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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | frr3 9801* | Law of general well-founded recursion, part three. Finally, we show that 𝐹 is unique. We do this by showing that any function 𝐻 with the same properties we proved of 𝐹 in frr1 9799 and frr2 9800 is identical to 𝐹. (Contributed by Scott Fenton, 11-Sep-2023.) |
| ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) ⇒ ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐻 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐻‘𝑧) = (𝑧𝐺(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝐹 = 𝐻) | ||
| Syntax | cr1 9802 | Extend class definition to include the cumulative hierarchy of sets function. |
| class 𝑅1 | ||
| Syntax | crnk 9803 | Extend class definition to include rank function. |
| class rank | ||
| Definition | df-r1 9804 | Define the cumulative hierarchy of sets function, using Takeuti and Zaring's notation (𝑅1). Starting with the empty set, this function builds up layers of sets where the next layer is the power set of the previous layer (and the union of previous layers when the argument is a limit ordinal). Using the Axiom of Regularity, we can show that any set whatsoever belongs to one of the layers of this hierarchy (see tz9.13 9831). Our definition expresses Definition 9.9 of [TakeutiZaring] p. 76 in a closed form, from which we derive the recursive definition as Theorems r10 9808, r1suc 9810, and r1lim 9812. Theorem r1val1 9826 shows a recursive definition that works for all values, and Theorems r1val2 9877 and r1val3 9878 show the value expressed in terms of rank. Other notations for this function are R with the argument as a subscript (Equation 3.1 of [BellMachover] p. 477), V with a subscript (Definition of [Enderton] p. 202), M with a subscript (Definition 15.19 of [Monk1] p. 113), the capital Greek letter psi (Definition of [Mendelson] p. 281), and bold-face R (Definition 2.1 of [Kunen] p. 95). (Contributed by NM, 2-Sep-2003.) |
| ⊢ 𝑅1 = rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) | ||
| Definition | df-rank 9805* | Define the rank function. See rankval 9856, rankval2 9858, rankval3 9880, or rankval4 9907 its value. The rank is a kind of "inverse" of the cumulative hierarchy of sets function 𝑅1: given a set, it returns an ordinal number telling us the smallest layer of the hierarchy to which the set belongs. Based on Definition 9.14 of [TakeutiZaring] p. 79. Theorem rankid 9873 illustrates the "inverse" concept. Another nice theorem showing the relationship is rankr1a 9876. (Contributed by NM, 11-Oct-2003.) |
| ⊢ rank = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)}) | ||
| Theorem | r1funlim 9806 | The cumulative hierarchy of sets function is a function on a limit ordinal. (This weak form of r1fnon 9807 avoids ax-rep 5279.) (Contributed by Mario Carneiro, 16-Nov-2014.) |
| ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | ||
| Theorem | r1fnon 9807 | The cumulative hierarchy of sets function is a function on the class of ordinal numbers. (Contributed by NM, 5-Oct-2003.) (Revised by Mario Carneiro, 10-Sep-2013.) |
| ⊢ 𝑅1 Fn On | ||
| Theorem | r10 9808 | Value of the cumulative hierarchy of sets function at ∅. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by NM, 2-Sep-2003.) (Revised by Mario Carneiro, 10-Sep-2013.) |
| ⊢ (𝑅1‘∅) = ∅ | ||
| Theorem | r1sucg 9809 | Value of the cumulative hierarchy of sets function at a successor ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by Mario Carneiro, 16-Nov-2014.) |
| ⊢ (𝐴 ∈ dom 𝑅1 → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1‘𝐴)) | ||
| Theorem | r1suc 9810 | Value of the cumulative hierarchy of sets function at a successor ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by NM, 2-Sep-2003.) (Revised by Mario Carneiro, 10-Sep-2013.) |
| ⊢ (𝐴 ∈ On → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1‘𝐴)) | ||
| Theorem | r1limg 9811* | Value of the cumulative hierarchy of sets function at a limit ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by Mario Carneiro, 16-Nov-2014.) |
| ⊢ ((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → (𝑅1‘𝐴) = ∪ 𝑥 ∈ 𝐴 (𝑅1‘𝑥)) | ||
| Theorem | r1lim 9812* | Value of the cumulative hierarchy of sets function at a limit ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by NM, 4-Oct-2003.) (Revised by Mario Carneiro, 16-Nov-2014.) |
| ⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → (𝑅1‘𝐴) = ∪ 𝑥 ∈ 𝐴 (𝑅1‘𝑥)) | ||
| Theorem | r1fin 9813 | The first ω levels of the cumulative hierarchy are all finite. (Contributed by Mario Carneiro, 15-May-2013.) |
| ⊢ (𝐴 ∈ ω → (𝑅1‘𝐴) ∈ Fin) | ||
| Theorem | r1sdom 9814 | Each stage in the cumulative hierarchy is strictly larger than the last. (Contributed by Mario Carneiro, 19-Apr-2013.) |
| ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → (𝑅1‘𝐵) ≺ (𝑅1‘𝐴)) | ||
| Theorem | r111 9815 | The cumulative hierarchy is a one-to-one function. (Contributed by Mario Carneiro, 19-Apr-2013.) |
| ⊢ 𝑅1:On–1-1→V | ||
| Theorem | r1tr 9816 | The cumulative hierarchy of sets is transitive. Lemma 7T of [Enderton] p. 202. (Contributed by NM, 8-Sep-2003.) (Revised by Mario Carneiro, 16-Nov-2014.) |
| ⊢ Tr (𝑅1‘𝐴) | ||
| Theorem | r1tr2 9817 | The union of a cumulative hierarchy of sets at ordinal 𝐴 is a subset of the hierarchy at 𝐴. JFM CLASSES1 th. 40. (Contributed by FL, 20-Apr-2011.) |
| ⊢ ∪ (𝑅1‘𝐴) ⊆ (𝑅1‘𝐴) | ||
| Theorem | r1ordg 9818 | 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 9819 | 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 9820 | 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 9821 | 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 9822 | 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 9823 | 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 9824 | 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 9825 | 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 9826* | 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 9827* | Lemma for tz9.12 9830. (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 9828* | Lemma for tz9.12 9830. (Contributed by NM, 22-Sep-2003.) |
| ⊢ 𝐴 ∈ V & ⊢ 𝐹 = (𝑧 ∈ V ↦ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)}) ⇒ ⊢ suc ∪ (𝐹 “ 𝐴) ∈ On | ||
| Theorem | tz9.12lem3 9829* | Lemma for tz9.12 9830. (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 9830* | 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 9827 through tz9.12lem3 9829. (Contributed by NM, 22-Sep-2003.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦) → ∃𝑦 ∈ On 𝐴 ∈ (𝑅1‘𝑦)) | ||
| Theorem | tz9.13 9831* | 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 9832* | Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13 9831 expresses the class existence requirement as an antecedent. (Contributed by NM, 4-Oct-2003.) |
| ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥)) | ||
| Theorem | rankwflemb 9833* | 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 9834 | The domain and codomain of the rank function. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 12-Sep-2013.) |
| ⊢ rank:∪ (𝑅1 “ On)⟶On | ||
| Theorem | rankon 9835 | 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 9836 | 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 9837* | Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 9856 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 9838 | One direction of rankr1a 9876. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 17-Nov-2014.) |
| ⊢ (𝐴 ∈ (𝑅1‘𝐵) → (rank‘𝐴) ∈ 𝐵) | ||
| Theorem | rankvaln 9839 | Value of the rank function at a non-well-founded set. (The antecedent is always false under Foundation, by unir1 9853, 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 9840 | 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 9841 | A rank is a member of the cumulative hierarchy. (Contributed by Mario Carneiro, 17-Nov-2014.) |
| ⊢ (rank‘𝐴) ∈ dom 𝑅1 | ||
| Theorem | rankr1ag 9842 | A version of rankr1a 9876 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 9843 | A relationship between rank and 𝑅1. See rankr1ag 9842 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 9844 | 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 9845 | The range of the 𝑅1 function is transitive. Lemma 2.10 of [Kunen] p. 97. One direction of r1elss 9846 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 9846 | 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 9847 | 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 9848 | 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 9849 | 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 9850 | 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 9851 | 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 9852 | 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 9853 | 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 9854 | 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 9855* | Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13g 9832 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 9856* | 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 9857* | Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 9856 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 9858* | 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 9859 | 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 9860 | Lemma for rankr1c 9861. (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 9861 | 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 9862 | 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 9863 | 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 9864 | 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 9865 | A well-founded set is not a member of itself. This proof does not require the axiom of regularity, unlike elirr 9637. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ¬ 𝐴 ∈ 𝐴) | ||
| Theorem | rankval3b 9866* | 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 9867 | 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 9868 | Lemma for rankonid 9869. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 22-Mar-2013.) |
| ⊢ (𝐴 ∈ dom 𝑅1 → (𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴)) | ||
| Theorem | rankonid 9869 | 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 9870 | 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 9871 | 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 9872 | 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 9873 | 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 9874 | 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 9875 | 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 9876 | A relationship between rank and 𝑅1, clearly equivalent to ssrankr1 9875 and friends through trichotomy, but in Raph's opinion considerably more intuitive. See rankr1b 9904 for the subset version. (Contributed by Raph Levien, 29-May-2004.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ (𝐵 ∈ On → (𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝐴) ∈ 𝐵)) | ||
| Theorem | r1val2 9877* | 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 9878* | 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 9879 | 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 9880* | 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 9881* | 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 9882* | 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 9883 | 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 9884 | 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 9885 | A stronger property of 𝑅1 than rankpw 9883. 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 9886 | Alternate shorter proof of r1pw 9885 based on the additional axioms ax-reg 9632 and ax-inf2 9681. (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 9887 | 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 9888 | 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 9889 | 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 9890 | 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 9891 | 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 9892 | 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 9893* | 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 9894 | 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 9895* | 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 9896 | 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 9897 | 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 9898 | 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 9899 | 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 9900 | A set is empty iff its rank is empty. (Contributed by Mario Carneiro, 17-Nov-2014.) |
| ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝐴 = ∅ ↔ (rank‘𝐴) = ∅)) | ||
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