P. 99 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 9801-9900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	prwf 9801	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 9802	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 9803	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 9804	Every set belongs to some stage of the cumulative hierarchy of sets, expressed using an indexed union. 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 9805*	Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13g 9782 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 9806*	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 9807*	Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 9806 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 9808*	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 9809	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 9810	Lemma for rankr1c 9811. (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 9811	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 9812	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 9813	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 9814	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 9815	A well-founded set is not a member of itself. This proof does not require the axiom of regularity, unlike elirr 9578. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ¬ 𝐴 ∈ 𝐴)
Theorem	rankval2b 9816*	Value of an alternate definition of the rank function. Definition of [BellMachover] p. 478. This variant of rankval2 9808 does not use Regularity, and so requires the assumption that 𝐴 is in the range of 𝑅1. (Contributed by BTernaryTau, 19-Jan-2026.)
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ⊆ (𝑅1‘𝑥)})
Theorem	rankval3b 9817*	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 9818	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 9819	Lemma for rankonid 9820. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 22-Mar-2013.)
⊢ (𝐴 ∈ dom 𝑅1 → (𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴))
Theorem	rankonid 9820	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 9821	The ordinals are all well-founded. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ On ⊆ ∪ (𝑅1 “ On)
Theorem	r1wf 9822	Each stage in the cumulative hierarchy is well-founded. (Contributed by BTernaryTau, 19-Jan-2026.)
⊢ (𝑅1‘𝐴) ∈ ∪ (𝑅1 “ On)
Theorem	elwf 9823	An element of a well-founded set is well-founded. (Contributed by BTernaryTau, 30-Dec-2025.)
⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ ∪ (𝑅1 “ On))
Theorem	onssr1 9824	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 9825	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 9826	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 9827	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 9828	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 9829	A relationship between rank and 𝑅1, clearly equivalent to ssrankr1 9828 and friends through trichotomy, but in Raph's opinion considerably more intuitive. See rankr1b 9862 for the subset version. (Contributed by Raph Levien, 29-May-2004.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐵 ∈ On → (𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝐴) ∈ 𝐵))
Theorem	r1val2 9830*	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 9831*	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 9832	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	rankelg 9833	The membership relation is inherited by the rank function. Closed form of rankel 9832. (Contributed by Scott Fenton, 16-Jul-2015.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝐵) → (rank‘𝐴) ∈ (rank‘𝐵))
Theorem	rankval3 9834*	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 9835*	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 9836*	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 9837	The rank of the powerset is the successor of the rank. 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	rankpwg 9838	The rank of the powerset is the successor of the rank. Closed form of rankpw 9837. (Contributed by Scott Fenton, 16-Jul-2015.)
⊢ (𝐴 ∈ 𝑉 → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
Theorem	ranklim 9839	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 9840	A set is in a given stage of the cumulative hierarchy of sets if and only if its powerset is in the successor stage. (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 9841	Alternate shorter proof of r1pw 9840 based on the additional axioms ax-reg 9570 and ax-inf2 9626. (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 9842	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 9843	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 9844	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 9845	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 9846	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 9847	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 9848*	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 9849	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 9850*	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 9851	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 9852	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 9853	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	rankung 9854	The rank of the union of two sets. Closed form of rankun 9851. (Contributed by Scott Fenton, 15-Jul-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (rank‘(𝐴 ∪ 𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))
Theorem	ranksng 9855	The rank of a singleton. Closed form of ranksn 9849. (Contributed by Scott Fenton, 15-Jul-2015.)
⊢ (𝐴 ∈ 𝑉 → (rank‘{𝐴}) = suc (rank‘𝐴))
Theorem	r1rankid 9856	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 9857	A set is empty iff its rank is empty. (Contributed by Mario Carneiro, 17-Nov-2014.)
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝐴 = ∅ ↔ (rank‘𝐴) = ∅))
Theorem	rankeq0 9858	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 9859	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 9860	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	rankval4b 9861*	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. This variant of rankval4 9865 does not use Regularity, and so requires the assumption that 𝐴 is in the range of 𝑅1. (Contributed by BTernaryTau, 19-Jan-2026.)
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∪ 𝑥 ∈ 𝐴 suc (rank‘𝑥))
Theorem	rankr1b 9862	A relationship between rank and 𝑅1. See rankr1a 9829 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 9863	The rank of a successor. (Contributed by NM, 18-Sep-2006.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (rank‘suc 𝐴) = suc (rank‘𝐴)
Theorem	rankuniss 9864	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 9865*	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 9866*	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 9867*	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 9868*	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 9869*	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 9870	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 9871	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 9872	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 9873	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 9874	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 9875	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 9876	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 9877	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 9880 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)))
Theorem	rankxplim2 9878	If the rank of a Cartesian product is a limit ordinal, so is the rank of the union of its arguments. (Contributed by NM, 19-Sep-2006.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (Lim (rank‘(𝐴 × 𝐵)) → Lim (rank‘(𝐴 ∪ 𝐵)))
Theorem	rankxplim3 9879	The rank of a Cartesian product is a limit ordinal iff its union is. (Contributed by NM, 19-Sep-2006.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (Lim (rank‘(𝐴 × 𝐵)) ↔ Lim ∪ (rank‘(𝐴 × 𝐵)))
Theorem	rankxpsuc 9880	The rank of a Cartesian product when the rank of the union of its arguments is a successor ordinal. Part of Exercise 4 of [Kunen] p. 107. See rankxplim 9877 for the limit ordinal case. (Contributed by NM, 19-Sep-2006.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (((rank‘(𝐴 ∪ 𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) = suc suc (rank‘(𝐴 ∪ 𝐵)))
Theorem	tcwf 9881	The transitive closure function is well-founded if its argument is. (Contributed by Mario Carneiro, 23-Jun-2013.)
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (TC‘𝐴) ∈ ∪ (𝑅1 “ On))
Theorem	tcrank 9882	This theorem expresses two different facts from the two subset implications in this equality. In the forward direction, it says that the transitive closure has members of every rank below 𝐴. Stated another way, to construct a set at a given rank, you have to climb the entire hierarchy of ordinals below (rank‘𝐴), constructing at least one set at each level in order to move up the ranks. In the reverse direction, it says that every member of (TC‘𝐴) has a rank below the rank of 𝐴, since intuitively it contains only the members of 𝐴 and the members of those and so on, but nothing "bigger" than 𝐴. (Contributed by Mario Carneiro, 23-Jun-2013.)
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = (rank “ (TC‘𝐴)))
Theorem	rankfilimbi 9883*	If all elements of a finite well-founded set have a rank less than a limit ordinal, then the rank of that set is also less than the limit ordinal. (Contributed by BTernaryTau, 19-Jan-2026.)
⊢ (((𝐴 ∈ Fin ∧ 𝐴 ∈ ∪ (𝑅1 “ On)) ∧ (∀𝑥 ∈ 𝐴 (rank‘𝑥) ∈ 𝐵 ∧ Lim 𝐵)) → (rank‘𝐴) ∈ 𝐵)
Theorem	r1filimi 9884*	If all elements of a finite set appear in the cumulative hierarchy prior to a limit ordinal, then that set also appears in the cumulative hierarchy prior to the limit ordinal. (Contributed by BTernaryTau, 19-Jan-2026.)
⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ (𝑅1 “ 𝐵) ∧ Lim 𝐵) → 𝐴 ∈ ∪ (𝑅1 “ 𝐵))
2.6.10  Hereditarily finite sets
Syntax	chf 9885	Extend class notation with the class of hereditarily finite sets.
class HF
Definition	df-hf 9886	Define the class of sets belonging to the finite stages of the cumulative hierarchy of sets. This is the class of sets of finite rank by elhf2 9891. They are called the hereditarily finite sets since they are the finite sets whose members are hereditarily finite, as proved in elhf3 9894. (Contributed by Scott Fenton, 9-Jul-2015.)
⊢ HF = ∪ (𝑅1 “ ω)
Theorem	dfhf2 9887	Alternate definition of the class of hereditarily finite sets as the value of the cumulative hierarchy of sets function at ω. This characterization is simpler but requires the axiom of infinity to hold. (Contributed by BTernaryTau, 25-Jan-2026.) Restate using the defined HF symbol. (Revised by Eric Schmidt, 24-Sep-2026.)
⊢ HF = (𝑅1‘ω)
Theorem	elhf 9888*	Membership in the hereditarily finite sets. (Contributed by Scott Fenton, 9-Jul-2015.) Reduce axiom usage and shorten proof. (Revised by BJ, 27-Sep-2026.)
⊢ (𝐴 ∈ HF ↔ ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥))
Theorem	elhfOLD 9889*	Obsolete version of elhf 9888 as of 27-Sep-2026. (Contributed by Scott Fenton, 9-Jul-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ HF ↔ ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1‘𝑥))
Theorem	hffi 9890	Hereditarily finite sets are finite sets. (Contributed by BTernaryTau, 30-Dec-2025.) Restate using the defined HF symbol. (Revised by Eric Schmidt, 8-Sep-2026.)
⊢ (𝐴 ∈ HF → 𝐴 ∈ Fin)
Theorem	elhf2 9891	Alternate form of membership in the hereditarily finite sets. (Contributed by Scott Fenton, 13-Jul-2015.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ HF ↔ (rank‘𝐴) ∈ ω)
Theorem	elhf2g 9892	Hereditarily finiteness via rank. Closed form of elhf2 9891. (Contributed by Scott Fenton, 15-Jul-2015.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ HF ↔ (rank‘𝐴) ∈ ω))
Theorem	elhf4 9893*	A set is hereditarily finite iff it is finite and all of its elements are hereditarily finite. (Contributed by BTernaryTau, 19-Jan-2026.) Use HF. (Revised by BTernaryTau, 17-Sep-2026.)
⊢ (𝐴 ∈ HF ↔ (𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝑥 ∈ HF ))
Theorem	elhf3 9894	A set is hereditarily finite if and only if it is finite and all its members are hereditarily finite. (Contributed by Eric Schmidt, 8-Sep-2026.) Avoid ax-reg 9570, ax-inf2 9626. (Revised by BTernaryTau, 17-Sep-2026.)
⊢ (𝐴 ∈ HF ↔ (𝐴 ∈ Fin ∧ 𝐴 ⊆ HF ))
Theorem	hfelhf 9895	Any member of a hereditarily finite set is itself a hereditarily finite set. (Contributed by Scott Fenton, 16-Jul-2015.) Avoid ax-reg 9570, ax-inf2 9626. (Revised by BTernaryTau, 17-Sep-2026.)
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ HF ) → 𝐴 ∈ HF )
Theorem	hfsshf 9896	Any subset of a hereditarily finite set is itself a hereditarily finite set. (Contributed by BTernaryTau, 17-Sep-2026.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ HF ) → 𝐴 ∈ HF )
Theorem	hfelhfOLD 9897	Obsolete version of elhf3 9894 as of 17-Sep-2026. (Contributed by Scott Fenton, 16-Jul-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ HF ) → 𝐴 ∈ HF )
Theorem	0hf 9898	The empty set is a hereditarily finite set. (Contributed by Scott Fenton, 9-Jul-2015.)
⊢ ∅ ∈ HF
Theorem	hfun 9899	The union of two hereditarily finite sets is a hereditarily finite set. (Contributed by Scott Fenton, 15-Jul-2015.) Avoid ax-reg 9570, ax-inf2 9626. (Revised by BTernaryTau, 17-Sep-2026.)
⊢ ((𝐴 ∈ HF ∧ 𝐵 ∈ HF ) → (𝐴 ∪ 𝐵) ∈ HF )
Theorem	hfunOLD 9900	Obsolete version of hfun 9899 as of 17-Sep-2026. (Contributed by Scott Fenton, 15-Jul-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐴 ∈ HF ∧ 𝐵 ∈ HF ) → (𝐴 ∪ 𝐵) ∈ HF )