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Theorem List for Metamath Proof Explorer - 9201-9300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremr1elwf 9201 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 9202* Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 9221 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 9203 One direction of rankr1a 9241. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
(𝐴 ∈ (𝑅1𝐵) → (rank‘𝐴) ∈ 𝐵)

Theoremrankvaln 9204 Value of the rank function at a non-well-founded set. (The antecedent is always false under Foundation, by unir1 9218, 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 9205 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 9206 A rank is a member of the cumulative hierarchy. (Contributed by Mario Carneiro, 17-Nov-2014.)
(rank‘𝐴) ∈ dom 𝑅1

Theoremrankr1ag 9207 A version of rankr1a 9241 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 9208 A relationship between rank and 𝑅1. See rankr1ag 9207 for the membership version. (Contributed by Mario Carneiro, 17-Nov-2014.)
((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ⊆ (𝑅1𝐵) ↔ (rank‘𝐴) ⊆ 𝐵))

Theoremr1rankidb 9209 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 9210 The range of the 𝑅1 function is transitive. Lemma 2.10 of [Kunen] p. 97. One direction of r1elss 9211 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 9211 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 9212 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 9213 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 9214 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 9215 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 9216 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 9217 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 9218 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 9219 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 9220* Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13g 9197 is useful in proofs of theorems about the rank function. (Contributed by NM, 4-Oct-2003.)
(𝐴𝑉 → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))

Theoremrankval 9221* 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 9222* Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 9221 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 9223* 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 9224 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 9225 Lemma for rankr1c 9226. (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 9226 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 9227 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 9228 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 9229 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 9230 A well-founded set is not a member of itself. This proof does not require the axiom of regularity, unlike elirr 9037. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴 (𝑅1 “ On) → ¬ 𝐴𝐴)

Theoremrankval3b 9231* 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 9232 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 9233 Lemma for rankonid 9234. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 22-Mar-2013.)
(𝐴 ∈ dom 𝑅1 → (𝐴 (𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴))

Theoremrankonid 9234 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 9235 The ordinals are all well-founded. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
On ⊆ (𝑅1 “ On)

Theoremonssr1 9236 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 9237 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 9238 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 9239 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 9240 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 9241 A relationship between rank and 𝑅1, clearly equivalent to ssrankr1 9240 and friends through trichotomy, but in Raph's opinion considerably more intuitive. See rankr1b 9269 for the subset version. (Contributed by Raph Levien, 29-May-2004.)
𝐴 ∈ V       (𝐵 ∈ On → (𝐴 ∈ (𝑅1𝐵) ↔ (rank‘𝐴) ∈ 𝐵))

Theoremr1val2 9242* 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 9243* 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 9244 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 9245* 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 9246* 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 9247* 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 9248 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 9249 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 9250 A stronger property of 𝑅1 than rankpw 9248. 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 9251 Alternate shorter proof of r1pw 9250 based on the additional axioms ax-reg 9032 and ax-inf2 9080. (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 9252 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 9253 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 9254 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 9255 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 9256 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 9257 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 9258* 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 9259 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 9260* 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 9261 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 9262 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 9263 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 9264 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 9265 A set is empty iff its rank is empty. (Contributed by Mario Carneiro, 17-Nov-2014.)
(𝐴 (𝑅1 “ On) → (𝐴 = ∅ ↔ (rank‘𝐴) = ∅))

Theoremrankeq0 9266 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 9267 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 9268 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 9269 A relationship between rank and 𝑅1. See rankr1a 9241 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 9270 The rank of a successor. (Contributed by NM, 18-Sep-2006.)
𝐴 ∈ V       (rank‘suc 𝐴) = suc (rank‘𝐴)

Theoremrankuniss 9271 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 9272* 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 9273* 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 9274* 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 9275* 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 9276* 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 9277 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 9278 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 9279 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 9280 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 9281 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 9282 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 9283 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 9284 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 9287 for the successor case. (Contributed by NM, 19-Sep-2006.)
𝐴 ∈ V    &   𝐵 ∈ V       ((Lim (rank‘(𝐴𝐵)) ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) = (rank‘(𝐴𝐵)))

Theoremrankxplim2 9285 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‘(𝐴𝐵)))

Theoremrankxplim3 9286 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)))

Theoremrankxpsuc 9287 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 9284 for the limit ordinal case. (Contributed by NM, 19-Sep-2006.)
𝐴 ∈ V    &   𝐵 ∈ V       (((rank‘(𝐴𝐵)) = suc 𝐶 ∧ (𝐴 × 𝐵) ≠ ∅) → (rank‘(𝐴 × 𝐵)) = suc suc (rank‘(𝐴𝐵)))

Theoremtcwf 9288 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))

Theoremtcrank 9289 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)))

2.6.6  Scott's trick; collection principle; Hilbert's epsilon

Theoremscottex 9290* Scott's trick collects all sets that have a certain property and are of the smallest possible rank. This theorem shows that the resulting collection, expressed as in Equation 9.3 of [Jech] p. 72, is a set. (Contributed by NM, 13-Oct-2003.)
{𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V

Theoremscott0 9291* Scott's trick collects all sets that have a certain property and are of the smallest possible rank. This theorem shows that the resulting collection, expressed as in Equation 9.3 of [Jech] p. 72, contains at least one representative with the property, if there is one. In other words, the collection is empty iff no set has the property (i.e. 𝐴 is empty). (Contributed by NM, 15-Oct-2003.)
(𝐴 = ∅ ↔ {𝑥𝐴 ∣ ∀𝑦𝐴 (rank‘𝑥) ⊆ (rank‘𝑦)} = ∅)

Theoremscottexs 9292* Theorem scheme version of scottex 9290. The collection of all 𝑥 of minimum rank such that 𝜑(𝑥) is true, is a set. (Contributed by NM, 13-Oct-2003.)
{𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ∈ V

Theoremscott0s 9293* Theorem scheme version of scott0 9291. The collection of all 𝑥 of minimum rank such that 𝜑(𝑥) is true, is not empty iff there is an 𝑥 such that 𝜑(𝑥) holds. (Contributed by NM, 13-Oct-2003.)
(∃𝑥𝜑 ↔ {𝑥 ∣ (𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ≠ ∅)

Theoremcplem1 9294* Lemma for the Collection Principle cp 9296. (Contributed by NM, 17-Oct-2003.)
𝐶 = {𝑦𝐵 ∣ ∀𝑧𝐵 (rank‘𝑦) ⊆ (rank‘𝑧)}    &   𝐷 = 𝑥𝐴 𝐶       𝑥𝐴 (𝐵 ≠ ∅ → (𝐵𝐷) ≠ ∅)

Theoremcplem2 9295* Lemma for the Collection Principle cp 9296. (Contributed by NM, 17-Oct-2003.)
𝐴 ∈ V       𝑦𝑥𝐴 (𝐵 ≠ ∅ → (𝐵𝑦) ≠ ∅)

Theoremcp 9296* Collection Principle. This remarkable theorem scheme is in effect a very strong generalization of the Axiom of Replacement. The proof makes use of Scott's trick scottex 9290 that collapses a proper class into a set of minimum rank. The wff 𝜑 can be thought of as 𝜑(𝑥, 𝑦). Scheme "Collection Principle" of [Jech] p. 72. (Contributed by NM, 17-Oct-2003.)
𝑤𝑥𝑧 (∃𝑦𝜑 → ∃𝑦𝑤 𝜑)

Theorembnd 9297* A very strong generalization of the Axiom of Replacement (compare zfrep6 7632), derived from the Collection Principle cp 9296. Its strength lies in the rather profound fact that 𝜑(𝑥, 𝑦) does not have to be a "function-like" wff, as it does in the standard Axiom of Replacement. This theorem is sometimes called the Boundedness Axiom. (Contributed by NM, 17-Oct-2004.)
(∀𝑥𝑧𝑦𝜑 → ∃𝑤𝑥𝑧𝑦𝑤 𝜑)

Theorembnd2 9298* A variant of the Boundedness Axiom bnd 9297 that picks a subset 𝑧 out of a possibly proper class 𝐵 in which a property is true. (Contributed by NM, 4-Feb-2004.)
𝐴 ∈ V       (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑧(𝑧𝐵 ∧ ∀𝑥𝐴𝑦𝑧 𝜑))

Theoremkardex 9299* The collection of all sets equinumerous to a set 𝐴 and having the least possible rank is a set. This is the part of the justification of the definition of kard of [Enderton] p. 222. (Contributed by NM, 14-Dec-2003.)
{𝑥 ∣ (𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ∈ V

Theoremkarden 9300* If we allow the Axiom of Regularity, we can avoid the Axiom of Choice by defining the cardinal number of a set as the set of all sets equinumerous to it and having the least possible rank. This theorem proves the equinumerosity relationship for this definition (compare carden 9949). The hypotheses correspond to the definition of kard of [Enderton] p. 222 (which we don't define separately since currently we do not use it elsewhere). This theorem along with kardex 9299 justify the definition of kard. The restriction to the least rank prevents the proper class that would result from {𝑥𝑥𝐴}. (Contributed by NM, 18-Dec-2003.) (Revised by AV, 12-Jul-2022.)
𝐴 ∈ V    &   𝐶 = {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))}    &   𝐷 = {𝑥 ∣ (𝑥𝐵 ∧ ∀𝑦(𝑦𝐵 → (rank‘𝑥) ⊆ (rank‘𝑦)))}       (𝐶 = 𝐷𝐴𝐵)

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