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Theorem List for Intuitionistic Logic Explorer - 6901-7000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremeqinftid 6901* Sufficient condition for an element to be equal to the infimum. (Contributed by Jim Kingdon, 16-Dec-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑𝐶𝐴)    &   ((𝜑𝑦𝐵) → ¬ 𝑦𝑅𝐶)    &   ((𝜑 ∧ (𝑦𝐴𝐶𝑅𝑦)) → ∃𝑧𝐵 𝑧𝑅𝑦)       (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)

Theoreminfvalti 6902* Alternate expression for the infimum. (Contributed by Jim Kingdon, 17-Dec-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))       (𝜑 → inf(𝐵, 𝐴, 𝑅) = (𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦))))

Theoreminfclti 6903* An infimum belongs to its base class (closure law). See also inflbti 6904 and infglbti 6905. (Contributed by Jim Kingdon, 17-Dec-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))       (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐴)

Theoreminflbti 6904* An infimum is a lower bound. See also infclti 6903 and infglbti 6905. (Contributed by Jim Kingdon, 18-Dec-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))       (𝜑 → (𝐶𝐵 → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅)))

Theoreminfglbti 6905* An infimum is the greatest lower bound. See also infclti 6903 and inflbti 6904. (Contributed by Jim Kingdon, 18-Dec-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))       (𝜑 → ((𝐶𝐴 ∧ inf(𝐵, 𝐴, 𝑅)𝑅𝐶) → ∃𝑧𝐵 𝑧𝑅𝐶))

Theoreminfnlbti 6906* A lower bound is not greater than the infimum. (Contributed by Jim Kingdon, 18-Dec-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))       (𝜑 → ((𝐶𝐴 ∧ ∀𝑧𝐵 ¬ 𝑧𝑅𝐶) → ¬ inf(𝐵, 𝐴, 𝑅)𝑅𝐶))

Theoreminfminti 6907* 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 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑𝐶𝐴)    &   (𝜑𝐶𝐵)    &   ((𝜑𝑦𝐵) → ¬ 𝑦𝑅𝐶)       (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)

Theoreminfmoti 6908* 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 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))       (𝜑 → ∃*𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))

Theoreminfeuti 6909* An infimum is unique. (Contributed by Jim Kingdon, 19-Dec-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))       (𝜑 → ∃!𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))

Theoreminfsnti 6910* The infimum of a singleton. (Contributed by Jim Kingdon, 19-Dec-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑𝐵𝐴)       (𝜑 → inf({𝐵}, 𝐴, 𝑅) = 𝐵)

Theoreminf00 6911 The infimum regarding an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
inf(𝐵, ∅, 𝑅) = ∅

Theoreminfisoti 6912* Image of an infimum under an isomorphism. (Contributed by Jim Kingdon, 19-Dec-2021.)
(𝜑𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   (𝜑𝐶𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐶 𝑧𝑅𝑦)))    &   ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))       (𝜑 → inf((𝐹𝐶), 𝐵, 𝑆) = (𝐹‘inf(𝐶, 𝐴, 𝑅)))

2.6.34  Ordinal isomorphism

Theoremordiso2 6913 Generalize ordiso 6914 to proper classes. (Contributed by Mario Carneiro, 24-Jun-2015.)
((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → 𝐴 = 𝐵)

Theoremordiso 6914* 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)))

2.6.35  Disjoint union

2.6.35.1  Disjoint union

Syntaxcdju 6915 Extend class notation to include disjoint union of two classes.
class (𝐴𝐵)

Definitiondf-dju 6916 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.)
(𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))

Theoremdjueq12 6917 Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))

Theoremdjueq1 6918 Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))

Theoremdjueq2 6919 Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
(𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))

Theoremnfdju 6920 Bound-variable hypothesis builder for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
𝑥𝐴    &   𝑥𝐵       𝑥(𝐴𝐵)

Theoremdjuex 6921 The disjoint union of sets is a set. See also the more precise djuss 6948. (Contributed by AV, 28-Jun-2022.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)

Theoremdjuexb 6922 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) ↔ (𝐴𝐵) ∈ V)

2.6.35.2  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).

Syntaxcinl 6923 Extend class notation to include left injection of a disjoint union.
class inl

Syntaxcinr 6924 Extend class notation to include right injection of a disjoint union.
class inr

Definitiondf-inl 6925 Left injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.)
inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)

Definitiondf-inr 6926 Right injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.)
inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)

Theoremdjulclr 6927 Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) (Revised by BJ, 6-Jul-2022.)
(𝐶𝐴 → ((inl ↾ 𝐴)‘𝐶) ∈ (𝐴𝐵))

Theoremdjurclr 6928 Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) (Revised by BJ, 6-Jul-2022.)
(𝐶𝐵 → ((inr ↾ 𝐵)‘𝐶) ∈ (𝐴𝐵))

Theoremdjulcl 6929 Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)
(𝐶𝐴 → (inl‘𝐶) ∈ (𝐴𝐵))

Theoremdjurcl 6930 Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)
(𝐶𝐵 → (inr‘𝐶) ∈ (𝐴𝐵))

Theoremdjuf1olem 6931* Lemma for djulf1o 6936 and djurf1o 6937. (Contributed by BJ and Jim Kingdon, 4-Jul-2022.)
𝑋 ∈ V    &   𝐹 = (𝑥𝐴 ↦ ⟨𝑋, 𝑥⟩)       𝐹:𝐴1-1-onto→({𝑋} × 𝐴)

Theoremdjuf1olemr 6932* Lemma for djulf1or 6934 and djurf1or 6935. For a version of this lemma with 𝐹 defined on 𝐴 and no restriction in the conclusion, see djuf1olem 6931. (Contributed by BJ and Jim Kingdon, 4-Jul-2022.)
𝑋 ∈ V    &   𝐹 = (𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩)       (𝐹𝐴):𝐴1-1-onto→({𝑋} × 𝐴)

Theoremdjulclb 6933 Left biconditional closure of disjoint union. (Contributed by Jim Kingdon, 2-Jul-2022.)
(𝐶𝑉 → (𝐶𝐴 ↔ (inl‘𝐶) ∈ (𝐴𝐵)))

Theoremdjulf1or 6934 The left injection function on all sets is one to one and onto. (Contributed by BJ and Jim Kingdon, 22-Jun-2022.)
(inl ↾ 𝐴):𝐴1-1-onto→({∅} × 𝐴)

Theoremdjurf1or 6935 The right injection function on all sets is one to one and onto. (Contributed by BJ and Jim Kingdon, 22-Jun-2022.)
(inr ↾ 𝐴):𝐴1-1-onto→({1o} × 𝐴)

Theoremdjulf1o 6936 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)

Theoremdjurf1o 6937 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)

Theoreminresflem 6938* Lemma for inlresf1 6939 and inrresf1 6940. (Contributed by BJ, 4-Jul-2022.)
𝐹:𝐴1-1-onto→({𝑋} × 𝐴)    &   (𝑥𝐴 → (𝐹𝑥) ∈ 𝐵)       𝐹:𝐴1-1𝐵

Theoreminlresf1 6939 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):𝐴1-1→(𝐴𝐵)

Theoreminrresf1 6940 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):𝐵1-1→(𝐴𝐵)

Theoremdjuinr 6941 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 6971 and djufun 6982) 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 6963). (Contributed by BJ and Jim Kingdon, 21-Jun-2022.)
(ran (inl ↾ 𝐴) ∩ ran (inr ↾ 𝐵)) = ∅

Theoremdjuin 6942 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)) = ∅

Theoreminl11 6943 Left injection is one-to-one. (Contributed by Jim Kingdon, 12-Jul-2023.)
((𝐴𝑉𝐵𝑊) → ((inl‘𝐴) = (inl‘𝐵) ↔ 𝐴 = 𝐵))

Theoremdjuunr 6944 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)) = (𝐴𝐵)

Theoremdjuun 6945 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)) = (𝐴𝐵)

Theoremeldju 6946* Element of a disjoint union. (Contributed by BJ and Jim Kingdon, 23-Jun-2022.)
(𝐶 ∈ (𝐴𝐵) ↔ (∃𝑥𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥)))

Theoremdjur 6947* 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 ∈ (𝐴𝐵) ↔ (∃𝑥𝐴 𝐶 = (inl‘𝑥) ∨ ∃𝑥𝐵 𝐶 = (inr‘𝑥)))

2.6.35.3  Universal property of the disjoint union

Theoremdjuss 6948 A disjoint union is a subset of a Cartesian product. (Contributed by AV, 25-Jun-2022.)
(𝐴𝐵) ⊆ ({∅, 1o} × (𝐴𝐵))

Theoremeldju1st 6949 The first component of an element of a disjoint union is either or 1o. (Contributed by AV, 26-Jun-2022.)
(𝑋 ∈ (𝐴𝐵) → ((1st𝑋) = ∅ ∨ (1st𝑋) = 1o))

Theoremeldju2ndl 6950 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 ∈ (𝐴𝐵) ∧ (1st𝑋) = ∅) → (2nd𝑋) ∈ 𝐴)

Theoremeldju2ndr 6951 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 ∈ (𝐴𝐵) ∧ (1st𝑋) ≠ ∅) → (2nd𝑋) ∈ 𝐵)

Theorem1stinl 6952 The first component of the value of a left injection is the empty set. (Contributed by AV, 27-Jun-2022.)
(𝑋𝑉 → (1st ‘(inl‘𝑋)) = ∅)

Theorem2ndinl 6953 The second component of the value of a left injection is its argument. (Contributed by AV, 27-Jun-2022.)
(𝑋𝑉 → (2nd ‘(inl‘𝑋)) = 𝑋)

Theorem1stinr 6954 The first component of the value of a right injection is 1o. (Contributed by AV, 27-Jun-2022.)
(𝑋𝑉 → (1st ‘(inr‘𝑋)) = 1o)

Theorem2ndinr 6955 The second component of the value of a right injection is its argument. (Contributed by AV, 27-Jun-2022.)
(𝑋𝑉 → (2nd ‘(inr‘𝑋)) = 𝑋)

Theoremdjune 6956 Left and right injection never produce equal values. (Contributed by Jim Kingdon, 2-Jul-2022.)
((𝐴𝑉𝐵𝑊) → (inl‘𝐴) ≠ (inr‘𝐵))

Theoremupdjudhf 6957* 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.)
(𝜑𝐹:𝐴𝐶)    &   (𝜑𝐺:𝐵𝐶)    &   𝐻 = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))       (𝜑𝐻:(𝐴𝐵)⟶𝐶)

Theoremupdjudhcoinlf 6958* 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.)
(𝜑𝐹:𝐴𝐶)    &   (𝜑𝐺:𝐵𝐶)    &   𝐻 = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))       (𝜑 → (𝐻 ∘ (inl ↾ 𝐴)) = 𝐹)

Theoremupdjudhcoinrg 6959* 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.)
(𝜑𝐹:𝐴𝐶)    &   (𝜑𝐺:𝐵𝐶)    &   𝐻 = (𝑥 ∈ (𝐴𝐵) ↦ if((1st𝑥) = ∅, (𝐹‘(2nd𝑥)), (𝐺‘(2nd𝑥))))       (𝜑 → (𝐻 ∘ (inr ↾ 𝐵)) = 𝐺)

Theoremupdjud 6960* Universal property of the disjoint union. (Proposed by BJ, 25-Jun-2022.) (Contributed by AV, 28-Jun-2022.)
(𝜑𝐹:𝐴𝐶)    &   (𝜑𝐺:𝐵𝐶)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → ∃!(:(𝐴𝐵)⟶𝐶 ∧ ( ∘ (inl ↾ 𝐴)) = 𝐹 ∧ ( ∘ (inr ↾ 𝐵)) = 𝐺))

Syntaxcdjucase 6961 Syntax for the "case" construction.
class case(𝑅, 𝑆)

Definitiondf-case 6962 The "case" construction: if 𝐹:𝐴𝑋 and 𝐺:𝐵𝑋 are functions, then case(𝐹, 𝐺):(𝐴𝐵)⟶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 6960. 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) = ((𝑅inl) ∪ (𝑆inr))

Theoremcasefun 6963 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))

Theoremcasedm 6964 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)

Theoremcaserel 6965 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))

Theoremcasef 6966 The "case" construction of two functions is a function on the disjoint union of their domains. (Contributed by BJ, 10-Jul-2022.)
(𝜑𝐹:𝐴𝑋)    &   (𝜑𝐺:𝐵𝑋)       (𝜑 → case(𝐹, 𝐺):(𝐴𝐵)⟶𝑋)

Theoremcaseinj 6967 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))

Theoremcasef1 6968 The "case" construction of two injective functions with disjoint ranges is an injective function. (Contributed by BJ, 10-Jul-2022.)
(𝜑𝐹:𝐴1-1𝑋)    &   (𝜑𝐺:𝐵1-1𝑋)    &   (𝜑 → (ran 𝐹 ∩ ran 𝐺) = ∅)       (𝜑 → case(𝐹, 𝐺):(𝐴𝐵)–1-1𝑋)

Theoremcaseinl 6969 Applying the "case" construction to a left injection. (Contributed by Jim Kingdon, 15-Mar-2023.)
(𝜑𝐹 Fn 𝐵)    &   (𝜑 → Fun 𝐺)    &   (𝜑𝐴𝐵)       (𝜑 → (case(𝐹, 𝐺)‘(inl‘𝐴)) = (𝐹𝐴))

Theoremcaseinr 6970 Applying the "case" construction to a right injection. (Contributed by Jim Kingdon, 12-Jul-2023.)
(𝜑 → Fun 𝐹)    &   (𝜑𝐺 Fn 𝐵)    &   (𝜑𝐴𝐵)       (𝜑 → (case(𝐹, 𝐺)‘(inr‘𝐴)) = (𝐺𝐴))

2.6.35.4  Dominance and equinumerosity properties of disjoint union

Theoremdjudom 6971 Dominance law for disjoint union. (Contributed by Jim Kingdon, 25-Jul-2022.)
((𝐴𝐵𝐶𝐷) → (𝐴𝐶) ≼ (𝐵𝐷))

Theoremomp1eomlem 6972* Lemma for omp1eom 6973. (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)

Theoremomp1eom 6973 Adding one to ω. (Contributed by Jim Kingdon, 10-Jul-2023.)
(ω ⊔ 1o) ≈ ω

Theoremendjusym 6974 Reversing right and left operands of a disjoint union produces an equinumerous result. (Contributed by Jim Kingdon, 10-Jul-2023.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ≈ (𝐵𝐴))

Theoremeninl 6975 Equinumerosity of a set and its image under left injection. (Contributed by Jim Kingdon, 30-Jul-2023.)
(𝐴𝑉 → (inl “ 𝐴) ≈ 𝐴)

Theoremeninr 6976 Equinumerosity of a set and its image under right injection. (Contributed by Jim Kingdon, 30-Jul-2023.)
(𝐴𝑉 → (inr “ 𝐴) ≈ 𝐴)

Theoremdifinfsnlem 6977* Lemma for difinfsn 6978. The case where we need to swap 𝐵 and (inr‘∅) in building the mapping 𝐺. (Contributed by Jim Kingdon, 9-Aug-2023.)
(𝜑 → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)    &   (𝜑𝐵𝐴)    &   (𝜑𝐹:(ω ⊔ 1o)–1-1𝐴)    &   (𝜑 → (𝐹‘(inr‘∅)) ≠ 𝐵)    &   𝐺 = (𝑛 ∈ ω ↦ if((𝐹‘(inl‘𝑛)) = 𝐵, (𝐹‘(inr‘∅)), (𝐹‘(inl‘𝑛))))       (𝜑𝐺:ω–1-1→(𝐴 ∖ {𝐵}))

Theoremdifinfsn 6978* An infinite set minus one element is infinite. We require that the set has decidable equality. (Contributed by Jim Kingdon, 8-Aug-2023.)
((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴𝐵𝐴) → ω ≼ (𝐴 ∖ {𝐵}))

Theoremdifinfinf 6979* An infinite set minus a finite subset is infinite. We require that the set has decidable equality. (Contributed by Jim Kingdon, 8-Aug-2023.)
(((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ω ≼ 𝐴) ∧ (𝐵𝐴𝐵 ∈ Fin)) → ω ≼ (𝐴𝐵))

2.6.35.5  Older definition temporarily kept for comparison, to be deleted

Syntaxcdjud 6980 Syntax for the domain-disjoint-union of two relations.
class (𝑅d 𝑆)

Definitiondf-djud 6981 The "domain-disjoint-union" of two relations: if 𝑅 ⊆ (𝐴 × 𝑋) and 𝑆 ⊆ (𝐵 × 𝑋) are two binary relations, then (𝑅d 𝑆) is the binary relation from (𝐴𝐵) to 𝑋 having the universal property of disjoint unions (see updjud 6960 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 6962, but they are explicitly written for clarity. (Contributed by MC and BJ, 10-Jul-2022.)

(𝑅d 𝑆) = ((𝑅(inl ↾ dom 𝑅)) ∪ (𝑆(inr ↾ dom 𝑆)))

Theoremdjufun 6982 The "domain-disjoint-union" of two functions is a function. (Contributed by BJ, 10-Jul-2022.)
(𝜑 → Fun 𝐹)    &   (𝜑 → Fun 𝐺)       (𝜑 → Fun (𝐹d 𝐺))

Theoremdjudm 6983 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 (𝐹d 𝐺) = (dom 𝐹 ⊔ dom 𝐺)

Theoremdjuinj 6984 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 (𝑅d 𝑆))

2.6.35.6  Countable sets

Theorem0ct 6985 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)

Theoremctmlemr 6986* Lemma for ctm 6987. One of the directions of the biconditional. (Contributed by Jim Kingdon, 16-Mar-2023.)
(∃𝑥 𝑥𝐴 → (∃𝑓 𝑓:ω–onto𝐴 → ∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o)))

Theoremctm 6987* Two equivalent definitions of countable for an inhabited set. Remark of [BauerSwan], p. 14:3. (Contributed by Jim Kingdon, 13-Mar-2023.)
(∃𝑥 𝑥𝐴 → (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) ↔ ∃𝑓 𝑓:ω–onto𝐴))

Theoremctssdclemn0 6988* Lemma for ctssdc 6991. The ¬ ∅ ∈ 𝑆 case. (Contributed by Jim Kingdon, 16-Aug-2023.)
(𝜑𝑆 ⊆ ω)    &   (𝜑 → ∀𝑛 ∈ ω DECID 𝑛𝑆)    &   (𝜑𝐹:𝑆onto𝐴)    &   (𝜑 → ¬ ∅ ∈ 𝑆)       (𝜑 → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))

Theoremctssdccl 6989* A mapping from a decidable subset of the natural numbers onto a countable set. This is similar to one direction of ctssdc 6991 but expressed in terms of classes rather than . (Contributed by Jim Kingdon, 30-Oct-2023.)
(𝜑𝐹:ω–onto→(𝐴 ⊔ 1o))    &   𝑆 = {𝑥 ∈ ω ∣ (𝐹𝑥) ∈ (inl “ 𝐴)}    &   𝐺 = (inl ∘ 𝐹)       (𝜑 → (𝑆 ⊆ ω ∧ 𝐺:𝑆onto𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛𝑆))

Theoremctssdclemr 6990* Lemma for ctssdc 6991. 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:𝑠onto𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛𝑠))

Theoremctssdc 6991* 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 7017. (Contributed by Jim Kingdon, 15-Aug-2023.)
(∃𝑠(𝑠 ⊆ ω ∧ ∃𝑓 𝑓:𝑠onto𝐴 ∧ ∀𝑛 ∈ ω DECID 𝑛𝑠) ↔ ∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o))

Theoremenumctlemm 6992* Lemma for enumct 6993. The case where 𝑁 is greater than zero. (Contributed by Jim Kingdon, 13-Mar-2023.)
(𝜑𝐹:𝑁onto𝐴)    &   (𝜑𝑁 ∈ ω)    &   (𝜑 → ∅ ∈ 𝑁)    &   𝐺 = (𝑘 ∈ ω ↦ if(𝑘𝑁, (𝐹𝑘), (𝐹‘∅)))       (𝜑𝐺:ω–onto𝐴)

Theoremenumct 6993* 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:𝑛onto𝐴 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:𝑛onto𝐴 → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))

Theoremfinct 6994* A finite set is countable. (Contributed by Jim Kingdon, 17-Mar-2023.)
(𝐴 ∈ Fin → ∃𝑔 𝑔:ω–onto→(𝐴 ⊔ 1o))

Theoremomct 6995 ω is countable. (Contributed by Jim Kingdon, 23-Dec-2023.)
𝑓 𝑓:ω–onto→(ω ⊔ 1o)

Theoremctfoex 6996* A countable class is a set. (Contributed by Jim Kingdon, 25-Dec-2023.)
(∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) → 𝐴 ∈ V)

2.6.36  Omniscient sets

Syntaxcomni 6997 Extend class definition to include the class of omniscient sets.
class Omni

Syntaxxnninf 6998 Set of nonincreasing sequences in 2o𝑚 ω.
class

Definitiondf-omni 6999* 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 → (∃𝑥𝑦 (𝑓𝑥) = ∅ ∨ ∀𝑥𝑦 (𝑓𝑥) = 1o))}

Definitiondf-nninf 7000* Define the set of nonincreasing sequences in 2o𝑚 ω. Definition in Section 3.1 of [Pierik], p. 15. If we assumed excluded middle, this would be essentially the same as 0* as defined at df-xnn0 9034 but in its absence the relationship between the two is more complicated. This definition would function much the same whether we used ω or 0, but the former allows us to take advantage of 2o = {∅, 1o} (df2o3 6320) so we adopt it. (Contributed by Jim Kingdon, 14-Jul-2022.)
= {𝑓 ∈ (2o𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓𝑖)}

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