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Theorem List for Intuitionistic Logic Explorer - 7001-7100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelfi2 7001* The empty intersection need not be considered in the set of finite intersections. (Contributed by Mario Carneiro, 21-Mar-2015.)
(𝐵𝑉 → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = 𝑥))
 
Theoremelfir 7002 Sufficient condition for an element of (fi‘𝐵). (Contributed by Mario Carneiro, 24-Nov-2013.)
((𝐵𝑉 ∧ (𝐴𝐵𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐴 ∈ (fi‘𝐵))
 
Theoremssfii 7003 Any element of a set 𝐴 is the intersection of a finite subset of 𝐴. (Contributed by FL, 27-Apr-2008.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
(𝐴𝑉𝐴 ⊆ (fi‘𝐴))
 
Theoremfi0 7004 The set of finite intersections of the empty set. (Contributed by Mario Carneiro, 30-Aug-2015.)
(fi‘∅) = ∅
 
Theoremfieq0 7005 A set is empty iff the class of all the finite intersections of that set is empty. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
(𝐴𝑉 → (𝐴 = ∅ ↔ (fi‘𝐴) = ∅))
 
Theoremfiss 7006 Subset relationship for function fi. (Contributed by Jeff Hankins, 7-Oct-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
((𝐵𝑉𝐴𝐵) → (fi‘𝐴) ⊆ (fi‘𝐵))
 
Theoremfiuni 7007 The union of the finite intersections of a set is simply the union of the set itself. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
(𝐴𝑉 𝐴 = (fi‘𝐴))
 
Theoremfipwssg 7008 If a set is a family of subsets of some base set, then so is its finite intersection. (Contributed by Stefan O'Rear, 2-Aug-2015.)
((𝐴𝑉𝐴 ⊆ 𝒫 𝑋) → (fi‘𝐴) ⊆ 𝒫 𝑋)
 
Theoremfifo 7009* Describe a surjection from nonempty finite sets to finite intersections. (Contributed by Mario Carneiro, 18-May-2015.)
𝐹 = (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ 𝑦)       (𝐴𝑉𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴))
 
Theoremdcfi 7010* Decidability of a family of propositions indexed by a finite set. (Contributed by Jim Kingdon, 30-Sep-2024.)
((𝐴 ∈ Fin ∧ ∀𝑥𝐴 DECID 𝜑) → DECID𝑥𝐴 𝜑)
 
2.6.34  Supremum and infimum
 
Syntaxcsup 7011 Extend class notation to include supremum of class 𝐴. Here 𝑅 is ordinarily a relation that strictly orders class 𝐵. For example, 𝑅 could be 'less than' and 𝐵 could be the set of real numbers.
class sup(𝐴, 𝐵, 𝑅)
 
Syntaxcinf 7012 Extend class notation to include infimum of class 𝐴. Here 𝑅 is ordinarily a relation that strictly orders class 𝐵. For example, 𝑅 could be 'less than' and 𝐵 could be the set of real numbers.
class inf(𝐴, 𝐵, 𝑅)
 
Definitiondf-sup 7013* Define the supremum of class 𝐴. It is meaningful when 𝑅 is a relation that strictly orders 𝐵 and when the supremum exists. (Contributed by NM, 22-May-1999.)
sup(𝐴, 𝐵, 𝑅) = {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))}
 
Definitiondf-inf 7014 Define the infimum of class 𝐴. It is meaningful when 𝑅 is a relation that strictly orders 𝐵 and when the infimum exists. For example, 𝑅 could be 'less than', 𝐵 could be the set of real numbers, and 𝐴 could be the set of all positive reals; in this case the infimum is 0. The infimum is defined as the supremum using the converse ordering relation. In the given example, 0 is the supremum of all reals (greatest real number) for which all positive reals are greater. (Contributed by AV, 2-Sep-2020.)
inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑅)
 
Theoremsupeq1 7015 Equality theorem for supremum. (Contributed by NM, 22-May-1999.)
(𝐵 = 𝐶 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅))
 
Theoremsupeq1d 7016 Equality deduction for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝐵 = 𝐶)       (𝜑 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅))
 
Theoremsupeq1i 7017 Equality inference for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
𝐵 = 𝐶       sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅)
 
Theoremsupeq2 7018 Equality theorem for supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐵 = 𝐶 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐶, 𝑅))
 
Theoremsupeq3 7019 Equality theorem for supremum. (Contributed by Scott Fenton, 13-Jun-2018.)
(𝑅 = 𝑆 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑆))
 
Theoremsupeq123d 7020 Equality deduction for supremum. (Contributed by Stefan O'Rear, 20-Jan-2015.)
(𝜑𝐴 = 𝐷)    &   (𝜑𝐵 = 𝐸)    &   (𝜑𝐶 = 𝐹)       (𝜑 → sup(𝐴, 𝐵, 𝐶) = sup(𝐷, 𝐸, 𝐹))
 
Theoremnfsup 7021 Hypothesis builder for supremum. (Contributed by Mario Carneiro, 20-Mar-2014.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝑅       𝑥sup(𝐴, 𝐵, 𝑅)
 
Theoremsupmoti 7022* Any class 𝐵 has at most one supremum in 𝐴 (where 𝑅 is interpreted as 'less than'). The hypothesis is satisfied by real numbers (see lttri3 8067) or other orders which correspond to tight apartnesses. (Contributed by Jim Kingdon, 23-Nov-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))       (𝜑 → ∃*𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
 
Theoremsupeuti 7023* A supremum is unique. Similar to Theorem I.26 of [Apostol] p. 24 (but for suprema in general). (Contributed by Jim Kingdon, 23-Nov-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))       (𝜑 → ∃!𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
 
Theoremsupval2ti 7024* Alternate expression for the supremum. (Contributed by Jim Kingdon, 23-Nov-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))       (𝜑 → sup(𝐵, 𝐴, 𝑅) = (𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))))
 
Theoremeqsupti 7025* Sufficient condition for an element to be equal to the supremum. (Contributed by Jim Kingdon, 23-Nov-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))       (𝜑 → ((𝐶𝐴 ∧ ∀𝑦𝐵 ¬ 𝐶𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝐶 → ∃𝑧𝐵 𝑦𝑅𝑧)) → sup(𝐵, 𝐴, 𝑅) = 𝐶))
 
Theoremeqsuptid 7026* Sufficient condition for an element to be equal to the supremum. (Contributed by Jim Kingdon, 24-Nov-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑𝐶𝐴)    &   ((𝜑𝑦𝐵) → ¬ 𝐶𝑅𝑦)    &   ((𝜑 ∧ (𝑦𝐴𝑦𝑅𝐶)) → ∃𝑧𝐵 𝑦𝑅𝑧)       (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶)
 
Theoremsupclti 7027* A supremum belongs to its base class (closure law). See also supubti 7028 and suplubti 7029. (Contributed by Jim Kingdon, 24-Nov-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))       (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
 
Theoremsupubti 7028* A supremum is an upper bound. See also supclti 7027 and suplubti 7029.

This proof demonstrates how to expand an iota-based definition (df-iota 5196) using riotacl2 5865.

(Contributed by Jim Kingdon, 24-Nov-2021.)

((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))       (𝜑 → (𝐶𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
 
Theoremsuplubti 7029* A supremum is the least upper bound. See also supclti 7027 and supubti 7028. (Contributed by Jim Kingdon, 24-Nov-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))       (𝜑 → ((𝐶𝐴𝐶𝑅sup(𝐵, 𝐴, 𝑅)) → ∃𝑧𝐵 𝐶𝑅𝑧))
 
Theoremsuplub2ti 7030* Bidirectional form of suplubti 7029. (Contributed by Jim Kingdon, 17-Jan-2022.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))    &   (𝜑𝑅 Or 𝐴)    &   (𝜑𝐵𝐴)       ((𝜑𝐶𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ ∃𝑧𝐵 𝐶𝑅𝑧))
 
Theoremsupelti 7031* Supremum membership in a set. (Contributed by Jim Kingdon, 16-Jan-2022.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑 → ∃𝑥𝐶 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))    &   (𝜑𝐶𝐴)       (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐶)
 
Theoremsup00 7032 The supremum under an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
sup(𝐵, ∅, 𝑅) = ∅
 
Theoremsupmaxti 7033* The greatest element of a set is its supremum. Note that the converse is not true; the supremum might not be an element of the set considered. (Contributed by Jim Kingdon, 24-Nov-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑𝐶𝐴)    &   (𝜑𝐶𝐵)    &   ((𝜑𝑦𝐵) → ¬ 𝐶𝑅𝑦)       (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶)
 
Theoremsupsnti 7034* The supremum of a singleton. (Contributed by Jim Kingdon, 26-Nov-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))    &   (𝜑𝐵𝐴)       (𝜑 → sup({𝐵}, 𝐴, 𝑅) = 𝐵)
 
Theoremisotilem 7035* Lemma for isoti 7036. (Contributed by Jim Kingdon, 26-Nov-2021.)
(𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑥𝐵𝑦𝐵 (𝑥 = 𝑦 ↔ (¬ 𝑥𝑆𝑦 ∧ ¬ 𝑦𝑆𝑥)) → ∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢))))
 
Theoremisoti 7036* An isomorphism preserves tightness. (Contributed by Jim Kingdon, 26-Nov-2021.)
(𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)) ↔ ∀𝑢𝐵𝑣𝐵 (𝑢 = 𝑣 ↔ (¬ 𝑢𝑆𝑣 ∧ ¬ 𝑣𝑆𝑢))))
 
Theoremsupisolem 7037* Lemma for supisoti 7039. (Contributed by Mario Carneiro, 24-Dec-2016.)
(𝜑𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   (𝜑𝐶𝐴)       ((𝜑𝐷𝐴) → ((∀𝑦𝐶 ¬ 𝐷𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝐷 → ∃𝑧𝐶 𝑦𝑅𝑧)) ↔ (∀𝑤 ∈ (𝐹𝐶) ¬ (𝐹𝐷)𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆(𝐹𝐷) → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣))))
 
Theoremsupisoex 7038* Lemma for supisoti 7039. (Contributed by Mario Carneiro, 24-Dec-2016.)
(𝜑𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   (𝜑𝐶𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐶 𝑦𝑅𝑧)))       (𝜑 → ∃𝑢𝐵 (∀𝑤 ∈ (𝐹𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹𝐶)𝑤𝑆𝑣)))
 
Theoremsupisoti 7039* Image of a supremum under an isomorphism. (Contributed by Jim Kingdon, 26-Nov-2021.)
(𝜑𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   (𝜑𝐶𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐶 𝑦𝑅𝑧)))    &   ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))       (𝜑 → sup((𝐹𝐶), 𝐵, 𝑆) = (𝐹‘sup(𝐶, 𝐴, 𝑅)))
 
Theoreminfeq1 7040 Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
(𝐵 = 𝐶 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))
 
Theoreminfeq1d 7041 Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
(𝜑𝐵 = 𝐶)       (𝜑 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))
 
Theoreminfeq1i 7042 Equality inference for infimum. (Contributed by AV, 2-Sep-2020.)
𝐵 = 𝐶       inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅)
 
Theoreminfeq2 7043 Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
(𝐵 = 𝐶 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐶, 𝑅))
 
Theoreminfeq3 7044 Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
(𝑅 = 𝑆 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐵, 𝑆))
 
Theoreminfeq123d 7045 Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
(𝜑𝐴 = 𝐷)    &   (𝜑𝐵 = 𝐸)    &   (𝜑𝐶 = 𝐹)       (𝜑 → inf(𝐴, 𝐵, 𝐶) = inf(𝐷, 𝐸, 𝐹))
 
Theoremnfinf 7046 Hypothesis builder for infimum. (Contributed by AV, 2-Sep-2020.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝑅       𝑥inf(𝐴, 𝐵, 𝑅)
 
Theoremcnvinfex 7047* Two ways of expressing existence of an infimum (one in terms of converse). (Contributed by Jim Kingdon, 17-Dec-2021.)
(𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))       (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
 
Theoremcnvti 7048* If a relation satisfies a condition corresponding to tightness of an apartness generated by an order, so does its converse. (Contributed by Jim Kingdon, 17-Dec-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))       ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
 
Theoremeqinfti 7049* Sufficient condition for an element to be equal to the infimum. (Contributed by Jim Kingdon, 16-Dec-2021.)
((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))       (𝜑 → ((𝐶𝐴 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝐶 ∧ ∀𝑦𝐴 (𝐶𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)) → inf(𝐵, 𝐴, 𝑅) = 𝐶))
 
Theoremeqinftid 7050* 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 7051* 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 7052* An infimum belongs to its base class (closure law). See also inflbti 7053 and infglbti 7054. (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 7053* An infimum is a lower bound. See also infclti 7052 and infglbti 7054. (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 7054* An infimum is the greatest lower bound. See also infclti 7052 and inflbti 7053. (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 7055* 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 7056* 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 7057* 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 7058* 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 7059* 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 7060 The infimum regarding an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
inf(𝐵, ∅, 𝑅) = ∅
 
Theoreminfisoti 7061* 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)))
 
Theoremsupex2g 7062 Existence of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐴𝐶 → sup(𝐵, 𝐴, 𝑅) ∈ V)
 
Theoreminfex2g 7063 Existence of infimum. (Contributed by Jim Kingdon, 1-Oct-2024.)
(𝐴𝐶 → inf(𝐵, 𝐴, 𝑅) ∈ V)
 
2.6.35  Ordinal isomorphism
 
Theoremordiso2 7064 Generalize ordiso 7065 to proper classes. (Contributed by Mario Carneiro, 24-Jun-2015.)
((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → 𝐴 = 𝐵)
 
Theoremordiso 7065* 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
 
Syntaxcdju 7066 Extend class notation to include disjoint union of two classes.
class (𝐴𝐵)
 
Definitiondf-dju 7067 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 7068 Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))
 
Theoremdjueq1 7069 Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
 
Theoremdjueq2 7070 Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
(𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
 
Theoremnfdju 7071 Bound-variable hypothesis builder for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
𝑥𝐴    &   𝑥𝐵       𝑥(𝐴𝐵)
 
Theoremdjuex 7072 The disjoint union of sets is a set. See also the more precise djuss 7099. (Contributed by AV, 28-Jun-2022.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
 
Theoremdjuexb 7073 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)
 
2.6.36.2  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).

 
Syntaxcinl 7074 Extend class notation to include left injection of a disjoint union.
class inl
 
Syntaxcinr 7075 Extend class notation to include right injection of a disjoint union.
class inr
 
Definitiondf-inl 7076 Left injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.)
inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
 
Definitiondf-inr 7077 Right injection of a disjoint union. (Contributed by Mario Carneiro, 21-Jun-2022.)
inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
 
Theoremdjulclr 7078 Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) (Revised by BJ, 6-Jul-2022.)
(𝐶𝐴 → ((inl ↾ 𝐴)‘𝐶) ∈ (𝐴𝐵))
 
Theoremdjurclr 7079 Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) (Revised by BJ, 6-Jul-2022.)
(𝐶𝐵 → ((inr ↾ 𝐵)‘𝐶) ∈ (𝐴𝐵))
 
Theoremdjulcl 7080 Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)
(𝐶𝐴 → (inl‘𝐶) ∈ (𝐴𝐵))
 
Theoremdjurcl 7081 Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.)
(𝐶𝐵 → (inr‘𝐶) ∈ (𝐴𝐵))
 
Theoremdjuf1olem 7082* Lemma for djulf1o 7087 and djurf1o 7088. (Contributed by BJ and Jim Kingdon, 4-Jul-2022.)
𝑋 ∈ V    &   𝐹 = (𝑥𝐴 ↦ ⟨𝑋, 𝑥⟩)       𝐹:𝐴1-1-onto→({𝑋} × 𝐴)
 
Theoremdjuf1olemr 7083* Lemma for djulf1or 7085 and djurf1or 7086. For a version of this lemma with 𝐹 defined on 𝐴 and no restriction in the conclusion, see djuf1olem 7082. (Contributed by BJ and Jim Kingdon, 4-Jul-2022.)
𝑋 ∈ V    &   𝐹 = (𝑥 ∈ V ↦ ⟨𝑋, 𝑥⟩)       (𝐹𝐴):𝐴1-1-onto→({𝑋} × 𝐴)
 
Theoremdjulclb 7084 Left biconditional closure of disjoint union. (Contributed by Jim Kingdon, 2-Jul-2022.)
(𝐶𝑉 → (𝐶𝐴 ↔ (inl‘𝐶) ∈ (𝐴𝐵)))
 
Theoremdjulf1or 7085 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 7086 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 7087 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 7088 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 7089* Lemma for inlresf1 7090 and inrresf1 7091. (Contributed by BJ, 4-Jul-2022.)
𝐹:𝐴1-1-onto→({𝑋} × 𝐴)    &   (𝑥𝐴 → (𝐹𝑥) ∈ 𝐵)       𝐹:𝐴1-1𝐵
 
Theoreminlresf1 7090 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 7091 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 7092 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 7122 and djufun 7133) 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 7114). (Contributed by BJ and Jim Kingdon, 21-Jun-2022.)
(ran (inl ↾ 𝐴) ∩ ran (inr ↾ 𝐵)) = ∅
 
Theoremdjuin 7093 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 7094 Left injection is one-to-one. (Contributed by Jim Kingdon, 12-Jul-2023.)
((𝐴𝑉𝐵𝑊) → ((inl‘𝐴) = (inl‘𝐵) ↔ 𝐴 = 𝐵))
 
Theoremdjuunr 7095 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 7096 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 7097* 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 7098* 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)))
 
2.6.36.3  Universal property of the disjoint union
 
Theoremdjuss 7099 A disjoint union is a subset of a Cartesian product. (Contributed by AV, 25-Jun-2022.)
(𝐴𝐵) ⊆ ({∅, 1o} × (𝐴𝐵))
 
Theoremeldju1st 7100 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))
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