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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | supeq2 8901 | Equality theorem for supremum. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝐵 = 𝐶 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐶, 𝑅)) | ||
Theorem | supeq3 8902 | Equality theorem for supremum. (Contributed by Scott Fenton, 13-Jun-2018.) |
⊢ (𝑅 = 𝑆 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑆)) | ||
Theorem | supeq123d 8903 | Equality deduction for supremum. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
⊢ (𝜑 → 𝐴 = 𝐷) & ⊢ (𝜑 → 𝐵 = 𝐸) & ⊢ (𝜑 → 𝐶 = 𝐹) ⇒ ⊢ (𝜑 → sup(𝐴, 𝐵, 𝐶) = sup(𝐷, 𝐸, 𝐹)) | ||
Theorem | nfsup 8904 | Hypothesis builder for supremum. (Contributed by Mario Carneiro, 20-Mar-2014.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ Ⅎ𝑥𝑅 ⇒ ⊢ Ⅎ𝑥sup(𝐴, 𝐵, 𝑅) | ||
Theorem | supmo 8905* | 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.) |
⊢ (𝜑 → 𝑅 Or 𝐴) ⇒ ⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) | ||
Theorem | supexd 8906 | A supremum is a set. (Contributed by NM, 22-May-1999.) (Revised by Mario Carneiro, 24-Dec-2016.) |
⊢ (𝜑 → 𝑅 Or 𝐴) ⇒ ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ V) | ||
Theorem | supeu 8907* | A supremum is unique. Similar to Theorem I.26 of [Apostol] p. 24 (but for suprema in general). (Contributed by NM, 12-Oct-2004.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ⇒ ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) | ||
Theorem | supval2 8908* | Alternate expression for the supremum. (Contributed by Mario Carneiro, 24-Dec-2016.) (Revised by Thierry Arnoux, 24-Sep-2017.) |
⊢ (𝜑 → 𝑅 Or 𝐴) ⇒ ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))) | ||
Theorem | eqsup 8909* | Sufficient condition for an element to be equal to the supremum. (Contributed by Mario Carneiro, 21-Apr-2015.) |
⊢ (𝜑 → 𝑅 Or 𝐴) ⇒ ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝐶𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝐶 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → sup(𝐵, 𝐴, 𝑅) = 𝐶)) | ||
Theorem | eqsupd 8910* | Sufficient condition for an element to be equal to the supremum. (Contributed by Mario Carneiro, 21-Apr-2015.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝐶𝑅𝑦) & ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝐶)) → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ⇒ ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶) | ||
Theorem | supcl 8911* | A supremum belongs to its base class (closure law). See also supub 8912 and suplub 8913. (Contributed by NM, 12-Oct-2004.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ⇒ ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴) | ||
Theorem | supub 8912* |
A supremum is an upper bound. See also supcl 8911 and suplub 8913.
This proof demonstrates how to expand an iota-based definition (df-iota 6308) using riotacl2 7119. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Mario Carneiro, 24-Dec-2016.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ⇒ ⊢ (𝜑 → (𝐶 ∈ 𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶)) | ||
Theorem | suplub 8913* | A supremum is the least upper bound. See also supcl 8911 and supub 8912. (Contributed by NM, 13-Oct-2004.) (Revised by Mario Carneiro, 24-Dec-2016.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ⇒ ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ 𝐶𝑅sup(𝐵, 𝐴, 𝑅)) → ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧)) | ||
Theorem | suplub2 8914* | Bidirectional form of suplub 8913. (Contributed by Mario Carneiro, 6-Sep-2014.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧)) | ||
Theorem | supnub 8915* | An upper bound is not less than the supremum. (Contributed by NM, 13-Oct-2004.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ⇒ ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐵 ¬ 𝐶𝑅𝑧) → ¬ 𝐶𝑅sup(𝐵, 𝐴, 𝑅))) | ||
Theorem | supex 8916 | A supremum is a set. (Contributed by NM, 22-May-1999.) |
⊢ 𝑅 Or 𝐴 ⇒ ⊢ sup(𝐵, 𝐴, 𝑅) ∈ V | ||
Theorem | sup00 8917 | The supremum under an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.) |
⊢ sup(𝐵, ∅, 𝑅) = ∅ | ||
Theorem | sup0riota 8918* | 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) = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥)) | ||
Theorem | sup0 8919* | 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 ∧ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑋) ∧ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥) → sup(∅, 𝐴, 𝑅) = 𝑋) | ||
Theorem | supmax 8920* | 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.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝐶𝑅𝑦) ⇒ ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶) | ||
Theorem | fisup2g 8921* | A finite set satisfies the conditions to have a supremum. (Contributed by Mario Carneiro, 28-Apr-2015.) |
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → ∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) | ||
Theorem | fisupcl 8922 | A nonempty finite set contains its supremum. (Contributed by Jeff Madsen, 9-May-2011.) |
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵) | ||
Theorem | supgtoreq 8923 | The supremum of a finite set is greater than or equal to all the elements of the set. (Contributed by AV, 1-Oct-2019.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) & ⊢ (𝜑 → 𝑆 = sup(𝐵, 𝐴, 𝑅)) ⇒ ⊢ (𝜑 → (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆)) | ||
Theorem | suppr 8924 | The supremum of a pair. (Contributed by NM, 17-Jun-2007.) (Proof shortened by Mario Carneiro, 24-Dec-2016.) |
⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → sup({𝐵, 𝐶}, 𝐴, 𝑅) = if(𝐶𝑅𝐵, 𝐵, 𝐶)) | ||
Theorem | supsn 8925 | The supremum of a singleton. (Contributed by NM, 2-Oct-2007.) |
⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → sup({𝐵}, 𝐴, 𝑅) = 𝐵) | ||
Theorem | supisolem 8926* | Lemma for supiso 8928. (Contributed by Mario Carneiro, 24-Dec-2016.) |
⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵)) & ⊢ (𝜑 → 𝐶 ⊆ 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝐷 ∈ 𝐴) → ((∀𝑦 ∈ 𝐶 ¬ 𝐷𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝐷 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧)) ↔ (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ (𝐹‘𝐷)𝑆𝑤 ∧ ∀𝑤 ∈ 𝐵 (𝑤𝑆(𝐹‘𝐷) → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣)))) | ||
Theorem | supisoex 8927* | Lemma for supiso 8928. (Contributed by Mario Carneiro, 24-Dec-2016.) |
⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵)) & ⊢ (𝜑 → 𝐶 ⊆ 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧))) ⇒ ⊢ (𝜑 → ∃𝑢 ∈ 𝐵 (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤 ∈ 𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣))) | ||
Theorem | supiso 8928* | Image of a supremum under an isomorphism. (Contributed by Mario Carneiro, 24-Dec-2016.) |
⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵)) & ⊢ (𝜑 → 𝐶 ⊆ 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧))) & ⊢ (𝜑 → 𝑅 Or 𝐴) ⇒ ⊢ (𝜑 → sup((𝐹 “ 𝐶), 𝐵, 𝑆) = (𝐹‘sup(𝐶, 𝐴, 𝑅))) | ||
Theorem | infeq1 8929 | Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.) |
⊢ (𝐵 = 𝐶 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅)) | ||
Theorem | infeq1d 8930 | Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.) |
⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅)) | ||
Theorem | infeq1i 8931 | Equality inference for infimum. (Contributed by AV, 2-Sep-2020.) |
⊢ 𝐵 = 𝐶 ⇒ ⊢ inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅) | ||
Theorem | infeq2 8932 | Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.) |
⊢ (𝐵 = 𝐶 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐶, 𝑅)) | ||
Theorem | infeq3 8933 | Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.) |
⊢ (𝑅 = 𝑆 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐵, 𝑆)) | ||
Theorem | infeq123d 8934 | Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.) |
⊢ (𝜑 → 𝐴 = 𝐷) & ⊢ (𝜑 → 𝐵 = 𝐸) & ⊢ (𝜑 → 𝐶 = 𝐹) ⇒ ⊢ (𝜑 → inf(𝐴, 𝐵, 𝐶) = inf(𝐷, 𝐸, 𝐹)) | ||
Theorem | nfinf 8935 | Hypothesis builder for infimum. (Contributed by AV, 2-Sep-2020.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ Ⅎ𝑥𝑅 ⇒ ⊢ Ⅎ𝑥inf(𝐴, 𝐵, 𝑅) | ||
Theorem | infexd 8936 | An infimum is a set. (Contributed by AV, 2-Sep-2020.) |
⊢ (𝜑 → 𝑅 Or 𝐴) ⇒ ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ V) | ||
Theorem | eqinf 8937* | Sufficient condition for an element to be equal to the infimum. (Contributed by AV, 2-Sep-2020.) |
⊢ (𝜑 → 𝑅 Or 𝐴) ⇒ ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝐶 ∧ ∀𝑦 ∈ 𝐴 (𝐶𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)) → inf(𝐵, 𝐴, 𝑅) = 𝐶)) | ||
Theorem | eqinfd 8938* | Sufficient condition for an element to be equal to the infimum. (Contributed by AV, 3-Sep-2020.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝑦𝑅𝐶) & ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝐶𝑅𝑦)) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦) ⇒ ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶) | ||
Theorem | infval 8939* | Alternate expression for the infimum. (Contributed by AV, 2-Sep-2020.) |
⊢ (𝜑 → 𝑅 Or 𝐴) ⇒ ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))) | ||
Theorem | infcllem 8940* | Lemma for infcl 8941, inflb 8942, infglb 8943, etc. (Contributed by AV, 3-Sep-2020.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))) | ||
Theorem | infcl 8941* | An infimum belongs to its base class (closure law). See also inflb 8942 and infglb 8943. (Contributed by AV, 3-Sep-2020.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) ⇒ ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐴) | ||
Theorem | inflb 8942* | An infimum is a lower bound. See also infcl 8941 and infglb 8943. (Contributed by AV, 3-Sep-2020.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) ⇒ ⊢ (𝜑 → (𝐶 ∈ 𝐵 → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅))) | ||
Theorem | infglb 8943* | An infimum is the greatest lower bound. See also infcl 8941 and inflb 8942. (Contributed by AV, 3-Sep-2020.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) ⇒ ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ inf(𝐵, 𝐴, 𝑅)𝑅𝐶) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝐶)) | ||
Theorem | infglbb 8944* | Bidirectional form of infglb 8943. (Contributed by AV, 3-Sep-2020.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (inf(𝐵, 𝐴, 𝑅)𝑅𝐶 ↔ ∃𝑧 ∈ 𝐵 𝑧𝑅𝐶)) | ||
Theorem | infnlb 8945* | A lower bound is not greater than the infimum. (Contributed by AV, 3-Sep-2020.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) ⇒ ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝐶) → ¬ inf(𝐵, 𝐴, 𝑅)𝑅𝐶)) | ||
Theorem | infex 8946 | An infimum is a set. (Contributed by AV, 3-Sep-2020.) |
⊢ 𝑅 Or 𝐴 ⇒ ⊢ inf(𝐵, 𝐴, 𝑅) ∈ V | ||
Theorem | infmin 8947* | 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.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝑦𝑅𝐶) ⇒ ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶) | ||
Theorem | infmo 8948* | Any class 𝐵 has at most one infimum in 𝐴 (where 𝑅 is interpreted as 'less than'). (Contributed by AV, 6-Oct-2020.) |
⊢ (𝜑 → 𝑅 Or 𝐴) ⇒ ⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) | ||
Theorem | infeu 8949* | An infimum is unique. (Contributed by AV, 6-Oct-2020.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (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 | fimin2g 8950* | A finite set has a minimum under a total order. (Contributed by AV, 6-Oct-2020.) |
⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥) | ||
Theorem | fiming 8951* | A finite set has a minimum under a total order. (Contributed by AV, 6-Oct-2020.) |
⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → 𝑥𝑅𝑦)) | ||
Theorem | fiinfg 8952* | Lemma showing existence and closure of infimum of a finite set. (Contributed by AV, 6-Oct-2020.) |
⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐴 𝑧𝑅𝑦))) | ||
Theorem | fiinf2g 8953* | A finite set satisfies the conditions to have an infimum. (Contributed by AV, 6-Oct-2020.) |
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → ∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) | ||
Theorem | fiinfcl 8954 | A nonempty finite set contains its infimum. (Contributed by AV, 3-Sep-2020.) |
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → inf(𝐵, 𝐴, 𝑅) ∈ 𝐵) | ||
Theorem | infltoreq 8955 | The infimum of a finite set is less than or equal to all the elements of the set. (Contributed by AV, 4-Sep-2020.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → 𝐶 ∈ 𝐵) & ⊢ (𝜑 → 𝑆 = inf(𝐵, 𝐴, 𝑅)) ⇒ ⊢ (𝜑 → (𝑆𝑅𝐶 ∨ 𝐶 = 𝑆)) | ||
Theorem | infpr 8956 | The infimum of a pair. (Contributed by AV, 4-Sep-2020.) |
⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → inf({𝐵, 𝐶}, 𝐴, 𝑅) = if(𝐵𝑅𝐶, 𝐵, 𝐶)) | ||
Theorem | infsupprpr 8957 | The infimum of a proper pair is less than the supremum of this pair. (Contributed by AV, 13-Mar-2023.) |
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶)) → inf({𝐵, 𝐶}, 𝐴, 𝑅)𝑅sup({𝐵, 𝐶}, 𝐴, 𝑅)) | ||
Theorem | infsn 8958 | The infimum of a singleton. (Contributed by NM, 2-Oct-2007.) |
⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → inf({𝐵}, 𝐴, 𝑅) = 𝐵) | ||
Theorem | inf00 8959 | The infimum regarding an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.) |
⊢ inf(𝐵, ∅, 𝑅) = ∅ | ||
Theorem | infempty 8960* | 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 ∧ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑋𝑅𝑦) ∧ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦) → inf(∅, 𝐴, 𝑅) = 𝑋) | ||
Theorem | infiso 8961* | Image of an infimum under an isomorphism. (Contributed by AV, 4-Sep-2020.) |
⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵)) & ⊢ (𝜑 → 𝐶 ⊆ 𝐴) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐶 𝑧𝑅𝑦))) & ⊢ (𝜑 → 𝑅 Or 𝐴) ⇒ ⊢ (𝜑 → inf((𝐹 “ 𝐶), 𝐵, 𝑆) = (𝐹‘inf(𝐶, 𝐴, 𝑅))) | ||
Syntax | coi 8962 | Extend class definition to include the canonical order isomorphism to an ordinal. |
class OrdIso(𝑅, 𝐴) | ||
Definition | df-oi 8963* | 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 𝐴 ∧ 𝑅 Se 𝐴), (recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) ↾ {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑥)𝑧𝑅𝑡}), ∅) | ||
Theorem | dfoi 8964* | Rewrite df-oi 8963 with abbreviations. (Contributed by Mario Carneiro, 24-Jun-2015.) |
⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} & ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣)) & ⊢ 𝐹 = recs(𝐺) ⇒ ⊢ OrdIso(𝑅, 𝐴) = if((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴), (𝐹 ↾ {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}), ∅) | ||
Theorem | oieq1 8965 | Equality theorem for ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.) |
⊢ (𝑅 = 𝑆 → OrdIso(𝑅, 𝐴) = OrdIso(𝑆, 𝐴)) | ||
Theorem | oieq2 8966 | Equality theorem for ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.) |
⊢ (𝐴 = 𝐵 → OrdIso(𝑅, 𝐴) = OrdIso(𝑅, 𝐵)) | ||
Theorem | nfoi 8967 | Hypothesis builder for ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.) (Revised by Mario Carneiro, 15-Oct-2016.) |
⊢ Ⅎ𝑥𝑅 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥OrdIso(𝑅, 𝐴) | ||
Theorem | ordiso2 8968 | Generalize ordiso 8969 to proper classes. (Contributed by Mario Carneiro, 24-Jun-2015.) |
⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → 𝐴 = 𝐵) | ||
Theorem | ordiso 8969* | 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))) | ||
Theorem | ordtypecbv 8970* | Lemma for ordtype 8985. (Contributed by Mario Carneiro, 26-Jun-2015.) |
⊢ 𝐹 = recs(𝐺) & ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} & ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣)) ⇒ ⊢ recs((𝑓 ∈ V ↦ (℩𝑠 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦}∀𝑟 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑖 ∈ ran 𝑓 𝑖𝑅𝑦} ¬ 𝑟𝑅𝑠))) = 𝐹 | ||
Theorem | ordtypelem1 8971* | Lemma for ordtype 8985. (Contributed by Mario Carneiro, 24-Jun-2015.) |
⊢ 𝐹 = recs(𝐺) & ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} & ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣)) & ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡} & ⊢ 𝑂 = OrdIso(𝑅, 𝐴) & ⊢ (𝜑 → 𝑅 We 𝐴) & ⊢ (𝜑 → 𝑅 Se 𝐴) ⇒ ⊢ (𝜑 → 𝑂 = (𝐹 ↾ 𝑇)) | ||
Theorem | ordtypelem2 8972* | Lemma for ordtype 8985. (Contributed by Mario Carneiro, 24-Jun-2015.) |
⊢ 𝐹 = recs(𝐺) & ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} & ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣)) & ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡} & ⊢ 𝑂 = OrdIso(𝑅, 𝐴) & ⊢ (𝜑 → 𝑅 We 𝐴) & ⊢ (𝜑 → 𝑅 Se 𝐴) ⇒ ⊢ (𝜑 → Ord 𝑇) | ||
Theorem | ordtypelem3 8973* | Lemma for ordtype 8985. (Contributed by Mario Carneiro, 24-Jun-2015.) |
⊢ 𝐹 = recs(𝐺) & ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} & ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣)) & ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡} & ⊢ 𝑂 = OrdIso(𝑅, 𝐴) & ⊢ (𝜑 → 𝑅 We 𝐴) & ⊢ (𝜑 → 𝑅 Se 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝑀 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹‘𝑀) ∈ {𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑀)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣}) | ||
Theorem | ordtypelem4 8974* | Lemma for ordtype 8985. (Contributed by Mario Carneiro, 24-Jun-2015.) |
⊢ 𝐹 = recs(𝐺) & ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} & ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣)) & ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡} & ⊢ 𝑂 = OrdIso(𝑅, 𝐴) & ⊢ (𝜑 → 𝑅 We 𝐴) & ⊢ (𝜑 → 𝑅 Se 𝐴) ⇒ ⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴) | ||
Theorem | ordtypelem5 8975* | Lemma for ordtype 8985. (Contributed by Mario Carneiro, 25-Jun-2015.) |
⊢ 𝐹 = recs(𝐺) & ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} & ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣)) & ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡} & ⊢ 𝑂 = OrdIso(𝑅, 𝐴) & ⊢ (𝜑 → 𝑅 We 𝐴) & ⊢ (𝜑 → 𝑅 Se 𝐴) ⇒ ⊢ (𝜑 → (Ord dom 𝑂 ∧ 𝑂:dom 𝑂⟶𝐴)) | ||
Theorem | ordtypelem6 8976* | Lemma for ordtype 8985. (Contributed by Mario Carneiro, 24-Jun-2015.) |
⊢ 𝐹 = recs(𝐺) & ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} & ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣)) & ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡} & ⊢ 𝑂 = OrdIso(𝑅, 𝐴) & ⊢ (𝜑 → 𝑅 We 𝐴) & ⊢ (𝜑 → 𝑅 Se 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝑀 ∈ dom 𝑂) → (𝑁 ∈ 𝑀 → (𝑂‘𝑁)𝑅(𝑂‘𝑀))) | ||
Theorem | ordtypelem7 8977* | Lemma for ordtype 8985. ran 𝑂 is an initial segment of 𝐴 under the well-order 𝑅. (Contributed by Mario Carneiro, 25-Jun-2015.) |
⊢ 𝐹 = recs(𝐺) & ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} & ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣)) & ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡} & ⊢ 𝑂 = OrdIso(𝑅, 𝐴) & ⊢ (𝜑 → 𝑅 We 𝐴) & ⊢ (𝜑 → 𝑅 Se 𝐴) ⇒ ⊢ (((𝜑 ∧ 𝑁 ∈ 𝐴) ∧ 𝑀 ∈ dom 𝑂) → ((𝑂‘𝑀)𝑅𝑁 ∨ 𝑁 ∈ ran 𝑂)) | ||
Theorem | ordtypelem8 8978* | Lemma for ordtype 8985. (Contributed by Mario Carneiro, 25-Jun-2015.) |
⊢ 𝐹 = recs(𝐺) & ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} & ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣)) & ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡} & ⊢ 𝑂 = OrdIso(𝑅, 𝐴) & ⊢ (𝜑 → 𝑅 We 𝐴) & ⊢ (𝜑 → 𝑅 Se 𝐴) ⇒ ⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂)) | ||
Theorem | ordtypelem9 8979* | Lemma for ordtype 8985. Either the function OrdIso is an isomorphism onto all of 𝐴, or OrdIso is not a set, which by oif 8983 implies that either ran 𝑂 ⊆ 𝐴 is a proper class or dom 𝑂 = On. (Contributed by Mario Carneiro, 25-Jun-2015.) |
⊢ 𝐹 = recs(𝐺) & ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} & ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣)) & ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡} & ⊢ 𝑂 = OrdIso(𝑅, 𝐴) & ⊢ (𝜑 → 𝑅 We 𝐴) & ⊢ (𝜑 → 𝑅 Se 𝐴) & ⊢ (𝜑 → 𝑂 ∈ V) ⇒ ⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴)) | ||
Theorem | ordtypelem10 8980* | Lemma for ordtype 8985. Using ax-rep 5182, 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 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} & ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣)) & ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡} & ⊢ 𝑂 = OrdIso(𝑅, 𝐴) & ⊢ (𝜑 → 𝑅 We 𝐴) & ⊢ (𝜑 → 𝑅 Se 𝐴) ⇒ ⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴)) | ||
Theorem | oi0 8981 | Definition of the ordinal isomorphism when its arguments are not meaningful. (Contributed by Mario Carneiro, 25-Jun-2015.) |
⊢ 𝐹 = OrdIso(𝑅, 𝐴) ⇒ ⊢ (¬ (𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 = ∅) | ||
Theorem | oicl 8982 | 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 | ||
Theorem | oif 8983 | 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 𝐹⟶𝐴 | ||
Theorem | oiiso2 8984 | The order isomorphism of the well-order 𝑅 on 𝐴 is an isomorphism onto ran 𝑂 (which is a subset of 𝐴 by oif 8983). (Contributed by Mario Carneiro, 25-Jun-2015.) |
⊢ 𝐹 = OrdIso(𝑅, 𝐴) ⇒ ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 Isom E , 𝑅 (dom 𝐹, ran 𝐹)) | ||
Theorem | ordtype 8985 | 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 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴)) | ||
Theorem | oiiniseg 8986 | 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 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑁 ∈ 𝐴 ∧ 𝑀 ∈ dom 𝐹)) → ((𝐹‘𝑀)𝑅𝑁 ∨ 𝑁 ∈ ran 𝐹)) | ||
Theorem | ordtype2 8987 | 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 𝐹 ⊆ 𝐴 is a proper class or dom 𝐹 = On. This weak version of ordtype 8985 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 25-Jun-2015.) |
⊢ 𝐹 = OrdIso(𝑅, 𝐴) ⇒ ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ 𝐹 ∈ V) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴)) | ||
Theorem | oiexg 8988 | The order isomorphism on a set is a set. (Contributed by Mario Carneiro, 25-Jun-2015.) |
⊢ 𝐹 = OrdIso(𝑅, 𝐴) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝐹 ∈ V) | ||
Theorem | oion 8989 | 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) ⇒ ⊢ (𝐴 ∈ 𝑉 → dom 𝐹 ∈ On) | ||
Theorem | oiiso 8990 | The order isomorphism of the well-order 𝑅 on 𝐴 is an isomorphism. (Contributed by Mario Carneiro, 23-May-2015.) |
⊢ 𝐹 = OrdIso(𝑅, 𝐴) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴)) | ||
Theorem | oien 8991 | The order type of a well-ordered set is equinumerous to the set. (Contributed by Mario Carneiro, 23-May-2015.) |
⊢ 𝐹 = OrdIso(𝑅, 𝐴) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴) → dom 𝐹 ≈ 𝐴) | ||
Theorem | oieu 8992 | 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 𝐴 ∧ 𝑅 Se 𝐴) → ((Ord 𝐵 ∧ 𝐺 Isom E , 𝑅 (𝐵, 𝐴)) ↔ (𝐵 = dom 𝐹 ∧ 𝐺 = 𝐹))) | ||
Theorem | oismo 8993 | 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 5182 (the second statement is trivial under ax-rep 5182). (Contributed by Mario Carneiro, 26-Jun-2015.) |
⊢ 𝐹 = OrdIso( E , 𝐴) ⇒ ⊢ (𝐴 ⊆ On → (Smo 𝐹 ∧ ran 𝐹 = 𝐴)) | ||
Theorem | oiid 8994 | 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)) | ||
Theorem | hartogslem1 8995* | Lemma for hartogs 8997. (Contributed by Mario Carneiro, 14-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.) |
⊢ 𝐹 = {〈𝑟, 𝑦〉 ∣ (((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} & ⊢ 𝑅 = {〈𝑠, 𝑡〉 ∣ ∃𝑤 ∈ 𝑦 ∃𝑧 ∈ 𝑦 ((𝑠 = (𝑓‘𝑤) ∧ 𝑡 = (𝑓‘𝑧)) ∧ 𝑤 E 𝑧)} ⇒ ⊢ (dom 𝐹 ⊆ 𝒫 (𝐴 × 𝐴) ∧ Fun 𝐹 ∧ (𝐴 ∈ 𝑉 → ran 𝐹 = {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴})) | ||
Theorem | hartogslem2 8996* | Lemma for hartogs 8997. (Contributed by Mario Carneiro, 14-Jan-2013.) |
⊢ 𝐹 = {〈𝑟, 𝑦〉 ∣ (((dom 𝑟 ⊆ 𝐴 ∧ ( I ↾ dom 𝑟) ⊆ 𝑟 ∧ 𝑟 ⊆ (dom 𝑟 × dom 𝑟)) ∧ (𝑟 ∖ I ) We dom 𝑟) ∧ 𝑦 = dom OrdIso((𝑟 ∖ I ), dom 𝑟))} & ⊢ 𝑅 = {〈𝑠, 𝑡〉 ∣ ∃𝑤 ∈ 𝑦 ∃𝑧 ∈ 𝑦 ((𝑠 = (𝑓‘𝑤) ∧ 𝑡 = (𝑓‘𝑧)) ∧ 𝑤 E 𝑧)} ⇒ ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ V) | ||
Theorem | hartogs 8997* | 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 9974- but this proof works in ZF.) (Contributed by Jeff Hankins, 22-Oct-2009.) (Revised by Mario Carneiro, 15-May-2015.) |
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ On) | ||
Theorem | wofib 8998 | The only sets which are well-ordered forwards and backwards are finite sets. (Contributed by Mario Carneiro, 30-Jan-2014.) (Revised by Mario Carneiro, 23-May-2015.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) ↔ (𝑅 We 𝐴 ∧ ◡𝑅 We 𝐴)) | ||
Theorem | wemaplem1 8999* | Value of the lexicographic order on a sequence space. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} ⇒ ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑄 ∈ 𝑊) → (𝑃𝑇𝑄 ↔ ∃𝑎 ∈ 𝐴 ((𝑃‘𝑎)𝑆(𝑄‘𝑎) ∧ ∀𝑏 ∈ 𝐴 (𝑏𝑅𝑎 → (𝑃‘𝑏) = (𝑄‘𝑏))))) | ||
Theorem | wemaplem2 9000* | Lemma for wemapso 9004. Transitivity. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} & ⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → 𝑃 ∈ (𝐵 ↑m 𝐴)) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m 𝐴)) & ⊢ (𝜑 → 𝑄 ∈ (𝐵 ↑m 𝐴)) & ⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → 𝑆 Po 𝐵) & ⊢ (𝜑 → 𝑎 ∈ 𝐴) & ⊢ (𝜑 → (𝑃‘𝑎)𝑆(𝑋‘𝑎)) & ⊢ (𝜑 → ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐))) & ⊢ (𝜑 → 𝑏 ∈ 𝐴) & ⊢ (𝜑 → (𝑋‘𝑏)𝑆(𝑄‘𝑏)) & ⊢ (𝜑 → ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐))) ⇒ ⊢ (𝜑 → 𝑃𝑇𝑄) |
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