by Jim Kingdon, 22-Jun-2022.)
⊢ inl:V–1-1-onto→({∅} × V)
Theorem	djurf1o 7036	The right injection function on all sets is one to one and onto. (Contributed by Jim Kingdon, 22-Jun-2022.)
⊢ inr:V–1-1-onto→({1o} × V)
Theorem	inresflem 7037*	Lemma for inlresf1 7038 and inrresf1 7039. (Contributed by BJ, 4-Jul-2022.)
⊢ 𝐹:𝐴–1-1-onto→({𝑋} × 𝐴)    &   ⊢ (𝑥 ∈ 𝐴 → (𝐹‘𝑥) ∈ 𝐵)    ⇒   ⊢ 𝐹:𝐴–1-1→𝐵
Theorem	inlresf1 7038	The left injection restricted to the left class of a disjoint union is an injective function from the left class into the disjoint union. (Contributed by AV, 28-Jun-2022.)
⊢ (inl ↾ 𝐴):𝐴–1-1→(𝐴 ⊔ 𝐵)
Theorem	inrresf1 7039	The right injection restricted to the right class of a disjoint union is an injective function from the right class into the disjoint union. (Contributed by AV, 28-Jun-2022.)
⊢ (inr ↾ 𝐵):𝐵–1-1→(𝐴 ⊔ 𝐵)
Theorem	djuinr 7040	The ranges of any left and right injections are disjoint. Remark: the extra generality offered by the two restrictions makes the theorem more readily usable (e.g., by djudom 7070 and djufun 7081) while the simpler statement ⊢ (ran inl ∩ ran inr) = ∅ is easily recovered from it by substituting V for both 𝐴 and 𝐵 as done in casefun 7062). (Contributed by BJ and Jim Kingdon, 21-Jun-2022.)
⊢ (ran (inl ↾ 𝐴) ∩ ran (inr ↾ 𝐵)) = ∅
Theorem	djuin 7041	The images of any classes under right and left injection produce disjoint sets. (Contributed by Jim Kingdon, 21-Jun-2022.) (Proof shortened by BJ, 9-Jul-2023.)
⊢ ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅
Theorem	inl11 7042	Left injection is one-to-one. (Contributed by Jim Kingdon, 12-Jul-2023.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((inl‘𝐴) = (inl‘𝐵) ↔ 𝐴 = 𝐵))
Theorem	djuunr 7043	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, 6-Jul-2022.)
⊢ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) = (𝐴 ⊔ 𝐵)
Theorem	djuun 7044	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 7045*	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 7046*	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)))
2.6.36.3  Universal property of the disjoint union
Theorem	djuss 7047	A disjoint union is a subset of a Cartesian product. (Contributed by AV, 25-Jun-2022.)
⊢ (𝐴 ⊔ 𝐵) ⊆ ({∅, 1o} × (𝐴 ∪ 𝐵))
Theorem	eldju1st 7048	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 7049	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 7050	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 7051	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 7052	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 7053	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 7054	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 7055	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 7056*	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 7057*	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 7058*	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 7059*	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 7060	Syntax for the "case" construction.
class case(𝑅, 𝑆)
Definition	df-case 7061	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 7059. 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 7062	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 7063	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 7064	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 7065	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 7066	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 7067	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 7068	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 7069	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))
2.6.36.4  Dominance and equinumerosity properties of disjoint union
Theorem	djudom 7070	Dominance law for disjoint union. (Contributed by Jim Kingdon, 25-Jul-2022.)
⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) → (𝐴 ⊔ 𝐶) ≼ (𝐵 ⊔ 𝐷))
Theorem	omp1eomlem 7071*	Lemma for omp1eom 7072. (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 7072	Adding one to ω. (Contributed by Jim Kingdon, 10-Jul-2023.)
⊢ (ω ⊔ 1o) ≈ ω
Theorem	endjusym 7073	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 7074	Equinumerosity of a set and its image under left injection. (Contributed by Jim Kingdon, 30-Jul-2023.)
⊢ (𝐴 ∈ 𝑉 → (inl “ 𝐴) ≈ 𝐴)
Theorem	eninr 7075	Equinumerosity of a set and its image under right injection. (Contributed by Jim Kingdon, 30-Jul-2023.)
⊢ (𝐴 ∈ 𝑉 → (inr “ 𝐴) ≈ 𝐴)
Theorem	difinfsnlem 7076*	Lemma for difinfsn 7077. 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 7077*	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 7078*	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))
2.6.36.5  Older definition temporarily kept for comparison, to be deleted
Syntax	cdjud 7079	Syntax for the domain-disjoint-union of two relations.
class (𝑅 ⊔d 𝑆)
Definition	df-djud 7080	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 7059 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 7061, 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 7081	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 7082	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 7083	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))
2.6.36.6  Countable sets
Theorem	0ct 7084	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 7085*	Lemma for ctm 7086. 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 7086*	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 7087*	Lemma for ctssdc 7090. 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 7088*	A mapping from a decidable subset of the natural numbers onto a countable set. This is similar to one direction of ctssdc 7090 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 7089*	Lemma for ctssdc 7090. 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 7090*	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 7126. (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 7091*	Lemma for enumct 7092. 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 7092*	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 7093*	A finite set is countable. (Contributed by Jim Kingdon, 17-Mar-2023.)
⊢ (𝐴 ∈ Fin → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))
Theorem	omct 7094	ω is countable. (Contributed by Jim Kingdon, 23-Dec-2023.)
⊢ ∃𝑓 𝑓:ω–onto→(ω ⊔ 1o)
Theorem	ctfoex 7095*	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 7096	Set of nonincreasing sequences in 2o ↑𝑚 ω.
class ℕ∞
Definition	df-nninf 7097*	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 9199 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 6409) 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 7098	ℕ∞ is a set. (Contributed by Jim Kingdon, 10-Aug-2022.)
⊢ ℕ∞ ∈ V
Theorem	nninff 7099	An element of ℕ∞ is a sequence of zeroes and ones. (Contributed by Jim Kingdon, 4-Aug-2022.)
⊢ (𝐴 ∈ ℕ∞ → 𝐴:ω⟶2o)
Theorem	infnninf 7100	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 4658 shows. (Contributed by Jim Kingdon, 14-Jul-2022.) Use maps-to notation. (Revised by BJ, 10-Aug-2024.)
⊢ (𝑖 ∈ ω ↦ 1o) ∈ ℕ∞