Home Metamath Proof ExplorerTheorem List (p. 87 of 424) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-27745) Hilbert Space Explorer (27746-29270) Users' Mathboxes (29271-42316)

Theorem List for Metamath Proof Explorer - 8601-8700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremtctr 8601 Defining property of the transitive closure function: it is transitive. (Contributed by Mario Carneiro, 23-Jun-2013.)
Tr (TC‘𝐴)

Theoremtcmin 8602 Defining property of the transitive closure function: it is a subset of any transitive class containing 𝐴. (Contributed by Mario Carneiro, 23-Jun-2013.)
(𝐴𝑉 → ((𝐴𝐵 ∧ Tr 𝐵) → (TC‘𝐴) ⊆ 𝐵))

Theoremtc2 8603* A variant of the definition of the transitive closure function, using instead the smallest transitive set containing 𝐴 as a member, gives almost the same set, except that 𝐴 itself must be added because it is not usually a member of (TC‘𝐴) (and it is never a member if 𝐴 is well-founded). (Contributed by Mario Carneiro, 23-Jun-2013.)
𝐴 ∈ V       ((TC‘𝐴) ∪ {𝐴}) = {𝑥 ∣ (𝐴𝑥 ∧ Tr 𝑥)}

Theoremtcsni 8604 The transitive closure of a singleton. Proof suggested by Gérard Lang. (Contributed by Mario Carneiro, 4-Jun-2015.)
𝐴 ∈ V       (TC‘{𝐴}) = ((TC‘𝐴) ∪ {𝐴})

Theoremtcss 8605 The transitive closure function inherits the subset relation. (Contributed by Mario Carneiro, 23-Jun-2013.)
𝐴 ∈ V       (𝐵𝐴 → (TC‘𝐵) ⊆ (TC‘𝐴))

Theoremtcel 8606 The transitive closure function converts the element relation to the subset relation. (Contributed by Mario Carneiro, 23-Jun-2013.)
𝐴 ∈ V       (𝐵𝐴 → (TC‘𝐵) ⊆ (TC‘𝐴))

Theoremtcidm 8607 The transitive closure function is idempotent. (Contributed by Mario Carneiro, 23-Jun-2013.)
(TC‘(TC‘𝐴)) = (TC‘𝐴)

Theoremtc0 8608 The transitive closure of the empty set. (Contributed by Mario Carneiro, 4-Jun-2015.)
(TC‘∅) = ∅

Theoremtc00 8609 The transitive closure is empty iff its argument is. Proof suggested by Gérard Lang. (Contributed by Mario Carneiro, 4-Jun-2015.)
(𝐴𝑉 → ((TC‘𝐴) = ∅ ↔ 𝐴 = ∅))

2.6.5  Rank

Syntaxcr1 8610 Extend class definition to include the cumulative hierarchy of sets function.
class 𝑅1

Syntaxcrnk 8611 Extend class definition to include rank function.
class rank

Definitiondf-r1 8612 Define the cumulative hierarchy of sets function, using Takeuti and Zaring's notation (𝑅1). Starting with the empty set, this function builds up layers of sets where the next layer is the power set of the previous layer (and the union of previous layers when the argument is a limit ordinal). Using the Axiom of Regularity, we can show that any set whatsoever belongs to one of the layers of this hierarchy (see tz9.13 8639). Our definition expresses Definition 9.9 of [TakeutiZaring] p. 76 in a closed form, from which we derive the recursive definition as theorems r10 8616, r1suc 8618, and r1lim 8620. Theorem r1val1 8634 shows a recursive definition that works for all values, and theorems r1val2 8685 and r1val3 8686 show the value expressed in terms of rank. Other notations for this function are R with the argument as a subscript (Equation 3.1 of [BellMachover] p. 477), V with a subscript (Definition of [Enderton] p. 202), M with a subscript (Definition 15.19 of [Monk1] p. 113), the capital Greek letter psi (Definition of [Mendelson] p. 281), and bold-face R (Definition 2.1 of [Kunen] p. 95). (Contributed by NM, 2-Sep-2003.)
𝑅1 = rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)

Definitiondf-rank 8613* Define the rank function. See rankval 8664, rankval2 8666, rankval3 8688, or rankval4 8715 its value. The rank is a kind of "inverse" of the cumulative hierarchy of sets function 𝑅1: given a set, it returns an ordinal number telling us the smallest layer of the hierarchy to which the set belongs. Based on Definition 9.14 of [TakeutiZaring] p. 79. Theorem rankid 8681 illustrates the "inverse" concept. Another nice theorem showing the relationship is rankr1a 8684. (Contributed by NM, 11-Oct-2003.)
rank = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})

Theoremr1funlim 8614 The cumulative hierarchy of sets function is a function on a limit ordinal. (This weak form of r1fnon 8615 avoids ax-rep 4762.) (Contributed by Mario Carneiro, 16-Nov-2014.)
(Fun 𝑅1 ∧ Lim dom 𝑅1)

Theoremr1fnon 8615 The cumulative hierarchy of sets function is a function on the class of ordinal numbers. (Contributed by NM, 5-Oct-2003.) (Revised by Mario Carneiro, 10-Sep-2013.)
𝑅1 Fn On

Theoremr10 8616 Value of the cumulative hierarchy of sets function at . Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by NM, 2-Sep-2003.) (Revised by Mario Carneiro, 10-Sep-2013.)
(𝑅1‘∅) = ∅

Theoremr1sucg 8617 Value of the cumulative hierarchy of sets function at a successor ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by Mario Carneiro, 16-Nov-2014.)
(𝐴 ∈ dom 𝑅1 → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1𝐴))

Theoremr1suc 8618 Value of the cumulative hierarchy of sets function at a successor ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by NM, 2-Sep-2003.) (Revised by Mario Carneiro, 10-Sep-2013.)
(𝐴 ∈ On → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1𝐴))

Theoremr1limg 8619* Value of the cumulative hierarchy of sets function at a limit ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by Mario Carneiro, 16-Nov-2014.)
((𝐴 ∈ dom 𝑅1 ∧ Lim 𝐴) → (𝑅1𝐴) = 𝑥𝐴 (𝑅1𝑥))

Theoremr1lim 8620* Value of the cumulative hierarchy of sets function at a limit ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by NM, 4-Oct-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)
((𝐴𝐵 ∧ Lim 𝐴) → (𝑅1𝐴) = 𝑥𝐴 (𝑅1𝑥))

Theoremr1fin 8621 The first ω levels of the cumulative hierarchy are all finite. (Contributed by Mario Carneiro, 15-May-2013.)
(𝐴 ∈ ω → (𝑅1𝐴) ∈ Fin)

Theoremr1sdom 8622 Each stage in the cumulative hierarchy is strictly larger than the last. (Contributed by Mario Carneiro, 19-Apr-2013.)
((𝐴 ∈ On ∧ 𝐵𝐴) → (𝑅1𝐵) ≺ (𝑅1𝐴))

Theoremr111 8623 The cumulative hierarchy is a one-to-one function. (Contributed by Mario Carneiro, 19-Apr-2013.)
𝑅1:On–1-1→V

Theoremr1tr 8624 The cumulative hierarchy of sets is transitive. Lemma 7T of [Enderton] p. 202. (Contributed by NM, 8-Sep-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)
Tr (𝑅1𝐴)

Theoremr1tr2 8625 The union of a cumulative hierarchy of sets at ordinal 𝐴 is a subset of the hierarchy at 𝐴. JFM CLASSES1 th. 40. (Contributed by FL, 20-Apr-2011.)
(𝑅1𝐴) ⊆ (𝑅1𝐴)

Theoremr1ordg 8626 Ordering relation for the cumulative hierarchy of sets. Part of Proposition 9.10(2) of [TakeutiZaring] p. 77. (Contributed by NM, 8-Sep-2003.)
(𝐵 ∈ dom 𝑅1 → (𝐴𝐵 → (𝑅1𝐴) ∈ (𝑅1𝐵)))

Theoremr1ord3g 8627 Ordering relation for the cumulative hierarchy of sets. Part of Theorem 3.3(i) of [BellMachover] p. 478. (Contributed by NM, 22-Sep-2003.)
((𝐴 ∈ dom 𝑅1𝐵 ∈ dom 𝑅1) → (𝐴𝐵 → (𝑅1𝐴) ⊆ (𝑅1𝐵)))

Theoremr1ord 8628 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 8629 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 8630 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 8631 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 8632 Each set of the cumulative hierarchy is closed under subsets. (Contributed by Mario Carneiro, 16-Nov-2014.)
(𝐴 ∈ (𝑅1𝐵) → 𝒫 𝐴 ⊆ (𝑅1𝐵))

Theoremr1sscl 8633 Each set of the cumulative hierarchy is closed under subsets. (Contributed by Mario Carneiro, 16-Nov-2014.)
((𝐴 ∈ (𝑅1𝐵) ∧ 𝐶𝐴) → 𝐶 ∈ (𝑅1𝐵))

Theoremr1val1 8634* 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 8635* Lemma for tz9.12 8638. (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 8636* Lemma for tz9.12 8638. (Contributed by NM, 22-Sep-2003.)
𝐴 ∈ V    &   𝐹 = (𝑧 ∈ V ↦ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1𝑣)})       suc (𝐹𝐴) ∈ On

Theoremtz9.12lem3 8637* Lemma for tz9.12 8638. (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 8638* 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 8635 through tz9.12lem3 8637. (Contributed by NM, 22-Sep-2003.)
𝐴 ∈ V       (∀𝑥𝐴𝑦 ∈ On 𝑥 ∈ (𝑅1𝑦) → ∃𝑦 ∈ On 𝐴 ∈ (𝑅1𝑦))

Theoremtz9.13 8639* 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 8640* Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13 8639 expresses the class existence requirement as an antecedent. (Contributed by NM, 4-Oct-2003.)
(𝐴𝑉 → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1𝑥))

Theoremrankwflemb 8641* 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 8642 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 8643 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 8644 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 8645* Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 8664 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 8646 One direction of rankr1a 8684. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
(𝐴 ∈ (𝑅1𝐵) → (rank‘𝐴) ∈ 𝐵)

Theoremrankvaln 8647 Value of the rank function at a non-well-founded set. (The antecedent is always false under Foundation, by unir1 8661, 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 8648 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 8649 A rank is a member of the cumulative hierarchy. (Contributed by Mario Carneiro, 17-Nov-2014.)
(rank‘𝐴) ∈ dom 𝑅1

Theoremrankr1ag 8650 A version of rankr1a 8684 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 8651 A relationship between rank and 𝑅1. See rankr1ag 8650 for the membership version. (Contributed by Mario Carneiro, 17-Nov-2014.)
((𝐴 (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ⊆ (𝑅1𝐵) ↔ (rank‘𝐴) ⊆ 𝐵))

Theoremr1rankidb 8652 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 8653 The range of the 𝑅1 function is transitive. Lemma 2.10 of [Kunen] p. 97. One direction of r1elss 8654 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 8654 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 8655 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 8656 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 8657 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 8658 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 8659 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 8660 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 8661 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 8662 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 8663* Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13g 8640 is useful in proofs of theorems about the rank function. (Contributed by NM, 4-Oct-2003.)
(𝐴𝑉 → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥))

Theoremrankval 8664* 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 8665* Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 8664 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 8666* 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 8667 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 8668 Lemma for rankr1c 8669. (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 8669 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 8670 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 8671 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 8672 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 8673 A well-founded set is not a member of itself. This proof does not require the axiom of regularity, unlike elirr 8490. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝐴 (𝑅1 “ On) → ¬ 𝐴𝐴)

Theoremrankval3b 8674* 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 8675 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 8676 Lemma for rankonid 8677. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 22-Mar-2013.)
(𝐴 ∈ dom 𝑅1 → (𝐴 (𝑅1 “ On) ∧ (rank‘𝐴) = 𝐴))

Theoremrankonid 8677 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 8678 The ordinals are all well-founded. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
On ⊆ (𝑅1 “ On)

Theoremonssr1 8679 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 8680 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 8681 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 8682 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 8683 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 8684 A relationship between rank and 𝑅1, clearly equivalent to ssrankr1 8683 and friends through trichotomy, but in Raph's opinion considerably more intuitive. See rankr1b 8712 for the subset version. (Contributed by Raph Levien, 29-May-2004.)
𝐴 ∈ V       (𝐵 ∈ On → (𝐴 ∈ (𝑅1𝐵) ↔ (rank‘𝐴) ∈ 𝐵))

Theoremr1val2 8685* 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 8686* 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 8687 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 8688* 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 8689* 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 8690* 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 8691 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 8692 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 8693 A stronger property of 𝑅1 than rankpw 8691. 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 8694 Alternate shorter proof of r1pw 8693 based on the additional axioms ax-reg 8482 and ax-inf2 8523. (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 8695 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 8696 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 8697 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 8698 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 8699 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 8700 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)))

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42316
 Copyright terms: Public domain < Previous  Next >