P. 440 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 43901-44000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	pr2el2 43901	If an unordered pair is equinumerous to ordinal two, then a part is a member. (Contributed by RP, 21-Oct-2023.)
⊢ ({𝐴, 𝐵} ≈ 2o → 𝐵 ∈ {𝐴, 𝐵})
Theorem	pr2cv2 43902	If an unordered pair is equinumerous to ordinal two, then a part is a set. (Contributed by RP, 21-Oct-2023.)
⊢ ({𝐴, 𝐵} ≈ 2o → 𝐵 ∈ V)
Theorem	pren2 43903	An unordered pair is equinumerous to ordinal two iff both parts are sets not equal to each other. (Contributed by RP, 8-Oct-2023.)
⊢ ({𝐴, 𝐵} ≈ 2o ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴 ≠ 𝐵))
Theorem	pr2eldif1 43904	If an unordered pair is equinumerous to ordinal two, then a part is an element of the difference of the pair and the singleton of the other part. (Contributed by RP, 21-Oct-2023.)
⊢ ({𝐴, 𝐵} ≈ 2o → 𝐴 ∈ ({𝐴, 𝐵} ∖ {𝐵}))
Theorem	pr2eldif2 43905	If an unordered pair is equinumerous to ordinal two, then a part is an element of the difference of the pair and the singleton of the other part. (Contributed by RP, 21-Oct-2023.)
⊢ ({𝐴, 𝐵} ≈ 2o → 𝐵 ∈ ({𝐴, 𝐵} ∖ {𝐴}))
Theorem	pren2d 43906	A pair of two distinct sets is equinumerous to ordinal two. (Contributed by RP, 21-Oct-2023.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    ⇒   ⊢ (𝜑 → {𝐴, 𝐵} ≈ 2o)
Theorem	aleph1min 43907	(ℵ‘1o) is the least uncountable ordinal. (Contributed by RP, 18-Nov-2023.)
⊢ (ℵ‘1o) = ∩ {𝑥 ∈ On ∣ ω ≺ 𝑥}
Theorem	alephiso2 43908	ℵ is a strictly order-preserving mapping of On onto the class of all infinite cardinal numbers. (Contributed by RP, 18-Nov-2023.)
⊢ ℵ Isom E , ≺ (On, {𝑥 ∈ ran card ∣ ω ⊆ 𝑥})
Theorem	alephiso3 43909	ℵ is a strictly order-preserving mapping of On onto the class of all infinite cardinal numbers. (Contributed by RP, 18-Nov-2023.)
⊢ ℵ Isom E , ≺ (On, (ran card ∖ ω))
21.36.4.5  Infinite Sets
Theorem	pwelg 43910*	The powerclass is an element of a class closed under union and powerclass operations iff the element is a member of that class. (Contributed by RP, 21-Mar-2020.)
⊢ (∀𝑥 ∈ 𝐵 (∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → (𝐴 ∈ 𝐵 ↔ 𝒫 𝐴 ∈ 𝐵))
Theorem	pwinfig 43911*	The powerclass of an infinite set is an infinite set, and vice-versa. Here 𝐵 is a class which is closed under both the union and the powerclass operations and which may have infinite sets as members. (Contributed by RP, 21-Mar-2020.)
⊢ (∀𝑥 ∈ 𝐵 (∪ 𝑥 ∈ 𝐵 ∧ 𝒫 𝑥 ∈ 𝐵) → (𝐴 ∈ (𝐵 ∖ Fin) ↔ 𝒫 𝐴 ∈ (𝐵 ∖ Fin)))
Theorem	pwinfi2 43912	The powerclass of an infinite set is an infinite set, and vice-versa. Here 𝑈 is a weak universe. (Contributed by RP, 21-Mar-2020.)
⊢ (𝑈 ∈ WUni → (𝐴 ∈ (𝑈 ∖ Fin) ↔ 𝒫 𝐴 ∈ (𝑈 ∖ Fin)))
Theorem	pwinfi3 43913	The powerclass of an infinite set is an infinite set, and vice-versa. Here 𝑇 is a transitive Tarski universe. (Contributed by RP, 21-Mar-2020.)
⊢ ((𝑇 ∈ Tarski ∧ Tr 𝑇) → (𝐴 ∈ (𝑇 ∖ Fin) ↔ 𝒫 𝐴 ∈ (𝑇 ∖ Fin)))
Theorem	pwinfi 43914	The powerclass of an infinite set is an infinite set, and vice-versa. (Contributed by RP, 21-Mar-2020.)
⊢ (𝐴 ∈ (V ∖ Fin) ↔ 𝒫 𝐴 ∈ (V ∖ Fin))
21.36.4.6  Finite intersection property While there is not yet a definition, the finite intersection property of a class is introduced by fiint 9239 where two textbook definitions are shown to be equivalent. This property is seen often with ordinal numbers (onin 6356, ordelinel 6428), chains of sets ordered by the proper subset relation (sorpssin 7686), various sets in the field of topology (inopn 22855, incld 22999, innei 23081, ... ) and "universal" classes like weak universes (wunin 10636, tskin 10682) and the class of all sets (inex1g 5266).
Theorem	fipjust 43915*	A definition of the finite intersection property of a class based on closure under pairwise intersection of its elements is independent of the dummy variables. (Contributed by RP, 1-Jan-2020.)
⊢ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 ∩ 𝑣) ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴)
Theorem	cllem0 43916*	The class of all sets with property 𝜑(𝑧) is closed under the binary operation on sets defined in 𝑅(𝑥, 𝑦). (Contributed by RP, 3-Jan-2020.)
⊢ 𝑉 = {𝑧 ∣ 𝜑}    &   ⊢ 𝑅 ∈ 𝑈    &   ⊢ (𝑧 = 𝑅 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑧 = 𝑥 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜃))    &   ⊢ ((𝜒 ∧ 𝜃) → 𝜓)    ⇒   ⊢ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 𝑅 ∈ 𝑉
Theorem	superficl 43917*	The class of all supersets of a class has the finite intersection property. (Contributed by RP, 1-Jan-2020.) (Proof shortened by RP, 3-Jan-2020.)
⊢ 𝐴 = {𝑧 ∣ 𝐵 ⊆ 𝑧}    ⇒   ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴
Theorem	superuncl 43918*	The class of all supersets of a class is closed under binary union. (Contributed by RP, 3-Jan-2020.)
⊢ 𝐴 = {𝑧 ∣ 𝐵 ⊆ 𝑧}    ⇒   ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∪ 𝑦) ∈ 𝐴
Theorem	ssficl 43919*	The class of all subsets of a class has the finite intersection property. (Contributed by RP, 1-Jan-2020.) (Proof shortened by RP, 3-Jan-2020.)
⊢ 𝐴 = {𝑧 ∣ 𝑧 ⊆ 𝐵}    ⇒   ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴
Theorem	ssuncl 43920*	The class of all subsets of a class is closed under binary union. (Contributed by RP, 3-Jan-2020.)
⊢ 𝐴 = {𝑧 ∣ 𝑧 ⊆ 𝐵}    ⇒   ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∪ 𝑦) ∈ 𝐴
Theorem	ssdifcl 43921*	The class of all subsets of a class is closed under class difference. (Contributed by RP, 3-Jan-2020.)
⊢ 𝐴 = {𝑧 ∣ 𝑧 ⊆ 𝐵}    ⇒   ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∖ 𝑦) ∈ 𝐴
Theorem	sssymdifcl 43922*	The class of all subsets of a class is closed under symmetric difference. (Contributed by RP, 3-Jan-2020.)
⊢ 𝐴 = {𝑧 ∣ 𝑧 ⊆ 𝐵}    ⇒   ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑥 ∖ 𝑦) ∪ (𝑦 ∖ 𝑥)) ∈ 𝐴
Theorem	fiinfi 43923*	If two classes have the finite intersection property, then so does their intersection. (Contributed by RP, 1-Jan-2020.)
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ∈ 𝐵)    &   ⊢ (𝜑 → 𝐶 = (𝐴 ∩ 𝐵))    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥 ∩ 𝑦) ∈ 𝐶)
21.36.4.7  RP ADDTO: Subclasses and subsets
Theorem	rababg 43924	Condition when restricted class is equal to unrestricted class. (Contributed by RP, 13-Aug-2020.)
⊢ (∀𝑥(𝜑 → 𝑥 ∈ 𝐴) ↔ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ 𝜑})
21.36.4.8  RP ADDTO: The intersection of a class
Theorem	elinintab 43925*	Two ways of saying a set is an element of the intersection of a class with the intersection of a class. (Contributed by RP, 13-Aug-2020.)
⊢ (𝐴 ∈ (𝐵 ∩ ∩ {𝑥 ∣ 𝜑}) ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥)))
Theorem	elmapintrab 43926*	Two ways to say a set is an element of the intersection of a class of images. (Contributed by RP, 16-Aug-2020.)
⊢ 𝐶 ∈ V    &   ⊢ 𝐶 ⊆ 𝐵    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ {𝑤 ∈ 𝒫 𝐵 ∣ ∃𝑥(𝑤 = 𝐶 ∧ 𝜑)} ↔ ((∃𝑥𝜑 → 𝐴 ∈ 𝐵) ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝐶))))
21.36.4.9  RP ADDTO: Theorems requiring subset and intersection existence
Theorem	elinintrab 43927*	Two ways of saying a set is an element of the intersection of a class with the intersection of a class. (Contributed by RP, 14-Aug-2020.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ {𝑤 ∈ 𝒫 𝐵 ∣ ∃𝑥(𝑤 = (𝐵 ∩ 𝑥) ∧ 𝜑)} ↔ ((∃𝑥𝜑 → 𝐴 ∈ 𝐵) ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥))))
Theorem	inintabss 43928*	Upper bound on intersection of class and the intersection of a class. (Contributed by RP, 13-Aug-2020.)
⊢ (𝐴 ∩ ∩ {𝑥 ∣ 𝜑}) ⊆ ∩ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴 ∩ 𝑥) ∧ 𝜑)}
Theorem	inintabd 43929*	Value of the intersection of class with the intersection of a nonempty class. (Contributed by RP, 13-Aug-2020.)
⊢ (𝜑 → ∃𝑥𝜓)    ⇒   ⊢ (𝜑 → (𝐴 ∩ ∩ {𝑥 ∣ 𝜓}) = ∩ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴 ∩ 𝑥) ∧ 𝜓)})
21.36.4.10  RP ADDTO: Relations
Theorem	xpinintabd 43930*	Value of the intersection of Cartesian product with the intersection of a nonempty class. (Contributed by RP, 12-Aug-2020.)
⊢ (𝜑 → ∃𝑥𝜓)    ⇒   ⊢ (𝜑 → ((𝐴 × 𝐵) ∩ ∩ {𝑥 ∣ 𝜓}) = ∩ {𝑤 ∈ 𝒫 (𝐴 × 𝐵) ∣ ∃𝑥(𝑤 = ((𝐴 × 𝐵) ∩ 𝑥) ∧ 𝜓)})
Theorem	relintabex 43931	If the intersection of a class is a relation, then the class is nonempty. (Contributed by RP, 12-Aug-2020.)
⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ∃𝑥𝜑)
Theorem	elcnvcnvintab 43932*	Two ways of saying a set is an element of the converse of the converse of the intersection of a class. (Contributed by RP, 20-Aug-2020.)
⊢ (𝐴 ∈ ◡◡∩ {𝑥 ∣ 𝜑} ↔ (𝐴 ∈ (V × V) ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥)))
Theorem	relintab 43933*	Value of the intersection of a class when it is a relation. (Contributed by RP, 12-Aug-2020.)
⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ 𝜑} = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜑)})
Theorem	nonrel 43934	A non-relation is equal to the base class with all ordered pairs removed. (Contributed by RP, 25-Oct-2020.)
⊢ (𝐴 ∖ ◡◡𝐴) = (𝐴 ∖ (V × V))
Theorem	elnonrel 43935	Only an ordered pair where not both entries are sets could be an element of the non-relation part of class. (Contributed by RP, 25-Oct-2020.)
⊢ (⟨𝑋, 𝑌⟩ ∈ (𝐴 ∖ ◡◡𝐴) ↔ (∅ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V)))
Theorem	cnvssb 43936	Subclass theorem for converse. (Contributed by RP, 22-Oct-2020.)
⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ◡𝐴 ⊆ ◡𝐵))
Theorem	relnonrel 43937	The non-relation part of a relation is empty. (Contributed by RP, 22-Oct-2020.)
⊢ (Rel 𝐴 ↔ (𝐴 ∖ ◡◡𝐴) = ∅)
Theorem	cnvnonrel 43938	The converse of the non-relation part of a class is empty. (Contributed by RP, 18-Oct-2020.)
⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅
Theorem	brnonrel 43939	A non-relation cannot relate any two classes. (Contributed by RP, 23-Oct-2020.)
⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) → ¬ 𝑋(𝐴 ∖ ◡◡𝐴)𝑌)
Theorem	dmnonrel 43940	The domain of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
⊢ dom (𝐴 ∖ ◡◡𝐴) = ∅
Theorem	rnnonrel 43941	The range of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
⊢ ran (𝐴 ∖ ◡◡𝐴) = ∅
Theorem	resnonrel 43942	A restriction of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) = ∅
Theorem	imanonrel 43943	An image under the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
⊢ ((𝐴 ∖ ◡◡𝐴) “ 𝐵) = ∅
Theorem	cononrel1 43944	Composition with the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
⊢ ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ∅
Theorem	cononrel2 43945	Composition with the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
⊢ (𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = ∅
21.36.4.11  RP ADDTO: Functions See also idssxp 6016 by Thierry Arnoux.
Theorem	elmapintab 43946*	Two ways to say a set is an element of mapped intersection of a class. Here 𝐹 maps elements of 𝐶 to elements of ∩ {𝑥 ∣ 𝜑} or 𝑥. (Contributed by RP, 19-Aug-2020.)
⊢ (𝐴 ∈ 𝐵 ↔ (𝐴 ∈ 𝐶 ∧ (𝐹‘𝐴) ∈ ∩ {𝑥 ∣ 𝜑}))    &   ⊢ (𝐴 ∈ 𝐸 ↔ (𝐴 ∈ 𝐶 ∧ (𝐹‘𝐴) ∈ 𝑥))    ⇒   ⊢ (𝐴 ∈ 𝐵 ↔ (𝐴 ∈ 𝐶 ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝐸)))
Theorem	fvnonrel 43947	The function value of any class under a non-relation is empty. (Contributed by RP, 23-Oct-2020.)
⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) = ∅
Theorem	elinlem 43948	Two ways to say a set is a member of an intersection. (Contributed by RP, 19-Aug-2020.)
⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ( I ‘𝐴) ∈ 𝐶))
Theorem	elcnvcnvlem 43949	Two ways to say a set is a member of the converse of the converse of a class. (Contributed by RP, 20-Aug-2020.)
⊢ (𝐴 ∈ ◡◡𝐵 ↔ (𝐴 ∈ (V × V) ∧ ( I ‘𝐴) ∈ 𝐵))
21.36.4.12  RP ADDTO: Finite induction (for finite ordinals) Original probably needs new subsection for Relation-related existence theorems.
Theorem	cnvcnvintabd 43950*	Value of the relationship content of the intersection of a class. (Contributed by RP, 20-Aug-2020.)
⊢ (𝜑 → ∃𝑥𝜓)    ⇒   ⊢ (𝜑 → ◡◡∩ {𝑥 ∣ 𝜓} = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜓)})
21.36.4.13  RP ADDTO: First and second members of an ordered pair
Theorem	elcnvlem 43951	Two ways to say a set is a member of the converse of a class. (Contributed by RP, 19-Aug-2020.)
⊢ 𝐹 = (𝑥 ∈ (V × V) ↦ ⟨(2nd ‘𝑥), (1st ‘𝑥)⟩)    ⇒   ⊢ (𝐴 ∈ ◡𝐵 ↔ (𝐴 ∈ (V × V) ∧ (𝐹‘𝐴) ∈ 𝐵))
Theorem	elcnvintab 43952*	Two ways of saying a set is an element of the converse of the intersection of a class. (Contributed by RP, 19-Aug-2020.)
⊢ (𝐴 ∈ ◡∩ {𝑥 ∣ 𝜑} ↔ (𝐴 ∈ (V × V) ∧ ∀𝑥(𝜑 → 𝐴 ∈ ◡𝑥)))
Theorem	cnvintabd 43953*	Value of the converse of the intersection of a nonempty class. (Contributed by RP, 20-Aug-2020.)
⊢ (𝜑 → ∃𝑥𝜓)    ⇒   ⊢ (𝜑 → ◡∩ {𝑥 ∣ 𝜓} = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡𝑥 ∧ 𝜓)})
21.36.4.14  RP ADDTO: The reflexive and transitive properties of relations
Theorem	undmrnresiss 43954*	Two ways of saying the identity relation restricted to the union of the domain and range of a relation is a subset of a relation. Generalization of reflexg 43955. (Contributed by RP, 26-Sep-2020.)
⊢ (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐵 ↔ ∀𝑥∀𝑦(𝑥𝐴𝑦 → (𝑥𝐵𝑥 ∧ 𝑦𝐵𝑦)))
Theorem	reflexg 43955*	Two ways of saying a relation is reflexive over its domain and range. (Contributed by RP, 4-Aug-2020.)
⊢ (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐴 ↔ ∀𝑥∀𝑦(𝑥𝐴𝑦 → (𝑥𝐴𝑥 ∧ 𝑦𝐴𝑦)))
Theorem	cnvssco 43956*	A condition weaker than reflexivity. (Contributed by RP, 3-Aug-2020.)
⊢ (◡𝐴 ⊆ ◡(𝐵 ∘ 𝐶) ↔ ∀𝑥∀𝑦∃𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧 ∧ 𝑧𝐵𝑦)))
Theorem	refimssco 43957	Reflexive relations are subsets of their self-composition. (Contributed by RP, 4-Aug-2020.)
⊢ (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐴 → ◡𝐴 ⊆ ◡(𝐴 ∘ 𝐴))
21.36.4.15  RP ADDTO: Basic properties of closures
Theorem	cleq2lem 43958	Equality implies bijection. (Contributed by RP, 24-Jul-2020.)
⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 = 𝐵 → ((𝑅 ⊆ 𝐴 ∧ 𝜑) ↔ (𝑅 ⊆ 𝐵 ∧ 𝜓)))
Theorem	cbvcllem 43959*	Change of bound variable in class of supersets of a with a property. (Contributed by RP, 24-Jul-2020.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜑)} = {𝑦 ∣ (𝑋 ⊆ 𝑦 ∧ 𝜓)}
Theorem	clublem 43960*	If a superset 𝑌 of 𝑋 possesses the property parameterized in 𝑥 in 𝜓, then 𝑌 is a superset of the closure of that property for the set 𝑋. (Contributed by RP, 23-Jul-2020.)
⊢ (𝜑 → 𝑌 ∈ V)    &   ⊢ (𝑥 = 𝑌 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑌)    &   ⊢ (𝜑 → 𝜒)    ⇒   ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} ⊆ 𝑌)
Theorem	clss2lem 43961*	The closure of a property is a superset of the closure of a less restrictive property. (Contributed by RP, 24-Jul-2020.)
⊢ (𝜑 → (𝜒 → 𝜓))    ⇒   ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} ⊆ ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜒)})
Theorem	dfid7 43962*	Definition of identity relation as the trivial closure. (Contributed by RP, 26-Jul-2020.)
⊢ I = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)})
Theorem	mptrcllem 43963*	Show two versions of a closure with reflexive properties are equal. (Contributed by RP, 19-Oct-2020.)
⊢ (𝑥 ∈ 𝑉 → ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} ∈ V)    &   ⊢ (𝑥 ∈ 𝑉 → ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓)} ∈ V)    &   ⊢ (𝑥 ∈ 𝑉 → 𝜒)    &   ⊢ (𝑥 ∈ 𝑉 → 𝜃)    &   ⊢ (𝑥 ∈ 𝑉 → 𝜏)    &   ⊢ (𝑦 = ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓)} → (𝜑 ↔ 𝜒))    &   ⊢ (𝑦 = ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓)} → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ↔ 𝜃))    &   ⊢ (𝑧 = ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} → (𝜓 ↔ 𝜏))    ⇒   ⊢ (𝑥 ∈ 𝑉 ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))}) = (𝑥 ∈ 𝑉 ↦ ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓)})
Theorem	cotrintab 43964	The intersection of a class is a transitive relation if membership in the class implies the member is a transitive relation. (Contributed by RP, 28-Oct-2020.)
⊢ (𝜑 → (𝑥 ∘ 𝑥) ⊆ 𝑥)    ⇒   ⊢ (∩ {𝑥 ∣ 𝜑} ∘ ∩ {𝑥 ∣ 𝜑}) ⊆ ∩ {𝑥 ∣ 𝜑}
Theorem	rclexi 43965*	The reflexive closure of a set exists. (Contributed by RP, 27-Oct-2020.)
⊢ 𝐴 ∈ 𝑉    ⇒   ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)} ∈ V
Theorem	rtrclexlem 43966	Existence of relation implies existence of union with Cartesian product of domain and range. (Contributed by RP, 1-Nov-2020.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∪ ((dom 𝑅 ∪ ran 𝑅) × (dom 𝑅 ∪ ran 𝑅))) ∈ V)
Theorem	rtrclex 43967*	The reflexive-transitive closure of a set exists. (Contributed by RP, 1-Nov-2020.)
⊢ (𝐴 ∈ V ↔ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ ((𝑥 ∘ 𝑥) ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))} ∈ V)
Theorem	trclubgNEW 43968*	If a relation exists then the transitive closure has an upper bound. (Contributed by RP, 24-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    ⇒   ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
Theorem	trclubNEW 43969*	If a relation exists then the transitive closure has an upper bound. (Contributed by RP, 24-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    &   ⊢ (𝜑 → Rel 𝑅)    ⇒   ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (dom 𝑅 × ran 𝑅))
Theorem	trclexi 43970*	The transitive closure of a set exists. (Contributed by RP, 27-Oct-2020.)
⊢ 𝐴 ∈ 𝑉    ⇒   ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ∈ V
Theorem	rtrclexi 43971*	The reflexive-transitive closure of a set exists. (Contributed by RP, 27-Oct-2020.)
⊢ 𝐴 ∈ 𝑉    ⇒   ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ ((𝑥 ∘ 𝑥) ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))} ∈ V
Theorem	clrellem 43972*	When the property 𝜓 holds for a relation substituted for 𝑥, then the closure on that property is a relation if the base set is a relation. (Contributed by RP, 30-Jul-2020.)
⊢ (𝜑 → 𝑌 ∈ V)    &   ⊢ (𝜑 → Rel 𝑋)    &   ⊢ (𝑥 = ◡◡𝑌 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑌)    &   ⊢ (𝜑 → 𝜒)    ⇒   ⊢ (𝜑 → Rel ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)})
Theorem	clcnvlem 43973*	When 𝐴, an upper bound of the closure, exists and certain substitutions hold the converse of the closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.)
⊢ ((𝜑 ∧ 𝑥 = (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋))) → (𝜒 → 𝜓))    &   ⊢ ((𝜑 ∧ 𝑦 = ◡𝑥) → (𝜓 → 𝜒))    &   ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃))    &   ⊢ (𝜑 → 𝑋 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ V)    &   ⊢ (𝜑 → 𝜃)    ⇒   ⊢ (𝜑 → ◡∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} = ∩ {𝑦 ∣ (◡𝑋 ⊆ 𝑦 ∧ 𝜒)})
Theorem	cnvtrucl0 43974*	The converse of the trivial closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.)
⊢ (𝑋 ∈ 𝑉 → ◡∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ ⊤)} = ∩ {𝑦 ∣ (◡𝑋 ⊆ 𝑦 ∧ ⊤)})
Theorem	cnvrcl0 43975*	The converse of the reflexive closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.)
⊢ (𝑋 ∈ 𝑉 → ◡∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)} = ∩ {𝑦 ∣ (◡𝑋 ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)})
Theorem	cnvtrcl0 43976*	The converse of the transitive closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.)
⊢ (𝑋 ∈ 𝑉 → ◡∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = ∩ {𝑦 ∣ (◡𝑋 ⊆ 𝑦 ∧ (𝑦 ∘ 𝑦) ⊆ 𝑦)})
Theorem	dmtrcl 43977*	The domain of the transitive closure is equal to the domain of its base relation. (Contributed by RP, 1-Nov-2020.)
⊢ (𝑋 ∈ 𝑉 → dom ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = dom 𝑋)
Theorem	rntrcl 43978*	The range of the transitive closure is equal to the range of its base relation. (Contributed by RP, 1-Nov-2020.)
⊢ (𝑋 ∈ 𝑉 → ran ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = ran 𝑋)
Theorem	dfrtrcl5 43979*	Definition of reflexive-transitive closure as a standard closure. (Contributed by RP, 1-Nov-2020.)
⊢ t* = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ∧ (𝑦 ∘ 𝑦) ⊆ 𝑦))})
21.36.4.16  RP REPLACE: Definitions and basic properties of transitive closures
Theorem	trcleq2lemRP 43980	Equality implies bijection. (Contributed by RP, 5-May-2020.) (Proof modification is discouraged.)
⊢ (𝐴 = 𝐵 → ((𝑅 ⊆ 𝐴 ∧ (𝐴 ∘ 𝐴) ⊆ 𝐴) ↔ (𝑅 ⊆ 𝐵 ∧ (𝐵 ∘ 𝐵) ⊆ 𝐵)))
21.36.4.17  Additions for square root; absolute value This is based on the observation that the real and imaginary parts of a complex number can be calculated from the number's absolute and real part and the sign of its imaginary part. Formalization of the formula in sqrtcval 43991 was motivated by a short Michael Penn video.
Theorem	sqrtcvallem1 43981	Two ways of saying a complex number does not lie on the positive real axis. Lemma for sqrtcval 43991. (Contributed by RP, 17-May-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (((ℑ‘𝐴) = 0 → (ℜ‘𝐴) ≤ 0) ↔ ¬ 𝐴 ∈ ℝ+))
Theorem	reabsifneg 43982	Alternate expression for the absolute value of a real number. Lemma for sqrtcval 43991. (Contributed by RP, 11-May-2024.)
⊢ (𝐴 ∈ ℝ → (abs‘𝐴) = if(𝐴 < 0, -𝐴, 𝐴))
Theorem	reabsifnpos 43983	Alternate expression for the absolute value of a real number. (Contributed by RP, 11-May-2024.)
⊢ (𝐴 ∈ ℝ → (abs‘𝐴) = if(𝐴 ≤ 0, -𝐴, 𝐴))
Theorem	reabsifpos 43984	Alternate expression for the absolute value of a real number. (Contributed by RP, 11-May-2024.)
⊢ (𝐴 ∈ ℝ → (abs‘𝐴) = if(0 < 𝐴, 𝐴, -𝐴))
Theorem	reabsifnneg 43985	Alternate expression for the absolute value of a real number. (Contributed by RP, 11-May-2024.)
⊢ (𝐴 ∈ ℝ → (abs‘𝐴) = if(0 ≤ 𝐴, 𝐴, -𝐴))
Theorem	reabssgn 43986	Alternate expression for the absolute value of a real number. (Contributed by RP, 22-May-2024.)
⊢ (𝐴 ∈ ℝ → (abs‘𝐴) = ((sgn‘𝐴) · 𝐴))
Theorem	sqrtcvallem2 43987	Equivalent to saying that the square of the imaginary component of the square root of a complex number is a nonnegative real number. Lemma for sqrtcval 43991. See imsqrtval 43994. (Contributed by RP, 11-May-2024.)
⊢ (𝐴 ∈ ℂ → 0 ≤ (((abs‘𝐴) − (ℜ‘𝐴)) / 2))
Theorem	sqrtcvallem3 43988	Equivalent to saying that the absolute value of the imaginary component of the square root of a complex number is a real number. Lemma for sqrtcval 43991, sqrtcval2 43992, resqrtval 43993, and imsqrtval 43994. (Contributed by RP, 11-May-2024.)
⊢ (𝐴 ∈ ℂ → (√‘(((abs‘𝐴) − (ℜ‘𝐴)) / 2)) ∈ ℝ)
Theorem	sqrtcvallem4 43989	Equivalent to saying that the square of the real component of the square root of a complex number is a nonnegative real number. Lemma for sqrtcval 43991. See resqrtval 43993. (Contributed by RP, 11-May-2024.)
⊢ (𝐴 ∈ ℂ → 0 ≤ (((abs‘𝐴) + (ℜ‘𝐴)) / 2))
Theorem	sqrtcvallem5 43990	Equivalent to saying that the real component of the square root of a complex number is a real number. Lemma for resqrtval 43993 and imsqrtval 43994. (Contributed by RP, 11-May-2024.)
⊢ (𝐴 ∈ ℂ → (√‘(((abs‘𝐴) + (ℜ‘𝐴)) / 2)) ∈ ℝ)
Theorem	sqrtcval 43991	Explicit formula for the complex square root in terms of the square root of nonnegative reals. The right-hand side is decomposed into real and imaginary parts in the format expected by crrei 15127 and crimi 15128. This formula can be found in section 3.7.27 of Handbook of Mathematical Functions, ed. M. Abramowitz and I. A. Stegun (1965, Dover Press). (Contributed by RP, 18-May-2024.)
⊢ (𝐴 ∈ ℂ → (√‘𝐴) = ((√‘(((abs‘𝐴) + (ℜ‘𝐴)) / 2)) + (i · (if((ℑ‘𝐴) < 0, -1, 1) · (√‘(((abs‘𝐴) − (ℜ‘𝐴)) / 2))))))
Theorem	sqrtcval2 43992	Explicit formula for the complex square root in terms of the square root of nonnegative reals. The right side is slightly more compact than sqrtcval 43991. (Contributed by RP, 18-May-2024.)
⊢ (𝐴 ∈ ℂ → (√‘𝐴) = ((√‘(((abs‘𝐴) + (ℜ‘𝐴)) / 2)) + (if((ℑ‘𝐴) < 0, -i, i) · (√‘(((abs‘𝐴) − (ℜ‘𝐴)) / 2)))))
Theorem	resqrtval 43993	Real part of the complex square root. (Contributed by RP, 18-May-2024.)
⊢ (𝐴 ∈ ℂ → (ℜ‘(√‘𝐴)) = (√‘(((abs‘𝐴) + (ℜ‘𝐴)) / 2)))
Theorem	imsqrtval 43994	Imaginary part of the complex square root. (Contributed by RP, 18-May-2024.)
⊢ (𝐴 ∈ ℂ → (ℑ‘(√‘𝐴)) = (if((ℑ‘𝐴) < 0, -1, 1) · (√‘(((abs‘𝐴) − (ℜ‘𝐴)) / 2))))
Theorem	resqrtvalex 43995	Example for resqrtval 43993. (Contributed by RP, 21-May-2024.)
⊢ (ℜ‘(√‘(;15 + (i · 8)))) = 4
Theorem	imsqrtvalex 43996	Example for imsqrtval 43994. (Contributed by RP, 21-May-2024.)
⊢ (ℑ‘(√‘(;15 + (i · 8)))) = 1
21.36.5  Additional statements on relations and subclasses
Theorem	al3im 43997	Version of ax-4 1811 for a nested implication. (Contributed by RP, 13-Apr-2020.)
⊢ (∀𝑥(𝜑 → (𝜓 → (𝜒 → 𝜃))) → (∀𝑥𝜑 → (∀𝑥𝜓 → (∀𝑥𝜒 → ∀𝑥𝜃))))
Theorem	intima0 43998*	Two ways of expressing the intersection of images of a class. (Contributed by RP, 13-Apr-2020.)
⊢ ∩ 𝑎 ∈ 𝐴 (𝑎 “ 𝐵) = ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)}
Theorem	elimaint 43999*	Element of image of intersection. (Contributed by RP, 13-Apr-2020.)
⊢ (𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
Theorem	cnviun 44000*	Converse of indexed union. (Contributed by RP, 20-Jun-2020.)
⊢ ◡∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 ◡𝐵