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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | sup00 9501 | The supremum under an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.) |
⊢ sup(𝐵, ∅, 𝑅) = ∅ | ||
Theorem | sup0riota 9502* | 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 9503* | 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 9504* | 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 9505* | 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 9506 | 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 9507 | 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 9508 | 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 9509 | The supremum of a singleton. (Contributed by NM, 2-Oct-2007.) |
⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → sup({𝐵}, 𝐴, 𝑅) = 𝐵) | ||
Theorem | supisolem 9510* | Lemma for supiso 9512. (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 9511* | Lemma for supiso 9512. (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 9512* | 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 9513 | Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.) |
⊢ (𝐵 = 𝐶 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅)) | ||
Theorem | infeq1d 9514 | Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.) |
⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅)) | ||
Theorem | infeq1i 9515 | Equality inference for infimum. (Contributed by AV, 2-Sep-2020.) |
⊢ 𝐵 = 𝐶 ⇒ ⊢ inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅) | ||
Theorem | infeq2 9516 | Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.) |
⊢ (𝐵 = 𝐶 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐶, 𝑅)) | ||
Theorem | infeq3 9517 | Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.) |
⊢ (𝑅 = 𝑆 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐵, 𝑆)) | ||
Theorem | infeq123d 9518 | 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 9519 | Hypothesis builder for infimum. (Contributed by AV, 2-Sep-2020.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ Ⅎ𝑥𝑅 ⇒ ⊢ Ⅎ𝑥inf(𝐴, 𝐵, 𝑅) | ||
Theorem | infexd 9520 | An infimum is a set. (Contributed by AV, 2-Sep-2020.) |
⊢ (𝜑 → 𝑅 Or 𝐴) ⇒ ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ V) | ||
Theorem | eqinf 9521* | 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 9522* | 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 9523* | 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 9524* | Lemma for infcl 9525, inflb 9526, infglb 9527, 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 9525* | An infimum belongs to its base class (closure law). See also inflb 9526 and infglb 9527. (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 9526* | An infimum is a lower bound. See also infcl 9525 and infglb 9527. (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 9527* | An infimum is the greatest lower bound. See also infcl 9525 and inflb 9526. (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 9528* | Bidirectional form of infglb 9527. (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 9529* | 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 9530 | An infimum is a set. (Contributed by AV, 3-Sep-2020.) |
⊢ 𝑅 Or 𝐴 ⇒ ⊢ inf(𝐵, 𝐴, 𝑅) ∈ V | ||
Theorem | infmin 9531* | 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 9532* | 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 9533* | 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 9534* | 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 9535* | 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 9536* | 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 9537* | 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 9538 | 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 9539 | 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 9540 | 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 9541 | 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 9542 | The infimum of a singleton. (Contributed by NM, 2-Oct-2007.) |
⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → inf({𝐵}, 𝐴, 𝑅) = 𝐵) | ||
Theorem | inf00 9543 | The infimum regarding an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.) |
⊢ inf(𝐵, ∅, 𝑅) = ∅ | ||
Theorem | infempty 9544* | 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 9545* | 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 9546 | Extend class definition to include the canonical order isomorphism to an ordinal. |
class OrdIso(𝑅, 𝐴) | ||
Definition | df-oi 9547* | 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 9548* | Rewrite df-oi 9547 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 9549 | Equality theorem for ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.) |
⊢ (𝑅 = 𝑆 → OrdIso(𝑅, 𝐴) = OrdIso(𝑆, 𝐴)) | ||
Theorem | oieq2 9550 | Equality theorem for ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.) |
⊢ (𝐴 = 𝐵 → OrdIso(𝑅, 𝐴) = OrdIso(𝑅, 𝐵)) | ||
Theorem | nfoi 9551 | 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 9552 | Generalize ordiso 9553 to proper classes. (Contributed by Mario Carneiro, 24-Jun-2015.) |
⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ Ord 𝐵) → 𝐴 = 𝐵) | ||
Theorem | ordiso 9553* | 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 9554* | Lemma for ordtype 9569. (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 9555* | Lemma for ordtype 9569. (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 9556* | Lemma for ordtype 9569. (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 9557* | Lemma for ordtype 9569. (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 9558* | Lemma for ordtype 9569. (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 9559* | Lemma for ordtype 9569. (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 9560* | Lemma for ordtype 9569. (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 9561* | Lemma for ordtype 9569. 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 9562* | Lemma for ordtype 9569. (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 9563* | Lemma for ordtype 9569. Either the function OrdIso is an isomorphism onto all of 𝐴, or OrdIso is not a set, which by oif 9567 implies that either ran 𝑂 ⊆ 𝐴 is a proper class or dom 𝑂 = On. (Contributed by Mario Carneiro, 25-Jun-2015.) (Revised by AV, 28-Jul-2024.) |
⊢ 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 9564* | Lemma for ordtype 9569. Using ax-rep 5284, 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 9565 | 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 9566 | 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 9567 | 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 9568 | The order isomorphism of the well-order 𝑅 on 𝐴 is an isomorphism onto ran 𝑂 (which is a subset of 𝐴 by oif 9567). (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 9569 | 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 9570 | 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 9571 | 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 9569 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 9572 | 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 9573 | 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 9574 | 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 9575 | 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 9576 | 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 9577 | 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 5284 (the second statement is trivial under ax-rep 5284). (Contributed by Mario Carneiro, 26-Jun-2015.) |
⊢ 𝐹 = OrdIso( E , 𝐴) ⇒ ⊢ (𝐴 ⊆ On → (Smo 𝐹 ∧ ran 𝐹 = 𝐴)) | ||
Theorem | oiid 9578 | 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 9579* | Lemma for hartogs 9581. (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 9580* | Lemma for hartogs 9581. (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 9581* | The class of ordinals dominated by a given set is an ordinal. A shorter (when taking into account lemmas hartogslem1 9579 and hartogslem2 9580) proof can be given using the axiom of choice, see ondomon 10600. As its label indicates, this result is used to justify the definition of the Hartogs function df-har 9594. (Contributed by Jeff Hankins, 22-Oct-2009.) (Revised by Mario Carneiro, 15-May-2015.) |
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ On) | ||
Theorem | wofib 9582 | 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 9583* | 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 9584* | Lemma for wemapso 9588. Transitivity. (Contributed by Stefan O'Rear, 17-Jan-2015.) (Revised by AV, 21-Jul-2024.) |
⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} & ⊢ (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) | ||
Theorem | wemaplem3 9585* | Lemma for wemapso 9588. Transitivity. (Contributed by Stefan O'Rear, 17-Jan-2015.) (Revised by AV, 21-Jul-2024.) |
⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} & ⊢ (𝜑 → 𝑃 ∈ (𝐵 ↑m 𝐴)) & ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑m 𝐴)) & ⊢ (𝜑 → 𝑄 ∈ (𝐵 ↑m 𝐴)) & ⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → 𝑆 Po 𝐵) & ⊢ (𝜑 → 𝑃𝑇𝑋) & ⊢ (𝜑 → 𝑋𝑇𝑄) ⇒ ⊢ (𝜑 → 𝑃𝑇𝑄) | ||
Theorem | wemappo 9586* |
Construct lexicographic order on a function space based on a
well-ordering of the indices and a total ordering of the values.
Without totality on the values or least differing indices, the best we can prove here is a partial order. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by AV, 21-Jul-2024.) |
⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} ⇒ ⊢ ((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) → 𝑇 Po (𝐵 ↑m 𝐴)) | ||
Theorem | wemapsolem 9587* | Lemma for wemapso 9588. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 21-Jul-2024.) |
⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} & ⊢ 𝑈 ⊆ (𝐵 ↑m 𝐴) & ⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → 𝑆 Or 𝐵) & ⊢ ((𝜑 ∧ ((𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈) ∧ 𝑎 ≠ 𝑏)) → ∃𝑐 ∈ dom (𝑎 ∖ 𝑏)∀𝑑 ∈ dom (𝑎 ∖ 𝑏) ¬ 𝑑𝑅𝑐) ⇒ ⊢ (𝜑 → 𝑇 Or 𝑈) | ||
Theorem | wemapso 9588* | Construct lexicographic order on a function space based on a well-ordering of the indices and a total ordering of the values. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 21-Jul-2024.) |
⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} ⇒ ⊢ ((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) → 𝑇 Or (𝐵 ↑m 𝐴)) | ||
Theorem | wemapso2lem 9589* | Lemma for wemapso2 9590. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 1-Jul-2019.) |
⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} & ⊢ 𝑈 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍} ⇒ ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑅 Or 𝐴 ∧ 𝑆 Or 𝐵) ∧ 𝑍 ∈ 𝑊) → 𝑇 Or 𝑈) | ||
Theorem | wemapso2 9590* | An alternative to having a well-order on 𝑅 in wemapso 9588 is to restrict the function set to finitely-supported functions. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 1-Jul-2019.) |
⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} & ⊢ 𝑈 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍} ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 Or 𝐴 ∧ 𝑆 Or 𝐵) → 𝑇 Or 𝑈) | ||
Theorem | card2on 9591* | The alternate definition of the cardinal of a set given in cardval2 10028 always gives a set, and indeed an ordinal. (Contributed by Mario Carneiro, 14-Jan-2013.) |
⊢ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴} ∈ On | ||
Theorem | card2inf 9592* | The alternate definition of the cardinal of a set given in cardval2 10028 has the curious property that for non-numerable sets (for which ndmfv 6941 yields ∅), it still evaluates to a nonempty set, and indeed it contains ω. (Contributed by Mario Carneiro, 15-Jan-2013.) (Revised by Mario Carneiro, 27-Apr-2015.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (¬ ∃𝑦 ∈ On 𝑦 ≈ 𝐴 → ω ⊆ {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}) | ||
Syntax | char 9593 | Class symbol for the Hartogs function. |
class har | ||
Definition | df-har 9594* |
Define the Hartogs function as mapping a set to the class of ordinals it
dominates. That class is an ordinal by hartogs 9581, which is used in
harf 9595.
The Hartogs number of a set is the least ordinal not dominated by that set. Theorem harval2 10034 proves that the Hartogs function actually gives the Hartogs number for well-orderable sets. The Hartogs number of an ordinal is its cardinal successor. This is proved for finite ordinal in harsucnn 10035. Traditionally, the Hartogs number of a set 𝑋 is written ℵ(𝑋), and its cardinal successor, 𝑋 +; we use functional notation for this, and cannot use the aleph symbol because it is taken for the enumerating function of the infinite initial ordinals df-aleph 9977. Some authors define the Hartogs number of a set to be the least *infinite* ordinal which does not inject into it, thus causing the range to consist only of alephs. We use the simpler definition where the value can be any successor cardinal. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
⊢ har = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦 ≼ 𝑥}) | ||
Theorem | harf 9595 | Functionality of the Hartogs function. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
⊢ har:V⟶On | ||
Theorem | harcl 9596 | Values of the Hartogs function are ordinals (closure of the Hartogs function in the ordinals). (Contributed by Stefan O'Rear, 11-Feb-2015.) |
⊢ (har‘𝑋) ∈ On | ||
Theorem | harval 9597* | Function value of the Hartogs function. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
⊢ (𝑋 ∈ 𝑉 → (har‘𝑋) = {𝑦 ∈ On ∣ 𝑦 ≼ 𝑋}) | ||
Theorem | elharval 9598 | The Hartogs number of a set contains exactly the ordinals that set dominates. Combined with harcl 9596, this implies that the Hartogs number of a set is greater than all ordinals that set dominates. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 15-May-2015.) |
⊢ (𝑌 ∈ (har‘𝑋) ↔ (𝑌 ∈ On ∧ 𝑌 ≼ 𝑋)) | ||
Theorem | harndom 9599 | The Hartogs number of a set does not inject into that set. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 15-May-2015.) |
⊢ ¬ (har‘𝑋) ≼ 𝑋 | ||
Theorem | harword 9600 | Weak ordering property of the Hartogs function. (Contributed by Stefan O'Rear, 14-Feb-2015.) |
⊢ (𝑋 ≼ 𝑌 → (har‘𝑋) ⊆ (har‘𝑌)) |
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