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Theorem List for Metamath Proof Explorer - 8101-8200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmarypha2lem2 8101* Lemma for marypha2 8104. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)
𝑇 = 𝑥𝐴 ({𝑥} × (𝐹𝑥))       𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
 
Theoremmarypha2lem3 8102* Lemma for marypha2 8104. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)
𝑇 = 𝑥𝐴 ({𝑥} × (𝐹𝑥))       ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐺𝑇 ↔ ∀𝑥𝐴 (𝐺𝑥) ∈ (𝐹𝑥)))
 
Theoremmarypha2lem4 8103* Lemma for marypha2 8104. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)
𝑇 = 𝑥𝐴 ({𝑥} × (𝐹𝑥))       ((𝐹 Fn 𝐴𝑋𝐴) → (𝑇𝑋) = (𝐹𝑋))
 
Theoremmarypha2 8104* Version of marypha1 8099 using a functional family of sets instead of a relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐹:𝐴⟶Fin)    &   ((𝜑𝑑𝐴) → 𝑑 (𝐹𝑑))       (𝜑 → ∃𝑔(𝑔:𝐴1-1→V ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ (𝐹𝑥)))
 
2.4.31  Supremum and infimum
 
Syntaxcsup 8105 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 8106 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 8107* Define the supremum of class 𝐴. It is meaningful when 𝑅 is a relation that strictly orders 𝐵 and when the supremum 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 whose square is less than 2; in this case the supremum is defined as the square root of 2 per sqrtval 13684. See dfsup2 8109 for alternate definition not requiring dummy variables. (Contributed by NM, 22-May-1999.)
sup(𝐴, 𝐵, 𝑅) = {𝑥𝐵 ∣ (∀𝑦𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐵 (𝑦𝑅𝑥 → ∃𝑧𝐴 𝑦𝑅𝑧))}
 
Definitiondf-inf 8108 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)
 
Theoremdfsup2 8109 Quantifier free definition of supremum. (Contributed by Scott Fenton, 19-Feb-2013.)
sup(𝐵, 𝐴, 𝑅) = (𝐴 ∖ ((𝑅𝐵) ∪ (𝑅 “ (𝐴 ∖ (𝑅𝐵)))))
 
Theoremsupeq1 8110 Equality theorem for supremum. (Contributed by NM, 22-May-1999.)
(𝐵 = 𝐶 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅))
 
Theoremsupeq1d 8111 Equality deduction for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝐵 = 𝐶)       (𝜑 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅))
 
Theoremsupeq1i 8112 Equality inference for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
𝐵 = 𝐶       sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅)
 
Theoremsupeq2 8113 Equality theorem for supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐵 = 𝐶 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐶, 𝑅))
 
Theoremsupeq3 8114 Equality theorem for supremum. (Contributed by Scott Fenton, 13-Jun-2018.)
(𝑅 = 𝑆 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑆))
 
Theoremsupeq123d 8115 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 8116 Hypothesis builder for supremum. (Contributed by Mario Carneiro, 20-Mar-2014.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝑅       𝑥sup(𝐴, 𝐵, 𝑅)
 
Theoremsupmo 8117* Any class 𝐵 has at most one supremum in 𝐴 (where 𝑅 is interpreted as 'less than'). (Contributed by NM, 5-May-1999.) (Revised by Mario Carneiro, 24-Dec-2016.)
(𝜑𝑅 Or 𝐴)       (𝜑 → ∃*𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
 
Theoremsupexd 8118 A supremum is a set. (Contributed by NM, 22-May-1999.) (Revised by Mario Carneiro, 24-Dec-2016.)
(𝜑𝑅 Or 𝐴)       (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ V)
 
Theoremsupeu 8119* A supremum is unique. Similar to Theorem I.26 of [Apostol] p. 24 (but for suprema in general). (Contributed by NM, 12-Oct-2004.)
(𝜑𝑅 Or 𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))       (𝜑 → ∃!𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
 
Theoremsupval2 8120* Alternate expression for the supremum. (Contributed by Mario Carneiro, 24-Dec-2016.) (Revised by Thierry Arnoux, 24-Sep-2017.)
(𝜑𝑅 Or 𝐴)       (𝜑 → sup(𝐵, 𝐴, 𝑅) = (𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))))
 
Theoremeqsup 8121* Sufficient condition for an element to be equal to the supremum. (Contributed by Mario Carneiro, 21-Apr-2015.)
(𝜑𝑅 Or 𝐴)       (𝜑 → ((𝐶𝐴 ∧ ∀𝑦𝐵 ¬ 𝐶𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝐶 → ∃𝑧𝐵 𝑦𝑅𝑧)) → sup(𝐵, 𝐴, 𝑅) = 𝐶))
 
Theoremeqsupd 8122* Sufficient condition for an element to be equal to the supremum. (Contributed by Mario Carneiro, 21-Apr-2015.)
(𝜑𝑅 Or 𝐴)    &   (𝜑𝐶𝐴)    &   ((𝜑𝑦𝐵) → ¬ 𝐶𝑅𝑦)    &   ((𝜑 ∧ (𝑦𝐴𝑦𝑅𝐶)) → ∃𝑧𝐵 𝑦𝑅𝑧)       (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶)
 
Theoremsupcl 8123* A supremum belongs to its base class (closure law). See also supub 8124 and suplub 8125. (Contributed by NM, 12-Oct-2004.)
(𝜑𝑅 Or 𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))       (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
 
Theoremsupub 8124* A supremum is an upper bound. See also supcl 8123 and suplub 8125.

This proof demonstrates how to expand an iota-based definition (df-iota 5653) using riotacl2 6401.

(Contributed by NM, 12-Oct-2004.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)

(𝜑𝑅 Or 𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))       (𝜑 → (𝐶𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
 
Theoremsuplub 8125* A supremum is the least upper bound. See also supcl 8123 and supub 8124. (Contributed by NM, 13-Oct-2004.) (Revised by Mario Carneiro, 24-Dec-2016.)
(𝜑𝑅 Or 𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))       (𝜑 → ((𝐶𝐴𝐶𝑅sup(𝐵, 𝐴, 𝑅)) → ∃𝑧𝐵 𝐶𝑅𝑧))
 
Theoremsuplub2 8126* Bidirectional form of suplub 8125. (Contributed by Mario Carneiro, 6-Sep-2014.)
(𝜑𝑅 Or 𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))    &   (𝜑𝐵𝐴)       ((𝜑𝐶𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ ∃𝑧𝐵 𝐶𝑅𝑧))
 
Theoremsupnub 8127* An upper bound is not less than the supremum. (Contributed by NM, 13-Oct-2004.)
(𝜑𝑅 Or 𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))       (𝜑 → ((𝐶𝐴 ∧ ∀𝑧𝐵 ¬ 𝐶𝑅𝑧) → ¬ 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
 
Theoremsupex 8128 A supremum is a set. (Contributed by NM, 22-May-1999.)
𝑅 Or 𝐴       sup(𝐵, 𝐴, 𝑅) ∈ V
 
Theoremsup00 8129 The supremum under an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
sup(𝐵, ∅, 𝑅) = ∅
 
Theoremsup0riota 8130* The supremum of an empty set is the smallest element of the base set. (Contributed by AV, 4-Sep-2020.)
(𝑅 Or 𝐴 → sup(∅, 𝐴, 𝑅) = (𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥))
 
Theoremsup0 8131* The supremum of an empty set under a base set which has a unique smallest element is the smallest element of the base set. (Contributed by AV, 4-Sep-2020.)
((𝑅 Or 𝐴 ∧ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) → sup(∅, 𝐴, 𝑅) = 𝑋)
 
Theoremsupmax 8132* 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 Jeff Hoffman, 17-Jun-2008.) (Proof shortened by OpenAI, 30-Mar-2020.)
(𝜑𝑅 Or 𝐴)    &   (𝜑𝐶𝐴)    &   (𝜑𝐶𝐵)    &   ((𝜑𝑦𝐵) → ¬ 𝐶𝑅𝑦)       (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶)
 
Theoremfisup2g 8133* A finite set satisfies the conditions to have a supremum. (Contributed by Mario Carneiro, 28-Apr-2015.)
((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵𝐴)) → ∃𝑥𝐵 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
 
Theoremfisupcl 8134 A nonempty finite set contains its supremum. (Contributed by Jeff Madsen, 9-May-2011.)
((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵𝐴)) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵)
 
Theoremsupgtoreq 8135 The supremum of a finite set is greater than or equal to all the elements of the set. (Contributed by AV, 1-Oct-2019.)
(𝜑𝑅 Or 𝐴)    &   (𝜑𝐵𝐴)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑𝐶𝐵)    &   (𝜑𝑆 = sup(𝐵, 𝐴, 𝑅))       (𝜑 → (𝐶𝑅𝑆𝐶 = 𝑆))
 
Theoremsuppr 8136 The supremum of a pair. (Contributed by NM, 17-Jun-2007.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → sup({𝐵, 𝐶}, 𝐴, 𝑅) = if(𝐶𝑅𝐵, 𝐵, 𝐶))
 
Theoremsupsn 8137 The supremum of a singleton. (Contributed by NM, 2-Oct-2007.)
((𝑅 Or 𝐴𝐵𝐴) → sup({𝐵}, 𝐴, 𝑅) = 𝐵)
 
Theoremsupisolem 8138* Lemma for supiso 8140. (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 8139* Lemma for supiso 8140. (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)))
 
Theoremsupiso 8140* Image of a supremum under an isomorphism. (Contributed by Mario Carneiro, 24-Dec-2016.)
(𝜑𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   (𝜑𝐶𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐶 𝑦𝑅𝑧)))    &   (𝜑𝑅 Or 𝐴)       (𝜑 → sup((𝐹𝐶), 𝐵, 𝑆) = (𝐹‘sup(𝐶, 𝐴, 𝑅)))
 
Theoreminfeq1 8141 Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
(𝐵 = 𝐶 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))
 
Theoreminfeq1d 8142 Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
(𝜑𝐵 = 𝐶)       (𝜑 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))
 
Theoreminfeq1i 8143 Equality inference for infimum. (Contributed by AV, 2-Sep-2020.)
𝐵 = 𝐶       inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅)
 
Theoreminfeq2 8144 Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
(𝐵 = 𝐶 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐶, 𝑅))
 
Theoreminfeq3 8145 Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
(𝑅 = 𝑆 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐵, 𝑆))
 
Theoreminfeq123d 8146 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 8147 Hypothesis builder for infimum. (Contributed by AV, 2-Sep-2020.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝑅       𝑥inf(𝐴, 𝐵, 𝑅)
 
Theoreminfexd 8148 An infimum is a set. (Contributed by AV, 2-Sep-2020.)
(𝜑𝑅 Or 𝐴)       (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ V)
 
Theoremeqinf 8149* Sufficient condition for an element to be equal to the infimum. (Contributed by AV, 2-Sep-2020.)
(𝜑𝑅 Or 𝐴)       (𝜑 → ((𝐶𝐴 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝐶 ∧ ∀𝑦𝐴 (𝐶𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)) → inf(𝐵, 𝐴, 𝑅) = 𝐶))
 
Theoremeqinfd 8150* Sufficient condition for an element to be equal to the infimum. (Contributed by AV, 3-Sep-2020.)
(𝜑𝑅 Or 𝐴)    &   (𝜑𝐶𝐴)    &   ((𝜑𝑦𝐵) → ¬ 𝑦𝑅𝐶)    &   ((𝜑 ∧ (𝑦𝐴𝐶𝑅𝑦)) → ∃𝑧𝐵 𝑧𝑅𝑦)       (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)
 
Theoreminfval 8151* Alternate expression for the infimum. (Contributed by AV, 2-Sep-2020.)
(𝜑𝑅 Or 𝐴)       (𝜑 → inf(𝐵, 𝐴, 𝑅) = (𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦))))
 
Theoreminfcllem 8152* Lemma for infcl 8153, inflb 8154, infglb 8155, etc. (Contributed by AV, 3-Sep-2020.)
(𝜑𝑅 Or 𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))       (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
 
Theoreminfcl 8153* An infimum belongs to its base class (closure law). See also inflb 8154 and infglb 8155. (Contributed by AV, 3-Sep-2020.)
(𝜑𝑅 Or 𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))       (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐴)
 
Theoreminflb 8154* An infimum is a lower bound. See also infcl 8153 and infglb 8155. (Contributed by AV, 3-Sep-2020.)
(𝜑𝑅 Or 𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))       (𝜑 → (𝐶𝐵 → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅)))
 
Theoreminfglb 8155* An infimum is the greatest lower bound. See also infcl 8153 and inflb 8154. (Contributed by AV, 3-Sep-2020.)
(𝜑𝑅 Or 𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))       (𝜑 → ((𝐶𝐴 ∧ inf(𝐵, 𝐴, 𝑅)𝑅𝐶) → ∃𝑧𝐵 𝑧𝑅𝐶))
 
Theoreminfglbb 8156* Bidirectional form of infglb 8155. (Contributed by AV, 3-Sep-2020.)
(𝜑𝑅 Or 𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))    &   (𝜑𝐵𝐴)       ((𝜑𝐶𝐴) → (inf(𝐵, 𝐴, 𝑅)𝑅𝐶 ↔ ∃𝑧𝐵 𝑧𝑅𝐶))
 
Theoreminfnlb 8157* A lower bound is not greater than the infimum. (Contributed by AV, 3-Sep-2020.)
(𝜑𝑅 Or 𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))       (𝜑 → ((𝐶𝐴 ∧ ∀𝑧𝐵 ¬ 𝑧𝑅𝐶) → ¬ inf(𝐵, 𝐴, 𝑅)𝑅𝐶))
 
Theoreminfex 8158 An infimum is a set. (Contributed by AV, 3-Sep-2020.)
𝑅 Or 𝐴       inf(𝐵, 𝐴, 𝑅) ∈ V
 
Theoreminfmin 8159* 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 AV, 3-Sep-2020.)
(𝜑𝑅 Or 𝐴)    &   (𝜑𝐶𝐴)    &   (𝜑𝐶𝐵)    &   ((𝜑𝑦𝐵) → ¬ 𝑦𝑅𝐶)       (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)
 
Theoreminfmo 8160* Any class 𝐵 has at most one infimum in 𝐴 (where 𝑅 is interpreted as 'less than'). (Contributed by AV, 6-Oct-2020.)
(𝜑𝑅 Or 𝐴)       (𝜑 → ∃*𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))
 
Theoreminfeu 8161* An infimum is unique. (Contributed by AV, 6-Oct-2020.)
(𝜑𝑅 Or 𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))       (𝜑 → ∃!𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))
 
Theoremfimin2g 8162* A finite set has a minimum under a total order. (Contributed by AV, 6-Oct-2020.)
((𝑅 Or 𝐴𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥)
 
Theoremfiming 8163* A finite set has a minimum under a total order. (Contributed by AV, 6-Oct-2020.)
((𝑅 Or 𝐴𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥𝑅𝑦))
 
Theoremfiinfg 8164* Lemma showing existence and closure of infimum of a finite set. (Contributed by AV, 6-Oct-2020.)
((𝑅 Or 𝐴𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴 (∀𝑦𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐴 𝑧𝑅𝑦)))
 
Theoremfiinf2g 8165* A finite set satisfies the conditions to have an infimum. (Contributed by AV, 6-Oct-2020.)
((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵𝐴)) → ∃𝑥𝐵 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))
 
Theoremfiinfcl 8166 A nonempty finite set contains its infimum. (Contributed by AV, 3-Sep-2020.)
((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵𝐴)) → inf(𝐵, 𝐴, 𝑅) ∈ 𝐵)
 
Theoreminfltoreq 8167 The infimum of a finite set is less than or equal to all the elements of the set. (Contributed by AV, 4-Sep-2020.)
(𝜑𝑅 Or 𝐴)    &   (𝜑𝐵𝐴)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑𝐶𝐵)    &   (𝜑𝑆 = inf(𝐵, 𝐴, 𝑅))       (𝜑 → (𝑆𝑅𝐶𝐶 = 𝑆))
 
Theoreminfpr 8168 The infimum of a pair. (Contributed by AV, 4-Sep-2020.)
((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → inf({𝐵, 𝐶}, 𝐴, 𝑅) = if(𝐵𝑅𝐶, 𝐵, 𝐶))
 
Theoreminfsn 8169 The infimum of a singleton. (Contributed by NM, 2-Oct-2007.)
((𝑅 Or 𝐴𝐵𝐴) → inf({𝐵}, 𝐴, 𝑅) = 𝐵)
 
Theoreminf00 8170 The infimum regarding an empty bas set is always the empty set. (Contributed by AV, 4-Sep-2020.)
inf(𝐵, ∅, 𝑅) = ∅
 
Theoreminfempty 8171* The infimum of an empty set under a base set which has a unique greatest element is the greatest element of the base set. (Contributed by AV, 4-Sep-2020.)
((𝑅 Or 𝐴 ∧ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑋𝑅𝑦) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑥𝑅𝑦) → inf(∅, 𝐴, 𝑅) = 𝑋)
 
Theoreminfiso 8172* Image of an infimum under an isomorphism. (Contributed by AV, 4-Sep-2020.)
(𝜑𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   (𝜑𝐶𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐶 𝑧𝑅𝑦)))    &   (𝜑𝑅 Or 𝐴)       (𝜑 → inf((𝐹𝐶), 𝐵, 𝑆) = (𝐹‘inf(𝐶, 𝐴, 𝑅)))
 
2.4.32  Ordinal isomorphism, Hartog's theorem
 
Syntaxcoi 8173 Extend class definition to include the canonical order isomorphism to an ordinal.
class OrdIso(𝑅, 𝐴)
 
Definitiondf-oi 8174* Define the canonical order isomorphism from the well-order 𝑅 on 𝐴 to an ordinal. (Contributed by Mario Carneiro, 23-May-2015.)
OrdIso(𝑅, 𝐴) = if((𝑅 We 𝐴𝑅 Se 𝐴), (recs(( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) ↾ {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (recs(( ∈ V ↦ (𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑥)𝑧𝑅𝑡}), ∅)
 
Theoremdfoi 8175* Rewrite df-oi 8174 with abbreviations. (Contributed by Mario Carneiro, 24-Jun-2015.)
𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}    &   𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))    &   𝐹 = recs(𝐺)       OrdIso(𝑅, 𝐴) = if((𝑅 We 𝐴𝑅 Se 𝐴), (𝐹 ↾ {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡}), ∅)
 
Theoremoieq1 8176 Equality theorem for ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.)
(𝑅 = 𝑆 → OrdIso(𝑅, 𝐴) = OrdIso(𝑆, 𝐴))
 
Theoremoieq2 8177 Equality theorem for ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.)
(𝐴 = 𝐵 → OrdIso(𝑅, 𝐴) = OrdIso(𝑅, 𝐵))
 
Theoremnfoi 8178 Hypothesis builder for ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝑅    &   𝑥𝐴       𝑥OrdIso(𝑅, 𝐴)
 
Theoremordiso2 8179 Generalize ordiso 8180 to proper classes. (Contributed by Mario Carneiro, 24-Jun-2015.)
((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → 𝐴 = 𝐵)
 
Theoremordiso 8180* 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)))
 
Theoremordtypecbv 8181* Lemma for ordtype 8196. (Contributed by Mario Carneiro, 26-Jun-2015.)
𝐹 = recs(𝐺)    &   𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}    &   𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))       recs((𝑓 ∈ V ↦ (𝑠 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))) = 𝐹
 
Theoremordtypelem1 8182* Lemma for ordtype 8196. (Contributed by Mario Carneiro, 24-Jun-2015.)
𝐹 = recs(𝐺)    &   𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}    &   𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))    &   𝑇 = {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡}    &   𝑂 = OrdIso(𝑅, 𝐴)    &   (𝜑𝑅 We 𝐴)    &   (𝜑𝑅 Se 𝐴)       (𝜑𝑂 = (𝐹𝑇))
 
Theoremordtypelem2 8183* Lemma for ordtype 8196. (Contributed by Mario Carneiro, 24-Jun-2015.)
𝐹 = recs(𝐺)    &   𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}    &   𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))    &   𝑇 = {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡}    &   𝑂 = OrdIso(𝑅, 𝐴)    &   (𝜑𝑅 We 𝐴)    &   (𝜑𝑅 Se 𝐴)       (𝜑 → Ord 𝑇)
 
Theoremordtypelem3 8184* Lemma for ordtype 8196. (Contributed by Mario Carneiro, 24-Jun-2015.)
𝐹 = recs(𝐺)    &   𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}    &   𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))    &   𝑇 = {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡}    &   𝑂 = OrdIso(𝑅, 𝐴)    &   (𝜑𝑅 We 𝐴)    &   (𝜑𝑅 Se 𝐴)       ((𝜑𝑀 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹𝑀) ∈ {𝑣 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤𝐴 ∣ ∀𝑗 ∈ (𝐹𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣})
 
Theoremordtypelem4 8185* Lemma for ordtype 8196. (Contributed by Mario Carneiro, 24-Jun-2015.)
𝐹 = recs(𝐺)    &   𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}    &   𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))    &   𝑇 = {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡}    &   𝑂 = OrdIso(𝑅, 𝐴)    &   (𝜑𝑅 We 𝐴)    &   (𝜑𝑅 Se 𝐴)       (𝜑𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)
 
Theoremordtypelem5 8186* Lemma for ordtype 8196. (Contributed by Mario Carneiro, 25-Jun-2015.)
𝐹 = recs(𝐺)    &   𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}    &   𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))    &   𝑇 = {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡}    &   𝑂 = OrdIso(𝑅, 𝐴)    &   (𝜑𝑅 We 𝐴)    &   (𝜑𝑅 Se 𝐴)       (𝜑 → (Ord dom 𝑂𝑂:dom 𝑂𝐴))
 
Theoremordtypelem6 8187* Lemma for ordtype 8196. (Contributed by Mario Carneiro, 24-Jun-2015.)
𝐹 = recs(𝐺)    &   𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}    &   𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))    &   𝑇 = {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡}    &   𝑂 = OrdIso(𝑅, 𝐴)    &   (𝜑𝑅 We 𝐴)    &   (𝜑𝑅 Se 𝐴)       ((𝜑𝑀 ∈ dom 𝑂) → (𝑁𝑀 → (𝑂𝑁)𝑅(𝑂𝑀)))
 
Theoremordtypelem7 8188* Lemma for ordtype 8196. ran 𝑂 is an initial segment of 𝐴 under the well-order 𝑅. (Contributed by Mario Carneiro, 25-Jun-2015.)
𝐹 = recs(𝐺)    &   𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}    &   𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))    &   𝑇 = {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡}    &   𝑂 = OrdIso(𝑅, 𝐴)    &   (𝜑𝑅 We 𝐴)    &   (𝜑𝑅 Se 𝐴)       (((𝜑𝑁𝐴) ∧ 𝑀 ∈ dom 𝑂) → ((𝑂𝑀)𝑅𝑁𝑁 ∈ ran 𝑂))
 
Theoremordtypelem8 8189* Lemma for ordtype 8196. (Contributed by Mario Carneiro, 25-Jun-2015.)
𝐹 = recs(𝐺)    &   𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}    &   𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))    &   𝑇 = {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡}    &   𝑂 = OrdIso(𝑅, 𝐴)    &   (𝜑𝑅 We 𝐴)    &   (𝜑𝑅 Se 𝐴)       (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂))
 
Theoremordtypelem9 8190* Lemma for ordtype 8196. Either the function OrdIso is an isomorphism onto all of 𝐴, or OrdIso is not a set, which by oif 8194 implies that either ran 𝑂𝐴 is a proper class or dom 𝑂 = On. (Contributed by Mario Carneiro, 25-Jun-2015.)
𝐹 = recs(𝐺)    &   𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}    &   𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))    &   𝑇 = {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡}    &   𝑂 = OrdIso(𝑅, 𝐴)    &   (𝜑𝑅 We 𝐴)    &   (𝜑𝑅 Se 𝐴)    &   (𝜑𝑂 ∈ V)       (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))
 
Theoremordtypelem10 8191* Lemma for ordtype 8196. Using ax-rep 4597, exclude the possibility that 𝑂 is a proper class and does not enumerate all of 𝐴. (Contributed by Mario Carneiro, 25-Jun-2015.)
𝐹 = recs(𝐺)    &   𝐶 = {𝑤𝐴 ∣ ∀𝑗 ∈ ran 𝑗𝑅𝑤}    &   𝐺 = ( ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑅𝑣))    &   𝑇 = {𝑥 ∈ On ∣ ∃𝑡𝐴𝑧 ∈ (𝐹𝑥)𝑧𝑅𝑡}    &   𝑂 = OrdIso(𝑅, 𝐴)    &   (𝜑𝑅 We 𝐴)    &   (𝜑𝑅 Se 𝐴)       (𝜑𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))
 
Theoremoi0 8192 Definition of the ordinal isomorphism when its arguments are not meaningful. (Contributed by Mario Carneiro, 25-Jun-2015.)
𝐹 = OrdIso(𝑅, 𝐴)       (¬ (𝑅 We 𝐴𝑅 Se 𝐴) → 𝐹 = ∅)
 
Theoremoicl 8193 The order type of the well-order 𝑅 on 𝐴 is an ordinal. (Contributed by Mario Carneiro, 23-May-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
𝐹 = OrdIso(𝑅, 𝐴)       Ord dom 𝐹
 
Theoremoif 8194 The order isomorphism of the well-order 𝑅 on 𝐴 is a function. (Contributed by Mario Carneiro, 23-May-2015.)
𝐹 = OrdIso(𝑅, 𝐴)       𝐹:dom 𝐹𝐴
 
Theoremoiiso2 8195 The order isomorphism of the well-order 𝑅 on 𝐴 is an isomorphism onto ran 𝑂 (which is a subset of 𝐴 by oif 8194). (Contributed by Mario Carneiro, 25-Jun-2015.)
𝐹 = OrdIso(𝑅, 𝐴)       ((𝑅 We 𝐴𝑅 Se 𝐴) → 𝐹 Isom E , 𝑅 (dom 𝐹, ran 𝐹))
 
Theoremordtype 8196 For any set-like well-ordered class, there is an isomorphic ordinal number called its order type. (Contributed by Jeff Hankins, 17-Oct-2009.) (Revised by Mario Carneiro, 25-Jun-2015.)
𝐹 = OrdIso(𝑅, 𝐴)       ((𝑅 We 𝐴𝑅 Se 𝐴) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴))
 
Theoremoiiniseg 8197 ran 𝐹 is an initial segment of 𝐴 under the well-order 𝑅. (Contributed by Mario Carneiro, 26-Jun-2015.)
𝐹 = OrdIso(𝑅, 𝐴)       (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ (𝑁𝐴𝑀 ∈ dom 𝐹)) → ((𝐹𝑀)𝑅𝑁𝑁 ∈ ran 𝐹))
 
Theoremordtype2 8198 For any set-like well-ordered class, if the order isomorphism exists (is a set), then it maps some ordinal onto 𝐴 isomorphically. Otherwise, 𝐹 is a proper class, which implies that either ran 𝐹𝐴 is a proper class or dom 𝐹 = On. This weak version of ordtype 8196 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 25-Jun-2015.)
𝐹 = OrdIso(𝑅, 𝐴)       ((𝑅 We 𝐴𝑅 Se 𝐴𝐹 ∈ V) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴))
 
Theoremoiexg 8199 The order isomorphism on a set is a set. (Contributed by Mario Carneiro, 25-Jun-2015.)
𝐹 = OrdIso(𝑅, 𝐴)       (𝐴𝑉𝐹 ∈ V)
 
Theoremoion 8200 The order type of the well-order 𝑅 on 𝐴 is an ordinal. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 23-May-2015.)
𝐹 = OrdIso(𝑅, 𝐴)       (𝐴𝑉 → dom 𝐹 ∈ On)
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