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Theorem List for Metamath Proof Explorer - 8901-9000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsupeq1 8901 Equality theorem for supremum. (Contributed by NM, 22-May-1999.)
(𝐵 = 𝐶 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅))

Theoremsupeq1d 8902 Equality deduction for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝐵 = 𝐶)       (𝜑 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅))

Theoremsupeq1i 8903 Equality inference for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
𝐵 = 𝐶       sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅)

Theoremsupeq2 8904 Equality theorem for supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐵 = 𝐶 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐶, 𝑅))

Theoremsupeq3 8905 Equality theorem for supremum. (Contributed by Scott Fenton, 13-Jun-2018.)
(𝑅 = 𝑆 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑆))

Theoremsupeq123d 8906 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 8907 Hypothesis builder for supremum. (Contributed by Mario Carneiro, 20-Mar-2014.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝑅       𝑥sup(𝐴, 𝐵, 𝑅)

Theoremsupmo 8908* 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 8909 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 8910* 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 8911* 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 8912* 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 8913* 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 8914* A supremum belongs to its base class (closure law). See also supub 8915 and suplub 8916. (Contributed by NM, 12-Oct-2004.)
(𝜑𝑅 Or 𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))       (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)

Theoremsupub 8915* A supremum is an upper bound. See also supcl 8914 and suplub 8916.

This proof demonstrates how to expand an iota-based definition (df-iota 6307) using riotacl2 7122.

(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 8916* A supremum is the least upper bound. See also supcl 8914 and supub 8915. (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 8917* Bidirectional form of suplub 8916. (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 8918* 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 8919 A supremum is a set. (Contributed by NM, 22-May-1999.)
𝑅 Or 𝐴       sup(𝐵, 𝐴, 𝑅) ∈ V

Theoremsup00 8920 The supremum under an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
sup(𝐵, ∅, 𝑅) = ∅

Theoremsup0riota 8921* 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 8922* 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 8923* 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 8924* 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 8925 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 8926 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 8927 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 8928 The supremum of a singleton. (Contributed by NM, 2-Oct-2007.)
((𝑅 Or 𝐴𝐵𝐴) → sup({𝐵}, 𝐴, 𝑅) = 𝐵)

Theoremsupisolem 8929* Lemma for supiso 8931. (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 8930* Lemma for supiso 8931. (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 8931* 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 8932 Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
(𝐵 = 𝐶 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))

Theoreminfeq1d 8933 Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
(𝜑𝐵 = 𝐶)       (𝜑 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))

Theoreminfeq1i 8934 Equality inference for infimum. (Contributed by AV, 2-Sep-2020.)
𝐵 = 𝐶       inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅)

Theoreminfeq2 8935 Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
(𝐵 = 𝐶 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐶, 𝑅))

Theoreminfeq3 8936 Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
(𝑅 = 𝑆 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐵, 𝑆))

Theoreminfeq123d 8937 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 8938 Hypothesis builder for infimum. (Contributed by AV, 2-Sep-2020.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝑅       𝑥inf(𝐴, 𝐵, 𝑅)

Theoreminfexd 8939 An infimum is a set. (Contributed by AV, 2-Sep-2020.)
(𝜑𝑅 Or 𝐴)       (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ V)

Theoremeqinf 8940* 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 8941* 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 8942* 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 8943* Lemma for infcl 8944, inflb 8945, infglb 8946, 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 8944* An infimum belongs to its base class (closure law). See also inflb 8945 and infglb 8946. (Contributed by AV, 3-Sep-2020.)
(𝜑𝑅 Or 𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))       (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐴)

Theoreminflb 8945* An infimum is a lower bound. See also infcl 8944 and infglb 8946. (Contributed by AV, 3-Sep-2020.)
(𝜑𝑅 Or 𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))       (𝜑 → (𝐶𝐵 → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅)))

Theoreminfglb 8946* An infimum is the greatest lower bound. See also infcl 8944 and inflb 8945. (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 8947* Bidirectional form of infglb 8946. (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 8948* 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 8949 An infimum is a set. (Contributed by AV, 3-Sep-2020.)
𝑅 Or 𝐴       inf(𝐵, 𝐴, 𝑅) ∈ V

Theoreminfmin 8950* 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 8951* 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 8952* 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 8953* A finite set has a minimum under a total order. (Contributed by AV, 6-Oct-2020.)
((𝑅 Or 𝐴𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥)

Theoremfiming 8954* A finite set has a minimum under a total order. (Contributed by AV, 6-Oct-2020.)
((𝑅 Or 𝐴𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥𝑅𝑦))

Theoremfiinfg 8955* 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 8956* 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 8957 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 8958 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 8959 The infimum of a pair. (Contributed by AV, 4-Sep-2020.)
((𝑅 Or 𝐴𝐵𝐴𝐶𝐴) → inf({𝐵, 𝐶}, 𝐴, 𝑅) = if(𝐵𝑅𝐶, 𝐵, 𝐶))

Theoreminfsupprpr 8960 The infimum of a proper pair is less than the supremum of this pair. (Contributed by AV, 13-Mar-2023.)
((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐵𝐶)) → inf({𝐵, 𝐶}, 𝐴, 𝑅)𝑅sup({𝐵, 𝐶}, 𝐴, 𝑅))

Theoreminfsn 8961 The infimum of a singleton. (Contributed by NM, 2-Oct-2007.)
((𝑅 Or 𝐴𝐵𝐴) → inf({𝐵}, 𝐴, 𝑅) = 𝐵)

Theoreminf00 8962 The infimum regarding an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
inf(𝐵, ∅, 𝑅) = ∅

Theoreminfempty 8963* 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 8964* 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)))

2.4.33  Ordinal isomorphism, Hartogs's theorem

Syntaxcoi 8965 Extend class definition to include the canonical order isomorphism to an ordinal.
class OrdIso(𝑅, 𝐴)

Definitiondf-oi 8966* 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 8967* Rewrite df-oi 8966 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 8968 Equality theorem for ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.)
(𝑅 = 𝑆 → OrdIso(𝑅, 𝐴) = OrdIso(𝑆, 𝐴))

Theoremoieq2 8969 Equality theorem for ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.)
(𝐴 = 𝐵 → OrdIso(𝑅, 𝐴) = OrdIso(𝑅, 𝐵))

Theoremnfoi 8970 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 8971 Generalize ordiso 8972 to proper classes. (Contributed by Mario Carneiro, 24-Jun-2015.)
((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → 𝐴 = 𝐵)

Theoremordiso 8972* 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 8973* Lemma for ordtype 8988. (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 8974* Lemma for ordtype 8988. (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 8975* Lemma for ordtype 8988. (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 8976* Lemma for ordtype 8988. (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 8977* Lemma for ordtype 8988. (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 8978* Lemma for ordtype 8988. (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 8979* Lemma for ordtype 8988. (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 8980* Lemma for ordtype 8988. 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 8981* Lemma for ordtype 8988. (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 8982* Lemma for ordtype 8988. Either the function OrdIso is an isomorphism onto all of 𝐴, or OrdIso is not a set, which by oif 8986 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 8983* Lemma for ordtype 8988. Using ax-rep 5181, 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 8984 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 8985 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 8986 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 8987 The order isomorphism of the well-order 𝑅 on 𝐴 is an isomorphism onto ran 𝑂 (which is a subset of 𝐴 by oif 8986). (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 8988 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 8989 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 8990 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 8988 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 8991 The order isomorphism on a set is a set. (Contributed by Mario Carneiro, 25-Jun-2015.)
𝐹 = OrdIso(𝑅, 𝐴)       (𝐴𝑉𝐹 ∈ V)

Theoremoion 8992 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)

Theoremoiiso 8993 The order isomorphism of the well-order 𝑅 on 𝐴 is an isomorphism. (Contributed by Mario Carneiro, 23-May-2015.)
𝐹 = OrdIso(𝑅, 𝐴)       ((𝐴𝑉𝑅 We 𝐴) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴))

Theoremoien 8994 The order type of a well-ordered set is equinumerous to the set. (Contributed by Mario Carneiro, 23-May-2015.)
𝐹 = OrdIso(𝑅, 𝐴)       ((𝐴𝑉𝑅 We 𝐴) → dom 𝐹𝐴)

Theoremoieu 8995 Uniqueness of the unique ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
𝐹 = OrdIso(𝑅, 𝐴)       ((𝑅 We 𝐴𝑅 Se 𝐴) → ((Ord 𝐵𝐺 Isom E , 𝑅 (𝐵, 𝐴)) ↔ (𝐵 = dom 𝐹𝐺 = 𝐹)))

Theoremoismo 8996 When 𝐴 is a subclass of On, 𝐹 is a strictly monotone ordinal functions, and it is also complete (it is an isomorphism onto all of 𝐴). The proof avoids ax-rep 5181 (the second statement is trivial under ax-rep 5181). (Contributed by Mario Carneiro, 26-Jun-2015.)
𝐹 = OrdIso( E , 𝐴)       (𝐴 ⊆ On → (Smo 𝐹 ∧ ran 𝐹 = 𝐴))

Theoremoiid 8997 The order type of an ordinal under the order is itself, and the order isomorphism is the identity function. (Contributed by Mario Carneiro, 26-Jun-2015.)
(Ord 𝐴 → OrdIso( E , 𝐴) = ( I ↾ 𝐴))

Theoremhartogslem1 8998* Lemma for hartogs 9000. (Contributed by Mario Carneiro, 14-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.)
𝐹 = {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}    &   𝑅 = {⟨𝑠, 𝑡⟩ ∣ ∃𝑤𝑦𝑧𝑦 ((𝑠 = (𝑓𝑤) ∧ 𝑡 = (𝑓𝑧)) ∧ 𝑤 E 𝑧)}       (dom 𝐹 ⊆ 𝒫 (𝐴 × 𝐴) ∧ Fun 𝐹 ∧ (𝐴𝑉 → ran 𝐹 = {𝑥 ∈ On ∣ 𝑥𝐴}))

Theoremhartogslem2 8999* Lemma for hartogs 9000. (Contributed by Mario Carneiro, 14-Jan-2013.)
𝐹 = {⟨𝑟, 𝑦⟩ ∣ (((dom 𝑟𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))}    &   𝑅 = {⟨𝑠, 𝑡⟩ ∣ ∃𝑤𝑦𝑧𝑦 ((𝑠 = (𝑓𝑤) ∧ 𝑡 = (𝑓𝑧)) ∧ 𝑤 E 𝑧)}       (𝐴𝑉 → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ V)

Theoremhartogs 9000* Given any set, the Hartogs number of the set is the least ordinal not dominated by that set. This theorem proves that there is always an ordinal which satisfies this. (This theorem can be proven trivially using the AC - see theorem ondomon 9977- but this proof works in ZF.) (Contributed by Jeff Hankins, 22-Oct-2009.) (Revised by Mario Carneiro, 15-May-2015.)
(𝐴𝑉 → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ On)

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