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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | r1ordg 9201 | Ordering relation for the cumulative hierarchy of sets. Part of Proposition 9.10(2) of [TakeutiZaring] p. 77. (Contributed by NM, 8-Sep-2003.) |
⊢ (𝐵 ∈ dom 𝑅1 → (𝐴 ∈ 𝐵 → (𝑅1‘𝐴) ∈ (𝑅1‘𝐵))) | ||
Theorem | r1ord3g 9202 | Ordering relation for the cumulative hierarchy of sets. Part of Theorem 3.3(i) of [BellMachover] p. 478. (Contributed by NM, 22-Sep-2003.) |
⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ⊆ 𝐵 → (𝑅1‘𝐴) ⊆ (𝑅1‘𝐵))) | ||
Theorem | r1ord 9203 | Ordering relation for the cumulative hierarchy of sets. Part of Proposition 9.10(2) of [TakeutiZaring] p. 77. (Contributed by NM, 8-Sep-2003.) (Revised by Mario Carneiro, 16-Nov-2014.) |
⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → (𝑅1‘𝐴) ∈ (𝑅1‘𝐵))) | ||
Theorem | r1ord2 9204 | Ordering relation for the cumulative hierarchy of sets. Part of Proposition 9.10(2) of [TakeutiZaring] p. 77. (Contributed by NM, 22-Sep-2003.) |
⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → (𝑅1‘𝐴) ⊆ (𝑅1‘𝐵))) | ||
Theorem | r1ord3 9205 | Ordering relation for the cumulative hierarchy of sets. Part of Theorem 3.3(i) of [BellMachover] p. 478. (Contributed by NM, 22-Sep-2003.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝑅1‘𝐴) ⊆ (𝑅1‘𝐵))) | ||
Theorem | r1sssuc 9206 | The value of the cumulative hierarchy of sets function is a subset of its value at the successor. JFM CLASSES1 Th. 39. (Contributed by FL, 20-Apr-2011.) |
⊢ (𝐴 ∈ On → (𝑅1‘𝐴) ⊆ (𝑅1‘suc 𝐴)) | ||
Theorem | r1pwss 9207 | Each set of the cumulative hierarchy is closed under subsets. (Contributed by Mario Carneiro, 16-Nov-2014.) |
⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝒫 𝐴 ⊆ (𝑅1‘𝐵)) | ||
Theorem | r1sscl 9208 | Each set of the cumulative hierarchy is closed under subsets. (Contributed by Mario Carneiro, 16-Nov-2014.) |
⊢ ((𝐴 ∈ (𝑅1‘𝐵) ∧ 𝐶 ⊆ 𝐴) → 𝐶 ∈ (𝑅1‘𝐵)) | ||
Theorem | r1val1 9209* | The value of the cumulative hierarchy of sets function expressed recursively. Theorem 7Q of [Enderton] p. 202. (Contributed by NM, 25-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐴 ∈ dom 𝑅1 → (𝑅1‘𝐴) = ∪ 𝑥 ∈ 𝐴 𝒫 (𝑅1‘𝑥)) | ||
Theorem | tz9.12lem1 9210* | Lemma for tz9.12 9213. (Contributed by NM, 22-Sep-2003.) (Revised by Mario Carneiro, 11-Sep-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐹 = (𝑧 ∈ V ↦ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)}) ⇒ ⊢ (𝐹 “ 𝐴) ⊆ On | ||
Theorem | tz9.12lem2 9211* | Lemma for tz9.12 9213. (Contributed by NM, 22-Sep-2003.) |
⊢ 𝐴 ∈ V & ⊢ 𝐹 = (𝑧 ∈ V ↦ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)}) ⇒ ⊢ suc ∪ (𝐹 “ 𝐴) ∈ On | ||
Theorem | tz9.12lem3 9212* | Lemma for tz9.12 9213. (Contributed by NM, 22-Sep-2003.) (Revised by Mario Carneiro, 11-Sep-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐹 = (𝑧 ∈ V ↦ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)}) ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦) → 𝐴 ∈ (𝑅1‘suc suc ∪ (𝐹 “ 𝐴))) | ||
Theorem | tz9.12 9213* | A set is well-founded if all of its elements are well-founded. Proposition 9.12 of [TakeutiZaring] p. 78. The main proof consists of tz9.12lem1 9210 through tz9.12lem3 9212. (Contributed by NM, 22-Sep-2003.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦) → ∃𝑦 ∈ On 𝐴 ∈ (𝑅1‘𝑦)) | ||
Theorem | tz9.13 9214* | Every set is well-founded, assuming the Axiom of Regularity. In other words, every set belongs to a layer of the cumulative hierarchy of sets. Proposition 9.13 of [TakeutiZaring] p. 78. (Contributed by NM, 23-Sep-2003.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥) | ||
Theorem | tz9.13g 9215* | Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13 9214 expresses the class existence requirement as an antecedent. (Contributed by NM, 4-Oct-2003.) |
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥)) | ||
Theorem | rankwflemb 9216* | Two ways of saying a set is well-founded. (Contributed by NM, 11-Oct-2003.) (Revised by Mario Carneiro, 16-Nov-2014.) |
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥)) | ||
Theorem | rankf 9217 | The domain and range of the rank function. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 12-Sep-2013.) |
⊢ rank:∪ (𝑅1 “ On)⟶On | ||
Theorem | rankon 9218 | The rank of a set is an ordinal number. Proposition 9.15(1) of [TakeutiZaring] p. 79. (Contributed by NM, 5-Oct-2003.) (Revised by Mario Carneiro, 12-Sep-2013.) |
⊢ (rank‘𝐴) ∈ On | ||
Theorem | r1elwf 9219 | Any member of the cumulative hierarchy is well-founded. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 16-Nov-2014.) |
⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On)) | ||
Theorem | rankvalb 9220* | Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 9239 does not use Regularity, and so requires the assumption that 𝐴 is in the range of 𝑅1. (Contributed by NM, 11-Oct-2003.) (Revised by Mario Carneiro, 10-Sep-2013.) |
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}) | ||
Theorem | rankr1ai 9221 | One direction of rankr1a 9259. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐴 ∈ (𝑅1‘𝐵) → (rank‘𝐴) ∈ 𝐵) | ||
Theorem | rankvaln 9222 | Value of the rank function at a non-well-founded set. (The antecedent is always false under Foundation, by unir1 9236, unless 𝐴 is a proper class.) (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 10-Sep-2013.) |
⊢ (¬ 𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∅) | ||
Theorem | rankidb 9223 | Identity law for the rank function. (Contributed by NM, 3-Oct-2003.) (Revised by Mario Carneiro, 22-Mar-2013.) |
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))) | ||
Theorem | rankdmr1 9224 | A rank is a member of the cumulative hierarchy. (Contributed by Mario Carneiro, 17-Nov-2014.) |
⊢ (rank‘𝐴) ∈ dom 𝑅1 | ||
Theorem | rankr1ag 9225 | A version of rankr1a 9259 that is suitable without assuming Regularity or Replacement. (Contributed by Mario Carneiro, 3-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝐴) ∈ 𝐵)) | ||
Theorem | rankr1bg 9226 | A relationship between rank and 𝑅1. See rankr1ag 9225 for the membership version. (Contributed by Mario Carneiro, 17-Nov-2014.) |
⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ⊆ (𝑅1‘𝐵) ↔ (rank‘𝐴) ⊆ 𝐵)) | ||
Theorem | r1rankidb 9227 | Any set is a subset of the hierarchy of its rank. (Contributed by Mario Carneiro, 3-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴))) | ||
Theorem | r1elssi 9228 | The range of the 𝑅1 function is transitive. Lemma 2.10 of [Kunen] p. 97. One direction of r1elss 9229 that doesn't need 𝐴 to be a set. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 16-Nov-2014.) |
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ ∪ (𝑅1 “ On)) | ||
Theorem | r1elss 9229 | The range of the 𝑅1 function is transitive. Lemma 2.10 of [Kunen] p. 97. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 16-Nov-2014.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝐴 ⊆ ∪ (𝑅1 “ On)) | ||
Theorem | pwwf 9230 | A power set is well-founded iff the base set is. (Contributed by Mario Carneiro, 8-Jun-2013.) (Revised by Mario Carneiro, 16-Nov-2014.) |
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝐴 ∈ ∪ (𝑅1 “ On)) | ||
Theorem | sswf 9231 | A subset of a well-founded set is well-founded. (Contributed by Mario Carneiro, 17-Nov-2014.) |
⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ ∪ (𝑅1 “ On)) | ||
Theorem | snwf 9232 | A singleton is well-founded if its element is. (Contributed by Mario Carneiro, 10-Jun-2013.) (Revised by Mario Carneiro, 16-Nov-2014.) |
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝐴} ∈ ∪ (𝑅1 “ On)) | ||
Theorem | unwf 9233 | A binary union is well-founded iff its elements are. (Contributed by Mario Carneiro, 10-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) ↔ (𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On)) | ||
Theorem | prwf 9234 | An unordered pair is well-founded if its elements are. (Contributed by Mario Carneiro, 10-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → {𝐴, 𝐵} ∈ ∪ (𝑅1 “ On)) | ||
Theorem | opwf 9235 | An ordered pair is well-founded if its elements are. (Contributed by Mario Carneiro, 10-Jun-2013.) |
⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → 〈𝐴, 𝐵〉 ∈ ∪ (𝑅1 “ On)) | ||
Theorem | unir1 9236 | The cumulative hierarchy of sets covers the universe. Proposition 4.45 (b) to (a) of [Mendelson] p. 281. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 8-Jun-2013.) |
⊢ ∪ (𝑅1 “ On) = V | ||
Theorem | jech9.3 9237 | Every set belongs to some value of the cumulative hierarchy of sets function 𝑅1, i.e. the indexed union of all values of 𝑅1 is the universe. Lemma 9.3 of [Jech] p. 71. (Contributed by NM, 4-Oct-2003.) (Revised by Mario Carneiro, 8-Jun-2013.) |
⊢ ∪ 𝑥 ∈ On (𝑅1‘𝑥) = V | ||
Theorem | rankwflem 9238* | Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13g 9215 is useful in proofs of theorems about the rank function. (Contributed by NM, 4-Oct-2003.) |
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥)) | ||
Theorem | rankval 9239* | Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). (Contributed by NM, 24-Sep-2003.) (Revised by Mario Carneiro, 10-Sep-2013.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)} | ||
Theorem | rankvalg 9240* | Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 9239 expresses the class existence requirement as an antecedent instead of a hypothesis. (Contributed by NM, 5-Oct-2003.) |
⊢ (𝐴 ∈ 𝑉 → (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)}) | ||
Theorem | rankval2 9241* | Value of an alternate definition of the rank function. Definition of [BellMachover] p. 478. (Contributed by NM, 8-Oct-2003.) |
⊢ (𝐴 ∈ 𝐵 → (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (𝑅1‘𝑥)}) | ||
Theorem | uniwf 9242 | A union is well-founded iff the base set is. (Contributed by Mario Carneiro, 8-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∪ 𝐴 ∈ ∪ (𝑅1 “ On)) | ||
Theorem | rankr1clem 9243 | Lemma for rankr1c 9244. (Contributed by NM, 6-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (¬ 𝐴 ∈ (𝑅1‘𝐵) ↔ 𝐵 ⊆ (rank‘𝐴))) | ||
Theorem | rankr1c 9244 | A relationship between the rank function and the cumulative hierarchy of sets function 𝑅1. Proposition 9.15(2) of [TakeutiZaring] p. 79. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝐵 = (rank‘𝐴) ↔ (¬ 𝐴 ∈ (𝑅1‘𝐵) ∧ 𝐴 ∈ (𝑅1‘suc 𝐵)))) | ||
Theorem | rankidn 9245 | A relationship between the rank function and the cumulative hierarchy of sets function 𝑅1. (Contributed by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ¬ 𝐴 ∈ (𝑅1‘(rank‘𝐴))) | ||
Theorem | rankpwi 9246 | The rank of a power set. Part of Exercise 30 of [Enderton] p. 207. (Contributed by Mario Carneiro, 3-Jun-2013.) |
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝒫 𝐴) = suc (rank‘𝐴)) | ||
Theorem | rankelb 9247 | The membership relation is inherited by the rank function. Proposition 9.16 of [TakeutiZaring] p. 79. (Contributed by NM, 4-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → (𝐴 ∈ 𝐵 → (rank‘𝐴) ∈ (rank‘𝐵))) | ||
Theorem | wfelirr 9248 | A well-founded set is not a member of itself. This proof does not require the axiom of regularity, unlike elirr 9055. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ¬ 𝐴 ∈ 𝐴) | ||
Theorem | rankval3b 9249* | The value of the rank function expressed recursively: the rank of a set is the smallest ordinal number containing the ranks of all members of the set. Proposition 9.17 of [TakeutiZaring] p. 79. (Contributed by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥}) | ||
Theorem | ranksnb 9250 | The rank of a singleton. Theorem 15.17(v) of [Monk1] p. 112. (Contributed by Mario Carneiro, 10-Jun-2013.) |
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘{𝐴}) = suc (rank‘𝐴)) | ||
Theorem | rankonidlem 9251 | Lemma for rankonid 9252. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 22-Mar-2013.) |
⊢ (𝐴 ∈ dom 𝑅1 → (𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴)) | ||
Theorem | rankonid 9252 | The rank of an ordinal number is itself. Proposition 9.18 of [TakeutiZaring] p. 79 and its converse. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐴 ∈ dom 𝑅1 ↔ (rank‘𝐴) = 𝐴) | ||
Theorem | onwf 9253 | The ordinals are all well-founded. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ On ⊆ ∪ (𝑅1 “ On) | ||
Theorem | onssr1 9254 | Initial segments of the ordinals are contained in initial segments of the cumulative hierarchy. (Contributed by FL, 20-Apr-2011.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐴 ∈ dom 𝑅1 → 𝐴 ⊆ (𝑅1‘𝐴)) | ||
Theorem | rankr1g 9255 | A relationship between the rank function and the cumulative hierarchy of sets function 𝑅1. Proposition 9.15(2) of [TakeutiZaring] p. 79. (Contributed by NM, 6-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐴 ∈ 𝑉 → (𝐵 = (rank‘𝐴) ↔ (¬ 𝐴 ∈ (𝑅1‘𝐵) ∧ 𝐴 ∈ (𝑅1‘suc 𝐵)))) | ||
Theorem | rankid 9256 | Identity law for the rank function. (Contributed by NM, 3-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐴 ∈ V ⇒ ⊢ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) | ||
Theorem | rankr1 9257 | A relationship between the rank function and the cumulative hierarchy of sets function 𝑅1. Proposition 9.15(2) of [TakeutiZaring] p. 79. (Contributed by NM, 6-Oct-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐵 = (rank‘𝐴) ↔ (¬ 𝐴 ∈ (𝑅1‘𝐵) ∧ 𝐴 ∈ (𝑅1‘suc 𝐵))) | ||
Theorem | ssrankr1 9258 | A relationship between an ordinal number less than or equal to a rank, and the cumulative hierarchy of sets 𝑅1. Proposition 9.15(3) of [TakeutiZaring] p. 79. (Contributed by NM, 8-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐵 ∈ On → (𝐵 ⊆ (rank‘𝐴) ↔ ¬ 𝐴 ∈ (𝑅1‘𝐵))) | ||
Theorem | rankr1a 9259 | A relationship between rank and 𝑅1, clearly equivalent to ssrankr1 9258 and friends through trichotomy, but in Raph's opinion considerably more intuitive. See rankr1b 9287 for the subset version. (Contributed by Raph Levien, 29-May-2004.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐵 ∈ On → (𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝐴) ∈ 𝐵)) | ||
Theorem | r1val2 9260* | The value of the cumulative hierarchy of sets function expressed in terms of rank. Definition 15.19 of [Monk1] p. 113. (Contributed by NM, 30-Nov-2003.) |
⊢ (𝐴 ∈ On → (𝑅1‘𝐴) = {𝑥 ∣ (rank‘𝑥) ∈ 𝐴}) | ||
Theorem | r1val3 9261* | The value of the cumulative hierarchy of sets function expressed in terms of rank. Theorem 15.18 of [Monk1] p. 113. (Contributed by NM, 30-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐴 ∈ On → (𝑅1‘𝐴) = ∪ 𝑥 ∈ 𝐴 𝒫 {𝑦 ∣ (rank‘𝑦) ∈ 𝑥}) | ||
Theorem | rankel 9262 | The membership relation is inherited by the rank function. Proposition 9.16 of [TakeutiZaring] p. 79. (Contributed by NM, 4-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ∈ 𝐵 → (rank‘𝐴) ∈ (rank‘𝐵)) | ||
Theorem | rankval3 9263* | The value of the rank function expressed recursively: the rank of a set is the smallest ordinal number containing the ranks of all members of the set. Proposition 9.17 of [TakeutiZaring] p. 79. (Contributed by NM, 11-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 (rank‘𝑦) ∈ 𝑥} | ||
Theorem | bndrank 9264* | Any class whose elements have bounded rank is a set. Proposition 9.19 of [TakeutiZaring] p. 80. (Contributed by NM, 13-Oct-2003.) |
⊢ (∃𝑥 ∈ On ∀𝑦 ∈ 𝐴 (rank‘𝑦) ⊆ 𝑥 → 𝐴 ∈ V) | ||
Theorem | unbndrank 9265* | The elements of a proper class have unbounded rank. Exercise 2 of [TakeutiZaring] p. 80. (Contributed by NM, 13-Oct-2003.) |
⊢ (¬ 𝐴 ∈ V → ∀𝑥 ∈ On ∃𝑦 ∈ 𝐴 𝑥 ∈ (rank‘𝑦)) | ||
Theorem | rankpw 9266 | The rank of a power set. Part of Exercise 30 of [Enderton] p. 207. (Contributed by NM, 22-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (rank‘𝒫 𝐴) = suc (rank‘𝐴) | ||
Theorem | ranklim 9267 | The rank of a set belongs to a limit ordinal iff the rank of its power set does. (Contributed by NM, 18-Sep-2006.) |
⊢ (Lim 𝐵 → ((rank‘𝐴) ∈ 𝐵 ↔ (rank‘𝒫 𝐴) ∈ 𝐵)) | ||
Theorem | r1pw 9268 | A stronger property of 𝑅1 than rankpw 9266. The latter merely proves that 𝑅1 of the successor is a power set, but here we prove that if 𝐴 is in the cumulative hierarchy, then 𝒫 𝐴 is in the cumulative hierarchy of the successor. (Contributed by Raph Levien, 29-May-2004.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐵 ∈ On → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵))) | ||
Theorem | r1pwALT 9269 | Alternate shorter proof of r1pw 9268 based on the additional axioms ax-reg 9050 and ax-inf2 9098. (Contributed by Raph Levien, 29-May-2004.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐵 ∈ On → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵))) | ||
Theorem | r1pwcl 9270 | The cumulative hierarchy of a limit ordinal is closed under power set. (Contributed by Raph Levien, 29-May-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2014.) |
⊢ (Lim 𝐵 → (𝐴 ∈ (𝑅1‘𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘𝐵))) | ||
Theorem | rankssb 9271 | The subset relation is inherited by the rank function. Exercise 1 of [TakeutiZaring] p. 80. (Contributed by NM, 25-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → (𝐴 ⊆ 𝐵 → (rank‘𝐴) ⊆ (rank‘𝐵))) | ||
Theorem | rankss 9272 | The subset relation is inherited by the rank function. Exercise 1 of [TakeutiZaring] p. 80. (Contributed by NM, 25-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ⊆ 𝐵 → (rank‘𝐴) ⊆ (rank‘𝐵)) | ||
Theorem | rankunb 9273 | The rank of the union of two sets. Theorem 15.17(iii) of [Monk1] p. 112. (Contributed by Mario Carneiro, 10-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵))) | ||
Theorem | rankprb 9274 | The rank of an unordered pair. Part of Exercise 30 of [Enderton] p. 207. (Contributed by Mario Carneiro, 10-Jun-2013.) |
⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘{𝐴, 𝐵}) = suc ((rank‘𝐴) ∪ (rank‘𝐵))) | ||
Theorem | rankopb 9275 | The rank of an ordered pair. Part of Exercise 4 of [Kunen] p. 107. (Contributed by Mario Carneiro, 10-Jun-2013.) |
⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (rank‘〈𝐴, 𝐵〉) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵))) | ||
Theorem | rankuni2b 9276* | The value of the rank function expressed recursively: the rank of a set is the smallest ordinal number containing the ranks of all members of the set. Proposition 9.17 of [TakeutiZaring] p. 79. (Contributed by Mario Carneiro, 8-Jun-2013.) |
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘∪ 𝐴) = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥)) | ||
Theorem | ranksn 9277 | The rank of a singleton. Theorem 15.17(v) of [Monk1] p. 112. (Contributed by NM, 28-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (rank‘{𝐴}) = suc (rank‘𝐴) | ||
Theorem | rankuni2 9278* | The rank of a union. Part of Theorem 15.17(iv) of [Monk1] p. 112. (Contributed by NM, 30-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (rank‘∪ 𝐴) = ∪ 𝑥 ∈ 𝐴 (rank‘𝑥) | ||
Theorem | rankun 9279 | The rank of the union of two sets. Theorem 15.17(iii) of [Monk1] p. 112. (Contributed by NM, 26-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)) | ||
Theorem | rankpr 9280 | The rank of an unordered pair. Part of Exercise 30 of [Enderton] p. 207. (Contributed by NM, 28-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (rank‘{𝐴, 𝐵}) = suc ((rank‘𝐴) ∪ (rank‘𝐵)) | ||
Theorem | rankop 9281 | The rank of an ordered pair. Part of Exercise 4 of [Kunen] p. 107. (Contributed by NM, 13-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (rank‘〈𝐴, 𝐵〉) = suc suc ((rank‘𝐴) ∪ (rank‘𝐵)) | ||
Theorem | r1rankid 9282 | Any set is a subset of the hierarchy of its rank. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (𝑅1‘(rank‘𝐴))) | ||
Theorem | rankeq0b 9283 | A set is empty iff its rank is empty. (Contributed by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝐴 = ∅ ↔ (rank‘𝐴) = ∅)) | ||
Theorem | rankeq0 9284 | A set is empty iff its rank is empty. (Contributed by NM, 18-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 = ∅ ↔ (rank‘𝐴) = ∅) | ||
Theorem | rankr1id 9285 | The rank of the hierarchy of an ordinal number is itself. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐴 ∈ dom 𝑅1 ↔ (rank‘(𝑅1‘𝐴)) = 𝐴) | ||
Theorem | rankuni 9286 | The rank of a union. Part of Exercise 4 of [Kunen] p. 107. (Contributed by NM, 15-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ (rank‘∪ 𝐴) = ∪ (rank‘𝐴) | ||
Theorem | rankr1b 9287 | A relationship between rank and 𝑅1. See rankr1a 9259 for the membership version. (Contributed by NM, 15-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐵 ∈ On → (𝐴 ⊆ (𝑅1‘𝐵) ↔ (rank‘𝐴) ⊆ 𝐵)) | ||
Theorem | ranksuc 9288 | The rank of a successor. (Contributed by NM, 18-Sep-2006.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (rank‘suc 𝐴) = suc (rank‘𝐴) | ||
Theorem | rankuniss 9289 | Upper bound of the rank of a union. Part of Exercise 30 of [Enderton] p. 207. (Contributed by NM, 30-Nov-2003.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (rank‘∪ 𝐴) ⊆ (rank‘𝐴) | ||
Theorem | rankval4 9290* | The rank of a set is the supremum of the successors of the ranks of its members. Exercise 9.1 of [Jech] p. 72. Also a special case of Theorem 7V(b) of [Enderton] p. 204. (Contributed by NM, 12-Oct-2003.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (rank‘𝐴) = ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥) | ||
Theorem | rankbnd 9291* | The rank of a set is bounded by a bound for the successor of its members. (Contributed by NM, 18-Sep-2006.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (∀𝑥 ∈ 𝐴 suc (rank‘𝑥) ⊆ 𝐵 ↔ (rank‘𝐴) ⊆ 𝐵) | ||
Theorem | rankbnd2 9292* | The rank of a set is bounded by the successor of a bound for its members. (Contributed by NM, 15-Sep-2006.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐵 ∈ On → (∀𝑥 ∈ 𝐴 (rank‘𝑥) ⊆ 𝐵 ↔ (rank‘𝐴) ⊆ suc 𝐵)) | ||
Theorem | rankc1 9293* | A relationship that can be used for computation of rank. (Contributed by NM, 16-Sep-2006.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ (rank‘∪ 𝐴) ↔ (rank‘𝐴) = (rank‘∪ 𝐴)) | ||
Theorem | rankc2 9294* | A relationship that can be used for computation of rank. (Contributed by NM, 16-Sep-2006.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (∃𝑥 ∈ 𝐴 (rank‘𝑥) = (rank‘∪ 𝐴) → (rank‘𝐴) = suc (rank‘∪ 𝐴)) | ||
Theorem | rankelun 9295 | Rank membership is inherited by union. (Contributed by NM, 18-Sep-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘(𝐴 ∪ 𝐵)) ∈ (rank‘(𝐶 ∪ 𝐷))) | ||
Theorem | rankelpr 9296 | Rank membership is inherited by unordered pairs. (Contributed by NM, 18-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘{𝐴, 𝐵}) ∈ (rank‘{𝐶, 𝐷})) | ||
Theorem | rankelop 9297 | Rank membership is inherited by ordered pairs. (Contributed by NM, 18-Sep-2006.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ (((rank‘𝐴) ∈ (rank‘𝐶) ∧ (rank‘𝐵) ∈ (rank‘𝐷)) → (rank‘〈𝐴, 𝐵〉) ∈ (rank‘〈𝐶, 𝐷〉)) | ||
Theorem | rankxpl 9298 | A lower bound on the rank of a Cartesian product. (Contributed by NM, 18-Sep-2006.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ((𝐴 × 𝐵) ≠ ∅ → (rank‘(𝐴 ∪ 𝐵)) ⊆ (rank‘(𝐴 × 𝐵))) | ||
Theorem | rankxpu 9299 | An upper bound on the rank of a Cartesian product. (Contributed by NM, 18-Sep-2006.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (rank‘(𝐴 × 𝐵)) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵)) | ||
Theorem | rankfu 9300 | An upper bound on the rank of a function. (Contributed by Gérard Lang, 5-Aug-2018.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐹:𝐴⟶𝐵 → (rank‘𝐹) ⊆ suc suc (rank‘(𝐴 ∪ 𝐵))) |
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