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Theorem List for Metamath Proof Explorer - 8801-8900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremr1tr 8801 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 (𝑅1𝐴)

Theoremr1tr2 8802 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.)
(𝑅1𝐴) ⊆ (𝑅1𝐴)

Theoremr1ordg 8803 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 → (𝐴𝐵 → (𝑅1𝐴) ∈ (𝑅1𝐵)))

Theoremr1ord3g 8804 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 𝑅1𝐵 ∈ dom 𝑅1) → (𝐴𝐵 → (𝑅1𝐴) ⊆ (𝑅1𝐵)))

Theoremr1ord 8805 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 → (𝐴𝐵 → (𝑅1𝐴) ∈ (𝑅1𝐵)))

Theoremr1ord2 8806 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 → (𝐴𝐵 → (𝑅1𝐴) ⊆ (𝑅1𝐵)))

Theoremr1ord3 8807 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) → (𝐴𝐵 → (𝑅1𝐴) ⊆ (𝑅1𝐵)))

Theoremr1sssuc 8808 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 → (𝑅1𝐴) ⊆ (𝑅1‘suc 𝐴))

Theoremr1pwss 8809 Each set of the cumulative hierarchy is closed under subsets. (Contributed by Mario Carneiro, 16-Nov-2014.)
(𝐴 ∈ (𝑅1𝐵) → 𝒫 𝐴 ⊆ (𝑅1𝐵))

Theoremr1sscl 8810 Each set of the cumulative hierarchy is closed under subsets. (Contributed by Mario Carneiro, 16-Nov-2014.)
((𝐴 ∈ (𝑅1𝐵) ∧ 𝐶𝐴) → 𝐶 ∈ (𝑅1𝐵))

Theoremr1val1 8811* 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 → (𝑅1𝐴) = 𝑥𝐴 𝒫 (𝑅1𝑥))

Theoremtz9.12lem1 8812* Lemma for tz9.12 8815. (Contributed by NM, 22-Sep-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)
𝐴 ∈ V    &   𝐹 = (𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)})       (𝐹𝐴) ⊆ On

Theoremtz9.12lem2 8813* Lemma for tz9.12 8815. (Contributed by NM, 22-Sep-2003.)
𝐴 ∈ V    &   𝐹 = (𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)})       suc (𝐹𝐴) ∈ On

Theoremtz9.12lem3 8814* Lemma for tz9.12 8815. (Contributed by NM, 22-Sep-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)
𝐴 ∈ V    &   𝐹 = (𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)})       (∀𝑥𝐴𝑦 ∈ On 𝑥 ∈ (𝑅1𝑦) → 𝐴 ∈ (𝑅1‘suc suc (𝐹𝐴)))

Theoremtz9.12 8815* 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 8812 through tz9.12lem3 8814. (Contributed by NM, 22-Sep-2003.)
𝐴 ∈ V       (∀𝑥𝐴𝑦 ∈ On 𝑥 ∈ (𝑅1𝑦) → ∃𝑦 ∈ On 𝐴 ∈ (𝑅1𝑦))

Theoremtz9.13 8816* 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 ∈ (𝑅1𝑥)

Theoremtz9.13g 8817* Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13 8816 expresses the class existence requirement as an antecedent. (Contributed by NM, 4-Oct-2003.)
(𝐴𝑉 → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥))

Theoremrankwflemb 8818* 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))

Theoremrankf 8819 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

Theoremrankon 8820 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

Theoremr1elwf 8821 Any member of the cumulative hierarchy is well-founded. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
(𝐴 ∈ (𝑅1𝐵) → 𝐴 (𝑅1 “ On))

Theoremrankvalb 8822* Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 8841 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)})

Theoremrankr1ai 8823 One direction of rankr1a 8861. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
(𝐴 ∈ (𝑅1𝐵) → (rank‘𝐴) ∈ 𝐵)

Theoremrankvaln 8824 Value of the rank function at a non-well-founded set. (The antecedent is always false under Foundation, by unir1 8838, 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) = ∅)

Theoremrankidb 8825 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)))

Theoremrankdmr1 8826 A rank is a member of the cumulative hierarchy. (Contributed by Mario Carneiro, 17-Nov-2014.)
(rank‘𝐴) ∈ dom 𝑅1

Theoremrankr1ag 8827 A version of rankr1a 8861 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 ∈ (𝑅1𝐵) ↔ (rank‘𝐴) ∈ 𝐵))

Theoremrankr1bg 8828 A relationship between rank and 𝑅1. See rankr1ag 8827 for the membership version. (Contributed by Mario Carneiro, 17-Nov-2014.)
((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ⊆ (𝑅1𝐵) ↔ (rank‘𝐴) ⊆ 𝐵))

Theoremr1rankidb 8829 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)))

Theoremr1elssi 8830 The range of the 𝑅1 function is transitive. Lemma 2.10 of [Kunen] p. 97. One direction of r1elss 8831 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))

Theoremr1elss 8831 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))

Theorempwwf 8832 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))

Theoremsswf 8833 A subset of a well-founded set is well-founded. (Contributed by Mario Carneiro, 17-Nov-2014.)
((𝐴 (𝑅1 “ On) ∧ 𝐵𝐴) → 𝐵 (𝑅1 “ On))

Theoremsnwf 8834 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))

Theoremunwf 8835 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)) ↔ (𝐴𝐵) ∈ (𝑅1 “ On))

Theoremprwf 8836 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))

Theoremopwf 8837 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))

Theoremunir1 8838 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

Theoremjech9.3 8839 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 (𝑅1𝑥) = V

Theoremrankwflem 8840* Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13g 8817 is useful in proofs of theorems about the rank function. (Contributed by NM, 4-Oct-2003.)
(𝐴𝑉 → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))

Theoremrankval 8841* 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)}

Theoremrankvalg 8842* Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 8841 expresses the class existence requirement as an antecedent instead of a hypothesis. (Contributed by NM, 5-Oct-2003.)
(𝐴𝑉 → (rank‘𝐴) = {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})

Theoremrankval2 8843* Value of an alternate definition of the rank function. Definition of [BellMachover] p. 478. (Contributed by NM, 8-Oct-2003.)
(𝐴𝐵 → (rank‘𝐴) = {𝑥 ∈ On ∣ 𝐴 ⊆ (𝑅1𝑥)})

Theoremuniwf 8844 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))

Theoremrankr1clem 8845 Lemma for rankr1c 8846. (Contributed by NM, 6-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)
((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (¬ 𝐴 ∈ (𝑅1𝐵) ↔ 𝐵 ⊆ (rank‘𝐴)))

Theoremrankr1c 8846 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 ∈ (𝑅1𝐵) ∧ 𝐴 ∈ (𝑅1‘suc 𝐵))))

Theoremrankidn 8847 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)))

Theoremrankpwi 8848 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))

Theoremrankelb 8849 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) → (𝐴𝐵 → (rank‘𝐴) ∈ (rank‘𝐵)))

Theoremwfelirr 8850 A well-founded set is not a member of itself. This proof does not require the axiom of regularity, unlike elirr 8656. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴 (𝑅1 “ On) → ¬ 𝐴𝐴)

Theoremrankval3b 8851* 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 ∣ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥})

Theoremranksnb 8852 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))

Theoremrankonidlem 8853 Lemma for rankonid 8854. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 22-Mar-2013.)
(𝐴 ∈ dom 𝑅1 → (𝐴 (𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴))

Theoremrankonid 8854 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)

Theoremonwf 8855 The ordinals are all well-founded. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
On ⊆ (𝑅1 “ On)

Theoremonssr1 8856 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 𝑅1𝐴 ⊆ (𝑅1𝐴))

Theoremrankr1g 8857 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.)
(𝐴𝑉 → (𝐵 = (rank‘𝐴) ↔ (¬ 𝐴 ∈ (𝑅1𝐵) ∧ 𝐴 ∈ (𝑅1‘suc 𝐵))))

Theoremrankid 8858 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))

Theoremrankr1 8859 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 ∈ (𝑅1𝐵) ∧ 𝐴 ∈ (𝑅1‘suc 𝐵)))

Theoremssrankr1 8860 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 ∈ (𝑅1𝐵)))

Theoremrankr1a 8861 A relationship between rank and 𝑅1, clearly equivalent to ssrankr1 8860 and friends through trichotomy, but in Raph's opinion considerably more intuitive. See rankr1b 8889 for the subset version. (Contributed by Raph Levien, 29-May-2004.)
𝐴 ∈ V       (𝐵 ∈ On → (𝐴 ∈ (𝑅1𝐵) ↔ (rank‘𝐴) ∈ 𝐵))

Theoremr1val2 8862* 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 → (𝑅1𝐴) = {𝑥 ∣ (rank‘𝑥) ∈ 𝐴})

Theoremr1val3 8863* 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 → (𝑅1𝐴) = 𝑥𝐴 𝒫 {𝑦 ∣ (rank‘𝑦) ∈ 𝑥})

Theoremrankel 8864 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       (𝐴𝐵 → (rank‘𝐴) ∈ (rank‘𝐵))

Theoremrankval3 8865* 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 ∣ ∀𝑦𝐴 (rank‘𝑦) ∈ 𝑥}

Theorembndrank 8866* 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 ∀𝑦𝐴 (rank‘𝑦) ⊆ 𝑥𝐴 ∈ V)

Theoremunbndrank 8867* 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 ∃𝑦𝐴 𝑥 ∈ (rank‘𝑦))

Theoremrankpw 8868 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)

Theoremranklim 8869 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))

Theoremr1pw 8870 A stronger property of 𝑅1 than rankpw 8868. 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 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵)))

Theoremr1pwALT 8871 Alternate shorter proof of r1pw 8870 based on the additional axioms ax-reg 8651 and ax-inf2 8700. (Contributed by Raph Levien, 29-May-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐵 ∈ On → (𝐴 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1‘suc 𝐵)))

Theoremr1pwcl 8872 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 ∈ (𝑅1𝐵) ↔ 𝒫 𝐴 ∈ (𝑅1𝐵)))

Theoremrankssb 8873 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) → (𝐴𝐵 → (rank‘𝐴) ⊆ (rank‘𝐵)))

Theoremrankss 8874 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       (𝐴𝐵 → (rank‘𝐴) ⊆ (rank‘𝐵))

Theoremrankunb 8875 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‘(𝐴𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))

Theoremrankprb 8876 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)))

Theoremrankopb 8877 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)))

Theoremrankuni2b 8878* 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) = 𝑥𝐴 (rank‘𝑥))

Theoremranksn 8879 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)

Theoremrankuni2 8880* 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) = 𝑥𝐴 (rank‘𝑥)

Theoremrankun 8881 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‘(𝐴𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵))

Theoremrankpr 8882 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))

Theoremrankop 8883 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))

Theoremr1rankid 8884 Any set is a subset of the hierarchy of its rank. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)
(𝐴𝑉𝐴 ⊆ (𝑅1‘(rank‘𝐴)))

Theoremrankeq0b 8885 A set is empty iff its rank is empty. (Contributed by Mario Carneiro, 17-Nov-2014.)
(𝐴 (𝑅1 “ On) → (𝐴 = ∅ ↔ (rank‘𝐴) = ∅))

Theoremrankeq0 8886 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) = ∅)

Theoremrankr1id 8887 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‘(𝑅1𝐴)) = 𝐴)

Theoremrankuni 8888 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)

Theoremrankr1b 8889 A relationship between rank and 𝑅1. See rankr1a 8861 for the membership version. (Contributed by NM, 15-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.)
𝐴 ∈ V       (𝐵 ∈ On → (𝐴 ⊆ (𝑅1𝐵) ↔ (rank‘𝐴) ⊆ 𝐵))

Theoremranksuc 8890 The rank of a successor. (Contributed by NM, 18-Sep-2006.)
𝐴 ∈ V       (rank‘suc 𝐴) = suc (rank‘𝐴)

Theoremrankuniss 8891 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)

Theoremrankval4 8892* 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) = 𝑥𝐴 suc (rank‘𝑥)

Theoremrankbnd 8893* The rank of a set is bounded by a bound for the successor of its members. (Contributed by NM, 18-Sep-2006.)
𝐴 ∈ V       (∀𝑥𝐴 suc (rank‘𝑥) ⊆ 𝐵 ↔ (rank‘𝐴) ⊆ 𝐵)

Theoremrankbnd2 8894* 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 → (∀𝑥𝐴 (rank‘𝑥) ⊆ 𝐵 ↔ (rank‘𝐴) ⊆ suc 𝐵))

Theoremrankc1 8895* A relationship that can be used for computation of rank. (Contributed by NM, 16-Sep-2006.)
𝐴 ∈ V       (∀𝑥𝐴 (rank‘𝑥) ∈ (rank‘ 𝐴) ↔ (rank‘𝐴) = (rank‘ 𝐴))

Theoremrankc2 8896* A relationship that can be used for computation of rank. (Contributed by NM, 16-Sep-2006.)
𝐴 ∈ V       (∃𝑥𝐴 (rank‘𝑥) = (rank‘ 𝐴) → (rank‘𝐴) = suc (rank‘ 𝐴))

Theoremrankelun 8897 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‘(𝐴𝐵)) ∈ (rank‘(𝐶𝐷)))

Theoremrankelpr 8898 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}))

Theoremrankelop 8899 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⟩))

Theoremrankxpl 8900 A lower bound on the rank of a Cartesian product. (Contributed by NM, 18-Sep-2006.)
𝐴 ∈ V    &   𝐵 ∈ V       ((𝐴 × 𝐵) ≠ ∅ → (rank‘(𝐴𝐵)) ⊆ (rank‘(𝐴 × 𝐵)))

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