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Theorem List for Metamath Proof Explorer - 8401-8500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremr1ord 8401 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 8402 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 8403 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 8404 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 8405 Each set of the cumulative hierarchy is closed under subsets. (Contributed by Mario Carneiro, 16-Nov-2014.)
(𝐴 ∈ (𝑅1𝐵) → 𝒫 𝐴 ⊆ (𝑅1𝐵))
 
Theoremr1sscl 8406 Each set of the cumulative hierarchy is closed under subsets. (Contributed by Mario Carneiro, 16-Nov-2014.)
((𝐴 ∈ (𝑅1𝐵) ∧ 𝐶𝐴) → 𝐶 ∈ (𝑅1𝐵))
 
Theoremr1val1 8407* 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 8408* Lemma for tz9.12 8411. (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 8409* Lemma for tz9.12 8411. (Contributed by NM, 22-Sep-2003.)
𝐴 ∈ V    &   𝐹 = (𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)})       suc (𝐹𝐴) ∈ On
 
Theoremtz9.12lem3 8410* Lemma for tz9.12 8411. (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 8411* 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 8408 through tz9.12lem3 8410. (Contributed by NM, 22-Sep-2003.)
𝐴 ∈ V       (∀𝑥𝐴𝑦 ∈ On 𝑥 ∈ (𝑅1𝑦) → ∃𝑦 ∈ On 𝐴 ∈ (𝑅1𝑦))
 
Theoremtz9.13 8412* 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 8413* Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13 8412 expresses the class existence requirement as an antecedent. (Contributed by NM, 4-Oct-2003.)
(𝐴𝑉 → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥))
 
Theoremrankwflemb 8414* 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 8415 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 8416 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 8417 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 8418* Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 8437 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 8419 One direction of rankr1a 8457. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
(𝐴 ∈ (𝑅1𝐵) → (rank‘𝐴) ∈ 𝐵)
 
Theoremrankvaln 8420 Value of the rank function at a non-well-founded set. (The antecedent is always false under Foundation, by unir1 8434, 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 8421 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 8422 A rank is a member of the cumulative hierarchy. (Contributed by Mario Carneiro, 17-Nov-2014.)
(rank‘𝐴) ∈ dom 𝑅1
 
Theoremrankr1ag 8423 A version of rankr1a 8457 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 8424 A relationship between rank and 𝑅1. See rankr1ag 8423 for the membership version. (Contributed by Mario Carneiro, 17-Nov-2014.)
((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ⊆ (𝑅1𝐵) ↔ (rank‘𝐴) ⊆ 𝐵))
 
Theoremr1rankidb 8425 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 8426 The range of the 𝑅1 function is transitive. Lemma 2.10 of [Kunen] p. 97. One direction of r1elss 8427 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 8427 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 8428 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 8429 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 8430 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 8431 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 8432 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 8433 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 8434 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 8435 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 8436* Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13g 8413 is useful in proofs of theorems about the rank function. (Contributed by NM, 4-Oct-2003.)
(𝐴𝑉 → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))
 
Theoremrankval 8437* 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 8438* Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 8437 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 8439* 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 8440 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 8441 Lemma for rankr1c 8442. (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 8442 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 8443 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 8444 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 8445 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 8446 A well-founded set is not a member of itself. This proof does not require the axiom of regularity, unlike elirr 8263. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴 (𝑅1 “ On) → ¬ 𝐴𝐴)
 
Theoremrankval3b 8447* 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 8448 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 8449 Lemma for rankonid 8450. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 22-Mar-2013.)
(𝐴 ∈ dom 𝑅1 → (𝐴 (𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴))
 
Theoremrankonid 8450 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 8451 The ordinals are all well-founded. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
On ⊆ (𝑅1 “ On)
 
Theoremonssr1 8452 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 8453 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 8454 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 8455 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 8456 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 8457 A relationship between rank and 𝑅1, clearly equivalent to ssrankr1 8456 and friends through trichotomy, but in Raph's opinion considerably more intuitive. See rankr1b 8485 for the subset version. (Contributed by Raph Levien, 29-May-2004.)
𝐴 ∈ V       (𝐵 ∈ On → (𝐴 ∈ (𝑅1𝐵) ↔ (rank‘𝐴) ∈ 𝐵))
 
Theoremr1val2 8458* 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 8459* 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 8460 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 8461* 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 8462* 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 8463* 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 8464 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 8465 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 8466 A stronger property of 𝑅1 than rankpw 8464. 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 8467 Alternate shorter proof of r1pw 8466 based on the additional axioms ax-reg 8255 and ax-inf2 8296. (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 8468 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 8469 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 8470 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 8471 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 8472 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 8473 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 8474* 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 8475 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 8476* 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 8477 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 8478 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 8479 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 8480 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 8481 A set is empty iff its rank is empty. (Contributed by Mario Carneiro, 17-Nov-2014.)
(𝐴 (𝑅1 “ On) → (𝐴 = ∅ ↔ (rank‘𝐴) = ∅))
 
Theoremrankeq0 8482 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 8483 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 8484 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 8485 A relationship between rank and 𝑅1. See rankr1a 8457 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 8486 The rank of a successor. (Contributed by NM, 18-Sep-2006.)
𝐴 ∈ V       (rank‘suc 𝐴) = suc (rank‘𝐴)
 
Theoremrankuniss 8487 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 8488* 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 8489* 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 8490* 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 8491* 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 8492* 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 8493 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 8494 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 8495 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 8496 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)))
 
Theoremrankxpu 8497 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‘(𝐴𝐵))
 
Theoremrankfu 8498 An upper bound on the rank of a function. (Contributed by Gérard Lang, 5-Aug-2018.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐹:𝐴𝐵 → (rank‘𝐹) ⊆ suc suc (rank‘(𝐴𝐵)))
 
Theoremrankmapu 8499 An upper bound on the rank of set exponentiation. (Contributed by Gérard Lang, 5-Aug-2018.)
𝐴 ∈ V    &   𝐵 ∈ V       (rank‘(𝐴𝑚 𝐵)) ⊆ suc suc suc (rank‘(𝐴𝐵))
 
Theoremrankxplim 8500 The rank of a Cartesian product when the rank of the union of its arguments is a limit ordinal. Part of Exercise 4 of [Kunen] p. 107. See rankxpsuc 8503 for the successor case. (Contributed by NM, 19-Sep-2006.)
𝐴 ∈ V    &   𝐵 ∈ V       ((Lim (rank‘(𝐴𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴𝐵)))
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