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	epfrs 9701*	The strong form of the Axiom of Regularity (no sethood requirement on 𝐴), with the axiom itself present as an antecedent. See also zfregs 9702. (Contributed by Mario Carneiro, 22-Mar-2013.)
⊢ (( E Fr 𝐴 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)
Theorem	zfregs 9702*	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 9701. (Contributed by NM, 17-Sep-2003.)
⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)
Theorem	zfregs2 9703*	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 9704	Extend class notation to include the transitive closure function.
class TC
Definition	df-tc 9705*	The transitive closure function. (Contributed by Mario Carneiro, 23-Jun-2013.)
⊢ TC = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ Tr 𝑦)})
Theorem	tcvalg 9706*	Value of the transitive closure function. (The fact that this intersection exists is a non-trivial fact that depends on ax-inf 9608; see tz9.1 9699.) (Contributed by Mario Carneiro, 23-Jun-2013.)
⊢ (𝐴 ∈ 𝑉 → (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
Theorem	tcid 9707	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 9708	Defining property of the transitive closure function: it is transitive. (Contributed by Mario Carneiro, 23-Jun-2013.)
⊢ Tr (TC‘𝐴)
Theorem	tcmin 9709	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 9710*	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 9711	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 9712	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 9713	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 9714	The transitive closure function is idempotent. (Contributed by Mario Carneiro, 23-Jun-2013.)
⊢ (TC‘(TC‘𝐴)) = (TC‘𝐴)
Theorem	tc0 9715	The transitive closure of the empty set. (Contributed by Mario Carneiro, 4-Jun-2015.)
⊢ (TC‘∅) = ∅
Theorem	tc00 9716	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 9717*	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 9718	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 9719*	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 9720*	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 9721*	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 9722*	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 6350 and tz7.5 6383. (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 9723*	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 6352 and tfi 7850, 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 9724*	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 9725*	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 9726*	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 9727*	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 9728*	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 9729*	Functions defined by well-founded recursion are identical up to relation, domain, and characteristic function. General version of frr3 9734. (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 9730*	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 9731*	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 9732	Law of general well-founded recursion, part one. This theorem and the following two drop the partial order requirement from fpr1 8301, fpr2 8302, and fpr3 8303, 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 9733	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 9734*	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 9732 and frr2 9733 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 9735	Extend class definition to include the cumulative hierarchy of sets function.
class 𝑅1
Syntax	crnk 9736	Extend class definition to include rank function.
class rank
Definition	df-r1 9737	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 9764). Our definition expresses Definition 9.9 of [TakeutiZaring] p. 76 in a closed form, from which we derive the recursive definition as Theorems r10 9741, r1suc 9743, and r1lim 9745. Theorem r1val1 9759 shows a recursive definition that works for all values, and Theorems r1val2 9810 and r1val3 9811 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), ∅)
Definition	df-rank 9738*	Define the rank function. See rankval 9789, rankval2 9791, rankval3 9813, or rankval4 9840 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 9806 illustrates the "inverse" concept. Another nice theorem showing the relationship is rankr1a 9809. (Contributed by NM, 11-Oct-2003.)
⊢ rank = (𝑥 ∈ V ↦ ∩ {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘suc 𝑦)})
Theorem	r1funlim 9739	The cumulative hierarchy of sets function is a function on a limit ordinal. (This weak form of r1fnon 9740 avoids ax-rep 5239.) (Contributed by Mario Carneiro, 16-Nov-2014.)
⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1)
Theorem	r1fnon 9740	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 9741	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 9742	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 9743	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 9744*	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 9745*	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.)
⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → (𝑅1‘𝐴) = ∪ 𝑥 ∈ 𝐴 (𝑅1‘𝑥))
Theorem	r1fin 9746	The first ω levels of the cumulative hierarchy are all finite. (Contributed by Mario Carneiro, 15-May-2013.)
⊢ (𝐴 ∈ ω → (𝑅1‘𝐴) ∈ Fin)
Theorem	r1sdom 9747	Each stage in the cumulative hierarchy is strictly larger than the last. (Contributed by Mario Carneiro, 19-Apr-2013.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → (𝑅1‘𝐵) ≺ (𝑅1‘𝐴))
Theorem	r111 9748	The cumulative hierarchy is a one-to-one function. (Contributed by Mario Carneiro, 19-Apr-2013.)
⊢ 𝑅1:On–1-1→V
Theorem	r1tr 9749	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 9750	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.)
⊢ ∪ (𝑅1‘𝐴) ⊆ (𝑅1‘𝐴)
Theorem	r1ordg 9751	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 9752	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 9753	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 9754	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 9755	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 9756	The value of the cumulative hierarchy of sets function is a subset of its value at the successor. JFM CLASSES1 Th. 39. (Contributed by FL, 20-Apr-2011.)
⊢ (𝐴 ∈ On → (𝑅1‘𝐴) ⊆ (𝑅1‘suc 𝐴))
Theorem	r1pwss 9757	Each set of the cumulative hierarchy is closed under subsets. (Contributed by Mario Carneiro, 16-Nov-2014.)
⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝒫 𝐴 ⊆ (𝑅1‘𝐵))
Theorem	r1sscl 9758	Each set of the cumulative hierarchy is closed under subsets. (Contributed by Mario Carneiro, 16-Nov-2014.)
⊢ ((𝐴 ∈ (𝑅1‘𝐵) ∧ 𝐶 ⊆ 𝐴) → 𝐶 ∈ (𝑅1‘𝐵))
Theorem	r1val1 9759*	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 9760*	Lemma for tz9.12 9763. (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 9761*	Lemma for tz9.12 9763. (Contributed by NM, 22-Sep-2003.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐹 = (𝑧 ∈ V ↦ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)})    ⇒   ⊢ suc ∪ (𝐹 “ 𝐴) ∈ On
Theorem	tz9.12lem3 9762*	Lemma for tz9.12 9763. (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 9763*	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 9760 through tz9.12lem3 9762. (Contributed by NM, 22-Sep-2003.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦) → ∃𝑦 ∈ On 𝐴 ∈ (𝑅1‘𝑦))
Theorem	tz9.13 9764*	Every set is well-founded, assuming the Axiom of Regularity. In other words, every set belongs to a layer of the cumulative hierarchy of sets. Proposition 9.13 of [TakeutiZaring] p. 78. (Contributed by NM, 23-Sep-2003.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥)
Theorem	tz9.13g 9765*	Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13 9764 expresses the class existence requirement as an antecedent. (Contributed by NM, 4-Oct-2003.)
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘𝑥))
Theorem	rankwflemb 9766*	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 9767	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 9768	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 9769	Any member of the cumulative hierarchy is well-founded. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))
Theorem	rankvalb 9770*	Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 9789 does not use Regularity, and so requires the assumption that 𝐴 is in the range of 𝑅1. (Contributed by NM, 11-Oct-2003.) (Revised by Mario Carneiro, 10-Sep-2013.)
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ (𝑅1‘suc 𝑥)})
Theorem	rankr1ai 9771	One direction of rankr1a 9809. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ (𝐴 ∈ (𝑅1‘𝐵) → (rank‘𝐴) ∈ 𝐵)
Theorem	rankvaln 9772	Value of the rank function at a non-well-founded set. (The antecedent is always false under Foundation, by unir1 9786, 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 9773	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 9774	A rank is a member of the cumulative hierarchy. (Contributed by Mario Carneiro, 17-Nov-2014.)
⊢ (rank‘𝐴) ∈ dom 𝑅1
Theorem	rankr1ag 9775	A version of rankr1a 9809 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 9776	A relationship between rank and 𝑅1. See rankr1ag 9775 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 9777	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 9778	The range of the 𝑅1 function is transitive. Lemma 2.10 of [Kunen] p. 97. One direction of r1elss 9779 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 9779	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 9780	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 9781	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 9782	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 9783	A binary union is well-founded iff its elements are. (Contributed by Mario Carneiro, 10-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) ↔ (𝐴 ∪ 𝐵) ∈ ∪ (𝑅1 “ On))
Theorem	prwf 9784	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 9785	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 9786	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 9787	Every set belongs to some value of the cumulative hierarchy of sets function 𝑅1, i.e. the indexed union of all values of 𝑅1 is the universe. Lemma 9.3 of [Jech] p. 71. (Contributed by NM, 4-Oct-2003.) (Revised by Mario Carneiro, 8-Jun-2013.)
⊢ ∪ 𝑥 ∈ On (𝑅1‘𝑥) = V
Theorem	rankwflem 9788*	Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13g 9765 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 9789*	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 9790*	Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 9789 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 9791*	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 9792	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 9793	Lemma for rankr1c 9794. (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 9794	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 9795	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 9796	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 9797	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 9798	A well-founded set is not a member of itself. This proof does not require the axiom of regularity, unlike elirr 9563. (Contributed by Mario Carneiro, 2-Jan-2017.)
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ¬ 𝐴 ∈ 𝐴)
Theorem	rankval3b 9799*	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 9800	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))