P. 91 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 9001-9100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	r1ord 9001	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 9002	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 9003	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 9004	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 9005	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 9006	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 9007*	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 9008*	Lemma for tz9.12 9011. (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 9009*	Lemma for tz9.12 9011. (Contributed by NM, 22-Sep-2003.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐹 = (𝑧 ∈ V ↦ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)})    ⇒   ⊢ suc ∪ (𝐹 “ 𝐴) ∈ On
Theorem	tz9.12lem3 9010*	Lemma for tz9.12 9011. (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 9011*	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 9008 through tz9.12lem3 9010. (Contributed by NM, 22-Sep-2003.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦) → ∃𝑦 ∈ On 𝐴 ∈ (𝑅1‘𝑦))
Theorem	tz9.13 9012*	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 9013*	Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13 9012 expresses the class existence requirement as an antecedent. (Contributed by NM, 4-Oct-2003.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥))
Theorem	rankwflemb 9014*	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 9015	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 9016	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 9017	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 9018*	Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 9037 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 9019	One direction of rankr1a 9057. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ (𝐴 ∈ (𝑅1‘𝐵) → (rank‘𝐴) ∈ 𝐵)
Theorem	rankvaln 9020	Value of the rank function at a non-well-founded set. (The antecedent is always false under Foundation, by unir1 9034, 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 9021	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 9022	A rank is a member of the cumulative hierarchy. (Contributed by Mario Carneiro, 17-Nov-2014.)
⊢ (rank‘𝐴) ∈ dom 𝑅1
Theorem	rankr1ag 9023	A version of rankr1a 9057 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 9024	A relationship between rank and 𝑅1. See rankr1ag 9023 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 9025	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 9026	The range of the 𝑅1 function is transitive. Lemma 2.10 of [Kunen] p. 97. One direction of r1elss 9027 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 9027	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 9028	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 9029	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 9030	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 9031	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 9032	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 9033	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 9034	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 9035	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 9036*	Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13g 9013 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 9037*	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 9038*	Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 9037 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 9039*	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 9040	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 9041	Lemma for rankr1c 9042. (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 9042	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 9043	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 9044	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 9045	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 9046	A well-founded set is not a member of itself. This proof does not require the axiom of regularity, unlike elirr 8854. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ¬ 𝐴 ∈ 𝐴)
Theorem	rankval3b 9047*	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 9048	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 9049	Lemma for rankonid 9050. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 22-Mar-2013.)
⊢ (𝐴 ∈ dom 𝑅1 → (𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴))
Theorem	rankonid 9050	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 9051	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 9052	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 9053	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 9054	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 9055	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 9056	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 9057	A relationship between rank and 𝑅1, clearly equivalent to ssrankr1 9056 and friends through trichotomy, but in Raph's opinion considerably more intuitive. See rankr1b 9085 for the subset version. (Contributed by Raph Levien, 29-May-2004.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐵 ∈ On → (𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝐴) ∈ 𝐵))
Theorem	r1val2 9058*	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 9059*	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 9060	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 9061*	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 9062*	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 9063*	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 9064	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 9065	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 9066	A stronger property of 𝑅1 than rankpw 9064. 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 9067	Alternate shorter proof of r1pw 9066 based on the additional axioms ax-reg 8849 and ax-inf2 8896. (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 9068	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 9069	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 9070	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 9071	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 9072	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 9073	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 9074*	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 9075	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 9076*	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 9077	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 9078	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 9079	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 9080	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 9081	A set is empty iff its rank is empty. (Contributed by Mario Carneiro, 17-Nov-2014.)
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝐴 = ∅ ↔ (rank‘𝐴) = ∅))
Theorem	rankeq0 9082	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 9083	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 9084	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 9085	A relationship between rank and 𝑅1. See rankr1a 9057 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 9086	The rank of a successor. (Contributed by NM, 18-Sep-2006.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (rank‘suc 𝐴) = suc (rank‘𝐴)
Theorem	rankuniss 9087	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 9088*	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 9089*	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 9090*	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 9091*	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 9092*	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 9093	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 9094	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 9095	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 9096	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 9097	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 9098	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)))
Theorem	rankmapu 9099	An upper bound on the rank of set exponentiation. (Contributed by Gérard Lang, 5-Aug-2018.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (rank‘(𝐴 ↑𝑚 𝐵)) ⊆ suc suc suc (rank‘(𝐴 ∪ 𝐵))
Theorem	rankxplim 9100	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 9103 for the successor case. (Contributed by NM, 19-Sep-2006.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((Lim (rank‘(𝐴 ∪ 𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴 ∪ 𝐵)))