P. 98 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 9701-9800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ttrcltr 9701	The transitive closure of a class is transitive. (Contributed by Scott Fenton, 17-Oct-2024.)
⊢ (t++𝑅 ∘ t++𝑅) ⊆ t++𝑅
Theorem	ttrclresv 9702	The transitive closure of 𝑅 restricted to V is the same as the transitive closure of 𝑅 itself. (Contributed by Scott Fenton, 20-Oct-2024.)
⊢ t++(𝑅 ↾ V) = t++𝑅
Theorem	ttrclco 9703	Composition law for the transitive closure of a relation. (Contributed by Scott Fenton, 20-Oct-2024.)
⊢ (t++𝑅 ∘ 𝑅) ⊆ t++𝑅
Theorem	cottrcl 9704	Composition law for the transitive closure of a relation. (Contributed by Scott Fenton, 20-Oct-2024.)
⊢ (𝑅 ∘ t++𝑅) ⊆ t++𝑅
Theorem	ttrclss 9705	If 𝑅 is a subclass of 𝑆 and 𝑆 is transitive, then the transitive closure of 𝑅 is a subclass of 𝑆. (Contributed by Scott Fenton, 20-Oct-2024.)
⊢ ((𝑅 ⊆ 𝑆 ∧ (𝑆 ∘ 𝑆) ⊆ 𝑆) → t++𝑅 ⊆ 𝑆)
Theorem	dmttrcl 9706	The domain of a transitive closure is the same as the domain of the original class. (Contributed by Scott Fenton, 26-Oct-2024.)
⊢ dom t++𝑅 = dom 𝑅
Theorem	rnttrcl 9707	The range of a transitive closure is the same as the range of the original class. (Contributed by Scott Fenton, 26-Oct-2024.)
⊢ ran t++𝑅 = ran 𝑅
Theorem	ttrclexg 9708	If 𝑅 is a set, then so is t++𝑅. (Contributed by Scott Fenton, 26-Oct-2024.)
⊢ (𝑅 ∈ 𝑉 → t++𝑅 ∈ V)
Theorem	dfttrcl2 9709*	When 𝑅 is a set and a relation, then its transitive closure can be defined by an intersection. (Contributed by Scott Fenton, 26-Oct-2024.)
⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → t++𝑅 = ∩ {𝑧 ∣ (𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
Theorem	ttrclselem1 9710*	Lemma for ttrclse 9712. Show that all finite ordinal function values of 𝐹 are subsets of 𝐴. (Contributed by Scott Fenton, 31-Oct-2024.)
⊢ 𝐹 = rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))    ⇒   ⊢ (𝑁 ∈ ω → (𝐹‘𝑁) ⊆ 𝐴)
Theorem	ttrclselem2 9711*	Lemma for ttrclse 9712. Show that a suc 𝑁 element long chain gives membership in the 𝑁-th predecessor class and vice-versa. (Contributed by Scott Fenton, 31-Oct-2024.)
⊢ 𝐹 = rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))    ⇒   ⊢ ((𝑁 ∈ ω ∧ 𝑅 Se 𝐴 ∧ 𝑋 ∈ 𝐴) → (∃𝑓(𝑓 Fn suc suc 𝑁 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓‘suc 𝑁) = 𝑋) ∧ ∀𝑎 ∈ suc 𝑁(𝑓‘𝑎)(𝑅 ↾ 𝐴)(𝑓‘suc 𝑎)) ↔ 𝑦 ∈ (𝐹‘𝑁)))
Theorem	ttrclse 9712	If 𝑅 is set-like over 𝐴, then the transitive closure of the restriction of 𝑅 to 𝐴 is set-like over 𝐴. This theorem requires the axioms of infinity and replacement for its proof. (Contributed by Scott Fenton, 31-Oct-2024.)
⊢ (𝑅 Se 𝐴 → t++(𝑅 ↾ 𝐴) Se 𝐴)
2.6.5  Transitive closure
Theorem	trcl 9713*	For any set 𝐴, show the properties of its transitive closure 𝐶. Similar to Theorem 9.1 of [TakeutiZaring] p. 73 except that we show an explicit expression for the transitive closure rather than just its existence. See tz9.1 9714 for an abbreviated version showing existence. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐹 = (rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)    &   ⊢ 𝐶 = ∪ 𝑦 ∈ ω (𝐹‘𝑦)    ⇒   ⊢ (𝐴 ⊆ 𝐶 ∧ Tr 𝐶 ∧ ∀𝑥((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → 𝐶 ⊆ 𝑥))
Theorem	tz9.1 9714*	Every set has a transitive closure (the smallest transitive extension). Theorem 9.1 of [TakeutiZaring] p. 73. See trcl 9713 for an explicit expression for the transitive closure. Apparently open problems are whether this theorem can be proved without the Axiom of Infinity; if not, then whether it implies Infinity; and if not, what is the "property" that Infinity has that the other axioms don't have that is weaker than Infinity itself? (Added 22-Mar-2011) The following article seems to answer the first question, that it can't be proved without Infinity, in the affirmative: Mancini, Antonella and Zambella, Domenico (2001). "A note on recursive models of set theories." Notre Dame Journal of Formal Logic, 42(2):109-115. (Thanks to Scott Fenton.) (Contributed by NM, 15-Sep-2003.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦))
Theorem	tz9.1c 9715*	Alternate expression for the existence of transitive closures tz9.1 9714: the intersection of all transitive sets containing 𝐴 is a set. (Contributed by Mario Carneiro, 22-Mar-2013.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V
Theorem	epfrs 9716*	The strong form of the Axiom of Regularity (no sethood requirement on 𝐴), with the axiom itself present as an antecedent. See also zfregs 9717. (Contributed by Mario Carneiro, 22-Mar-2013.)
⊢ (( E Fr 𝐴 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)
Theorem	zfregs 9717*	The strong form of the Axiom of Regularity, which does not require that 𝐴 be a set. Axiom 6' of [TakeutiZaring] p. 21. See also epfrs 9716. (Contributed by NM, 17-Sep-2003.)
⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)
Theorem	zfregs2 9718*	Alternate strong form of the Axiom of Regularity. Not every element of a nonempty class contains some element of that class. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by Wolf Lammen, 27-Sep-2013.)
⊢ (𝐴 ≠ ∅ → ¬ ∀𝑥 ∈ 𝐴 ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥))
Syntax	ctc 9719	Extend class notation to include the transitive closure function.
class TC
Definition	df-tc 9720*	The transitive closure function. (Contributed by Mario Carneiro, 23-Jun-2013.)
⊢ TC = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ Tr 𝑦)})
Theorem	tcvalg 9721*	Value of the transitive closure function. (The fact that this intersection exists is a non-trivial fact that depends on ax-inf 9623; see tz9.1 9714.) (Contributed by Mario Carneiro, 23-Jun-2013.)
⊢ (𝐴 ∈ 𝑉 → (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
Theorem	tcid 9722	Defining property of the transitive closure function: it contains its argument as a subset. (Contributed by Mario Carneiro, 23-Jun-2013.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (TC‘𝐴))
Theorem	tctr 9723	Defining property of the transitive closure function: it is transitive. (Contributed by Mario Carneiro, 23-Jun-2013.)
⊢ Tr (TC‘𝐴)
Theorem	tcmin 9724	Defining property of the transitive closure function: it is a subset of any transitive class containing 𝐴. (Contributed by Mario Carneiro, 23-Jun-2013.)
⊢ (𝐴 ∈ 𝑉 → ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → (TC‘𝐴) ⊆ 𝐵))
Theorem	tc2 9725*	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 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)}
Theorem	tcsni 9726	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})
Theorem	tcss 9727	The transitive closure function inherits the subset relation. (Contributed by Mario Carneiro, 23-Jun-2013.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐵 ⊆ 𝐴 → (TC‘𝐵) ⊆ (TC‘𝐴))
Theorem	tcel 9728	The transitive closure function converts the element relation to the subset relation. (Contributed by Mario Carneiro, 23-Jun-2013.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐵 ∈ 𝐴 → (TC‘𝐵) ⊆ (TC‘𝐴))
Theorem	tcidm 9729	The transitive closure function is idempotent. (Contributed by Mario Carneiro, 23-Jun-2013.)
⊢ (TC‘(TC‘𝐴)) = (TC‘𝐴)
Theorem	tc0 9730	The transitive closure of the empty set. (Contributed by Mario Carneiro, 4-Jun-2015.)
⊢ (TC‘∅) = ∅
Theorem	tc00 9731	The transitive closure is empty iff its argument is. Proof suggested by Gérard Lang. (Contributed by Mario Carneiro, 4-Jun-2015.)
⊢ (𝐴 ∈ 𝑉 → ((TC‘𝐴) = ∅ ↔ 𝐴 = ∅))
2.6.6  Set induction (or epsilon induction)
Theorem	setind 9732*	Set (epsilon) induction. Theorem 5.22 of [TakeutiZaring] p. 21. (Contributed by NM, 17-Sep-2003.)
⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → 𝐴 = V)
Theorem	setind2 9733	Set (epsilon) induction, stated compactly. Given as a homework problem in 1992 by George Boolos (1940-1996). (Contributed by NM, 17-Sep-2003.)
⊢ (𝒫 𝐴 ⊆ 𝐴 → 𝐴 = V)
Theorem	setinds 9734*	Principle of set induction (or E-induction). If a property passes from all elements of 𝑥 to 𝑥 itself, then it holds for all 𝑥. (Contributed by Scott Fenton, 10-Mar-2011.)
⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑)    ⇒   ⊢ 𝜑
Theorem	setinds2f 9735*	E induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (∀𝑦 ∈ 𝑥 𝜓 → 𝜑)    ⇒   ⊢ 𝜑
Theorem	setinds2 9736*	E induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (∀𝑦 ∈ 𝑥 𝜓 → 𝜑)    ⇒   ⊢ 𝜑
2.6.7  Well-Founded Induction
Theorem	frmin 9737*	Every (possibly proper) subclass of a class 𝐴 with a well-founded set-like relation 𝑅 has a minimal element. This is a very strong generalization of tz6.26 6343 and tz7.5 6376. (Contributed by Scott Fenton, 4-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Revised by Scott Fenton, 27-Nov-2024.)
⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
Theorem	frind 9738*	A subclass of a well-founded class 𝐴 with the property that whenever it contains all predecessors of an element of 𝐴 it also contains that element, is equal to 𝐴. Compare wfi 6345 and tfi 7853, which are special cases of this theorem that do not require the axiom of infinity. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → 𝑦 ∈ 𝐵))) → 𝐴 = 𝐵)
Theorem	frinsg 9739*	Well-Founded Induction Schema. If a property passes from all elements less than 𝑦 of a well-founded class 𝐴 to 𝑦 itself (induction hypothesis), then the property holds for all elements of 𝐴. Theorem 5.6(ii) of [Levy] p. 64. (Contributed by Scott Fenton, 7-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 → 𝜑))    ⇒   ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑)
Theorem	frins 9740*	Well-Founded Induction Schema. If a property passes from all elements less than 𝑦 of a well-founded class 𝐴 to 𝑦 itself (induction hypothesis), then the property holds for all elements of 𝐴. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
⊢ 𝑅 Fr 𝐴    &   ⊢ 𝑅 Se 𝐴    &   ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 → 𝜑))    ⇒   ⊢ (𝑦 ∈ 𝐴 → 𝜑)
Theorem	frins2f 9741*	Well-Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 7-Feb-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)
⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑))    &   ⊢ Ⅎ𝑦𝜓    &   ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑)
Theorem	frins2 9742*	Well-Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 8-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑))    &   ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑)
Theorem	frins3 9743*	Well-Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐵 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑))    ⇒   ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝐵 ∈ 𝐴) → 𝜒)
2.6.8  Well-Founded Recursion
Theorem	frr3g 9744*	Functions defined by well-founded recursion are identical up to relation, domain, and characteristic function. General version of frr3 9749. (Contributed by Scott Fenton, 10-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → 𝐹 = 𝐺)
Theorem	frrlem15 9745*	Lemma for general well-founded recursion. Two acceptable functions are compatible. (Contributed by Scott Fenton, 11-Sep-2023.)
⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}    &   ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺)    ⇒   ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣))
Theorem	frrlem16 9746*	Lemma for general well-founded recursion. Establish a subset relation. (Contributed by Scott Fenton, 11-Sep-2023.) Revised notion of transitive closure. (Revised by Scott Fenton, 1-Dec-2024.)
⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑧 ∈ 𝐴) → ∀𝑤 ∈ Pred (t++(𝑅 ↾ 𝐴), 𝐴, 𝑧)Pred(𝑅, 𝐴, 𝑤) ⊆ Pred(t++(𝑅 ↾ 𝐴), 𝐴, 𝑧))
Theorem	frr1 9747	Law of general well-founded recursion, part one. This theorem and the following two drop the partial order requirement from fpr1 8305, fpr2 8306, and fpr3 8307, which requires using the axiom of infinity. (Contributed by Scott Fenton, 11-Sep-2023.)
⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺)    ⇒   ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 Fn 𝐴)
Theorem	frr2 9748	Law of general well-founded recursion, part two. Now we establish the value of 𝐹 within 𝐴. (Contributed by Scott Fenton, 11-Sep-2023.)
⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺)    ⇒   ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))
Theorem	frr3 9749*	Law of general well-founded recursion, part three. Finally, we show that 𝐹 is unique. We do this by showing that any function 𝐻 with the same properties we proved of 𝐹 in frr1 9747 and frr2 9748 is identical to 𝐹. (Contributed by Scott Fenton, 11-Sep-2023.)
⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺)    ⇒   ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐻 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐻‘𝑧) = (𝑧𝐺(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝐹 = 𝐻)
2.6.9  Rank
Syntax	cr1 9750	Extend class definition to include the cumulative hierarchy of sets function.
class 𝑅1
Syntax	crnk 9751	Extend class definition to include rank function.
class rank
Definition	df-r1 9752	Define the cumulative hierarchy of sets function, using Takeuti and Zaring's notation (𝑅1). It is also called the Von Neumann hierarchy of sets function. 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 all previous layers when the argument is a limit ordinal). Using the axiom of regularity, we can show that every set belongs to one of the layers of this hierarchy (see tz9.13 9781). Our definition expresses Definition 9.9 of [TakeutiZaring] p. 76 in a closed form, from which we derive the recursive definition as Theorems r10 9758, r1suc 9760, and r1lim 9762. Theorem r1val1 9776 shows a recursive definition that works for all values, and Theorems r1val2 9830 and r1val3 9831 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). The values of this function, that is, the layers of this hierarchy, are called "stages" to emphasize the process-like nature of this function. Some sources also use the term "level". (Contributed by NM, 2-Sep-2003.)
⊢ 𝑅1 = rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)
Definition	df-rank 9753*	Define the rank function. The rank of a set is the smallest ordinal such that the stage of the cumulative hierarchy of sets at that ordinal includes that set, or equivalently the smallest ordinal such that the stage of the cumulative hierarchy of sets at the successor of that ordinal contains that set. This definition uses that second characterization, while the first is proven in rankval2 9808. See rankval 9806, rankval2 9808, rankval3 9834, or rankval4 9865 for its value. The rank is therefore a kind of "inverse" of the cumulative hierarchy of sets function, in a sense made precise in rankid 9826 and rankr1a 9829. Based on Definition 9.14 of [TakeutiZaring] p. 79. (Contributed by NM, 11-Oct-2003.)
⊢ rank = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
Theorem	r1funlimOLD 9754	The cumulative hierarchy of sets function is a function on a limit ordinal. (This weak form of r1fnon 9757 avoids ax-rep 5232.) (Contributed by Mario Carneiro, 16-Nov-2014.) (Proof modification is discouraged.) Use r1fun 9755 or r1dmlim 9756 instead. (New usage is discouraged.)
⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
Theorem	r1fun 9755	The cumulative hierarchy of sets function is a function. (Contributed by Mario Carneiro, 16-Nov-2014.) Extract this statement from its conjunction with r1dmlim 9756. (Revised by BJ, 27-Sep-2026.)
⊢ Fun 𝑅1
Theorem	r1dmlim 9756	The domain of the cumulative hierarchy of sets function is a limit ordinal. This weak form of r1fnon 9757 avoids ax-rep 5232. (Contributed by Mario Carneiro, 16-Nov-2014.) Extract this statement from its conjunction with r1fun 9755. (Revised by BJ, 27-Sep-2026.)
⊢ Lim dom 𝑅1
Theorem	r1fnon 9757	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
Theorem	r10 9758	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‘∅) = ∅
Theorem	r1sucg 9759	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 (𝑅1‘𝐴))
Theorem	r1suc 9760	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 (𝑅1‘𝐴))
Theorem	r1limg 9761*	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) → (𝑅1‘𝐴) = ∪ 𝑥 ∈ 𝐴 (𝑅1‘𝑥))
Theorem	r1lim 9762*	Value of the cumulative hierarchy of sets function at a limit ordinal. Use r1limg 9761 (which does not require ax-rep 5232) when sufficient. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by NM, 4-Oct-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)
⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → (𝑅1‘𝐴) = ∪ 𝑥 ∈ 𝐴 (𝑅1‘𝑥))
Theorem	r1fin 9763	The first ω stages of the cumulative hierarchy of sets are all finite. (Contributed by Mario Carneiro, 15-May-2013.)
⊢ (𝐴 ∈ ω → (𝑅1‘𝐴) ∈ Fin)
Theorem	r1sdom 9764	Each stage of the cumulative hierarchy of sets is strictly larger than any previous stage. (Contributed by Mario Carneiro, 19-Apr-2013.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → (𝑅1‘𝐵) ≺ (𝑅1‘𝐴))
Theorem	r111 9765	The cumulative hierarchy of sets is a one-to-one function. Use r1fun 9755 or r1fnon 9757 (which require fewer axioms) when sufficient. (Contributed by Mario Carneiro, 19-Apr-2013.)
⊢ 𝑅1:On–1-1→V
Theorem	r1tr 9766	Each stage of 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 (𝑅1‘𝐴)
Theorem	r1tr2 9767	Each stage of the cumulative hierarchy of sets includes its union, that is, is transitive. JFM CLASSES1 th. 40. (Contributed by FL, 20-Apr-2011.)
⊢ ∪ (𝑅1‘𝐴) ⊆ (𝑅1‘𝐴)
Theorem	r1ordg 9768	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 → (𝐴 ∈ 𝐵 → (𝑅1‘𝐴) ∈ (𝑅1‘𝐵)))
Theorem	r1ord3g 9769	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 𝑅1 ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ⊆ 𝐵 → (𝑅1‘𝐴) ⊆ (𝑅1‘𝐵)))
Theorem	r1ord 9770	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 9771	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 9772	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 9773	A stage of the cumulative hierarchy of sets is included in its successor stage. JFM CLASSES1 Th. 39. (Contributed by FL, 20-Apr-2011.)
⊢ (𝐴 ∈ On → (𝑅1‘𝐴) ⊆ (𝑅1‘suc 𝐴))
Theorem	r1pwss 9774	Each stage of the cumulative hierarchy of sets is closed under subsets. (Contributed by Mario Carneiro, 16-Nov-2014.)
⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝒫 𝐴 ⊆ (𝑅1‘𝐵))
Theorem	r1sscl 9775	Each stage of the cumulative hierarchy of sets is closed under subsets. (Contributed by Mario Carneiro, 16-Nov-2014.)
⊢ ((𝐴 ∈ (𝑅1‘𝐵) ∧ 𝐶 ⊆ 𝐴) → 𝐶 ∈ (𝑅1‘𝐵))
Theorem	r1val1 9776*	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 9777*	Lemma for tz9.12 9780. (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 9778*	Lemma for tz9.12 9780. (Contributed by NM, 22-Sep-2003.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐹 = (𝑧 ∈ V ↦ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)})    ⇒   ⊢ suc ∪ (𝐹 “ 𝐴) ∈ On
Theorem	tz9.12lem3 9779*	Lemma for tz9.12 9780. (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 9780*	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 9777 through tz9.12lem3 9779. (Contributed by NM, 22-Sep-2003.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦) → ∃𝑦 ∈ On 𝐴 ∈ (𝑅1‘𝑦))
Theorem	tz9.13 9781*	Every set is well-founded, assuming the Axiom of Regularity. In other words, every set belongs to a stage 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 9782*	Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13 9781 expresses the class existence requirement as an antecedent. (Contributed by NM, 4-Oct-2003.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥))
Theorem	rankwflemb 9783*	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 9784	The domain and codomain of the rank function. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 12-Sep-2013.)
⊢ rank:∪ (𝑅1 “ On)⟶On
Theorem	rankon 9785	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 9786	Any element of (any stage of) the cumulative hierarchy of sets is well-founded (recall that ∪ (𝑅1 “ On) is the class of well-founded sets). (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))
Theorem	rankvalb 9787*	Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 9806 does not use the axiom of 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 9788	One direction of rankr1a 9829. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ (𝐴 ∈ (𝑅1‘𝐵) → (rank‘𝐴) ∈ 𝐵)
Theorem	rankvaln 9789	Value of the rank function at a non-well-founded set. (The antecedent is always false under Foundation, by unir1 9803, 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 9790	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 9791	A rank is a member of the cumulative hierarchy of sets. (Contributed by Mario Carneiro, 17-Nov-2014.)
⊢ (rank‘𝐴) ∈ dom 𝑅1
Theorem	rankr1ag 9792	A version of rankr1a 9829 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 9793	A relationship between rank and 𝑅1. See rankr1ag 9792 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 9794	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 9795	The range of the 𝑅1 function is transitive. Lemma 2.10 of [Kunen] p. 97. One direction of r1elss 9796 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 9796	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 9797	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 9798	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 9799	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 9800	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))