P. 436 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 43501-43600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ifpororb 43501	Factor conditional logic operator over disjunction in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
⊢ (if-(𝜑, (𝜓 ∨ 𝜒), (𝜃 ∨ 𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) ∨ if-(𝜑, 𝜒, 𝜏)))
Theorem	ifpananb 43502	Factor conditional logic operator over conjunction in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
⊢ (if-(𝜑, (𝜓 ∧ 𝜒), (𝜃 ∧ 𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) ∧ if-(𝜑, 𝜒, 𝜏)))
Theorem	ifpnannanb 43503	Factor conditional logic operator over nand in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
⊢ (if-(𝜑, (𝜓 ⊼ 𝜒), (𝜃 ⊼ 𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) ⊼ if-(𝜑, 𝜒, 𝜏)))
Theorem	ifpor123g 43504	Disjunction of conditional logical operators. (Contributed by RP, 18-Apr-2020.)
⊢ ((if-(𝜑, 𝜒, 𝜏) ∨ if-(𝜓, 𝜃, 𝜂)) ↔ ((((𝜑 → ¬ 𝜓) ∨ (𝜒 ∨ 𝜃)) ∧ ((𝜓 → 𝜑) ∨ (𝜏 ∨ 𝜃))) ∧ (((𝜑 → 𝜓) ∨ (𝜒 ∨ 𝜂)) ∧ ((¬ 𝜓 → 𝜑) ∨ (𝜏 ∨ 𝜂)))))
Theorem	ifpimim 43505	Consequnce of implication. (Contributed by RP, 17-Apr-2020.)
⊢ (if-(𝜑, (𝜓 → 𝜒), (𝜃 → 𝜏)) → (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏)))
Theorem	ifpbibib 43506	Factor conditional logic operator over biconditional in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
⊢ (if-(𝜑, (𝜓 ↔ 𝜒), (𝜃 ↔ 𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) ↔ if-(𝜑, 𝜒, 𝜏)))
Theorem	ifpxorxorb 43507	Factor conditional logic operator over xor in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
⊢ (if-(𝜑, (𝜓 ⊻ 𝜒), (𝜃 ⊻ 𝜏)) ↔ (if-(𝜑, 𝜓, 𝜃) ⊻ if-(𝜑, 𝜒, 𝜏)))
21.36.4.2  Sophisms
Theorem	rp-fakeimass 43508	A special case where implication appears to conform to a mixed associative law. (Contributed by RP, 29-Feb-2020.)
⊢ ((𝜑 ∨ 𝜒) ↔ (((𝜑 → 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 → 𝜒))))
Theorem	rp-fakeanorass 43509	A special case where a mixture of and and or appears to conform to a mixed associative law. (Contributed by RP, 26-Feb-2020.)
⊢ ((𝜒 → 𝜑) ↔ (((𝜑 ∧ 𝜓) ∨ 𝜒) ↔ (𝜑 ∧ (𝜓 ∨ 𝜒))))
Theorem	rp-fakeoranass 43510	A special case where a mixture of or and and appears to conform to a mixed associative law. (Contributed by RP, 29-Feb-2020.)
⊢ ((𝜑 → 𝜒) ↔ (((𝜑 ∨ 𝜓) ∧ 𝜒) ↔ (𝜑 ∨ (𝜓 ∧ 𝜒))))
Theorem	rp-fakeinunass 43511	A special case where a mixture of intersection and union appears to conform to a mixed associative law. (Contributed by RP, 26-Feb-2020.)
⊢ (𝐶 ⊆ 𝐴 ↔ ((𝐴 ∩ 𝐵) ∪ 𝐶) = (𝐴 ∩ (𝐵 ∪ 𝐶)))
Theorem	rp-fakeuninass 43512	A special case where a mixture of union and intersection appears to conform to a mixed associative law. (Contributed by RP, 29-Feb-2020.)
⊢ (𝐴 ⊆ 𝐶 ↔ ((𝐴 ∪ 𝐵) ∩ 𝐶) = (𝐴 ∪ (𝐵 ∩ 𝐶)))
21.36.4.3  Finite Sets Membership in the class of finite sets can be expressed in many ways.
Theorem	rp-isfinite5 43513*	A set is said to be finite if it can be put in one-to-one correspondence with all the natural numbers between 1 and some 𝑛 ∈ ℕ0. (Contributed by RP, 3-Mar-2020.)
⊢ (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴)
Theorem	rp-isfinite6 43514*	A set is said to be finite if it is either empty or it can be put in one-to-one correspondence with all the natural numbers between 1 and some 𝑛 ∈ ℕ. (Contributed by RP, 10-Mar-2020.)
⊢ (𝐴 ∈ Fin ↔ (𝐴 = ∅ ∨ ∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴))
21.36.4.4  General Observations
Theorem	intabssd 43515*	When for each element 𝑦 there is a subset 𝐴 which may substituted for 𝑥 such that 𝑦 satisfying 𝜒 implies 𝑥 satisfies 𝜓 then the intersection of all 𝑥 that satisfy 𝜓 is a subclass the intersection of all 𝑦 that satisfy 𝜒. (Contributed by RP, 17-Oct-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓))    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑦)    ⇒   ⊢ (𝜑 → ∩ {𝑥 ∣ 𝜓} ⊆ ∩ {𝑦 ∣ 𝜒})
Theorem	eu0 43516*	There is only one empty set. (Contributed by RP, 1-Oct-2023.)
⊢ (∀𝑥 ¬ 𝑥 ∈ ∅ ∧ ∃!𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)
Theorem	epelon2 43517	Over the ordinal numbers, one may define the relation 𝐴 E 𝐵 iff 𝐴 ∈ 𝐵 and one finds that, under this ordering, On is a well-ordered class, see epweon 7754. This is a weak form of epelg 5542 which only requires that we know 𝐵 to be a set. (Contributed by RP, 27-Sep-2023.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵))
Theorem	ontric3g 43518*	For all 𝑥, 𝑦 ∈ On, one and only one of the following hold: 𝑥 ∈ 𝑦, 𝑦 = 𝑥, or 𝑦 ∈ 𝑥. This is a transparent strict trichotomy. (Contributed by RP, 27-Sep-2023.)
⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On ((𝑥 ∈ 𝑦 ↔ ¬ (𝑦 = 𝑥 ∨ 𝑦 ∈ 𝑥)) ∧ (𝑦 = 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥)) ∧ (𝑦 ∈ 𝑥 ↔ ¬ (𝑥 ∈ 𝑦 ∨ 𝑦 = 𝑥)))
Theorem	dfsucon 43519*	𝐴 is called a successor ordinal if it is not a limit ordinal and not the empty set. (Contributed by RP, 11-Nov-2023.)
⊢ ((Ord 𝐴 ∧ ¬ Lim 𝐴 ∧ 𝐴 ≠ ∅) ↔ ∃𝑥 ∈ On 𝐴 = suc 𝑥)
Theorem	snen1g 43520	A singleton is equinumerous to ordinal one iff its content is a set. (Contributed by RP, 8-Oct-2023.)
⊢ ({𝐴} ≈ 1o ↔ 𝐴 ∈ V)
Theorem	snen1el 43521	A singleton is equinumerous to ordinal one if its content is an element of it. (Contributed by RP, 8-Oct-2023.)
⊢ ({𝐴} ≈ 1o ↔ 𝐴 ∈ {𝐴})
Theorem	sn1dom 43522	A singleton is dominated by ordinal one. (Contributed by RP, 29-Oct-2023.)
⊢ {𝐴} ≼ 1o
Theorem	pr2dom 43523	An unordered pair is dominated by ordinal two. (Contributed by RP, 29-Oct-2023.)
⊢ {𝐴, 𝐵} ≼ 2o
Theorem	tr3dom 43524	An unordered triple is dominated by ordinal three. (Contributed by RP, 29-Oct-2023.)
⊢ {𝐴, 𝐵, 𝐶} ≼ 3o
Theorem	ensucne0 43525	A class equinumerous to a successor is never empty. (Contributed by RP, 11-Nov-2023.) (Proof shortened by SN, 16-Nov-2023.)
⊢ (𝐴 ≈ suc 𝐵 → 𝐴 ≠ ∅)
Theorem	ensucne0OLD 43526	A class equinumerous to a successor is never empty. (Contributed by RP, 11-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ≈ suc 𝐵 → 𝐴 ≠ ∅)
Theorem	dfom6 43527	Let ω be defined to be the union of the set of all finite ordinals. (Contributed by RP, 27-Sep-2023.)
⊢ ω = ∪ (On ∩ Fin)
Theorem	infordmin 43528	ω is the smallest infinite ordinal. (Contributed by RP, 27-Sep-2023.)
⊢ ∀𝑥 ∈ (On ∖ Fin)ω ⊆ 𝑥
Theorem	iscard4 43529	Two ways to express the property of being a cardinal number. (Contributed by RP, 8-Nov-2023.)
⊢ ((card‘𝐴) = 𝐴 ↔ 𝐴 ∈ ran card)
Theorem	minregex 43530*	Given any cardinal number 𝐴, there exists an argument 𝑥, which yields the least regular uncountable value of ℵ which is greater to or equal to 𝐴. This proof uses AC. (Contributed by RP, 23-Nov-2023.)
⊢ (𝐴 ∈ (ran card ∖ ω) → ∃𝑥 ∈ On 𝑥 = ∩ {𝑦 ∈ On ∣ (∅ ∈ 𝑦 ∧ 𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))})
Theorem	minregex2 43531*	Given any cardinal number 𝐴, there exists an argument 𝑥, which yields the least regular uncountable value of ℵ which dominates 𝐴. This proof uses AC. (Contributed by RP, 24-Nov-2023.)
⊢ (𝐴 ∈ (ran card ∖ ω) → ∃𝑥 ∈ On 𝑥 = ∩ {𝑦 ∈ On ∣ (∅ ∈ 𝑦 ∧ 𝐴 ≼ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))})
Theorem	iscard5 43532*	Two ways to express the property of being a cardinal number. (Contributed by RP, 8-Nov-2023.)
⊢ ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ≈ 𝐴))
Theorem	elrncard 43533*	Let us define a cardinal number to be an element 𝐴 ∈ On such that 𝐴 is not equipotent with any 𝑥 ∈ 𝐴. (Contributed by RP, 1-Oct-2023.)
⊢ (𝐴 ∈ ran card ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ≈ 𝐴))
Theorem	harval3 43534*	(har‘𝐴) is the least cardinal that is greater than 𝐴. (Contributed by RP, 4-Nov-2023.)
⊢ (𝐴 ∈ dom card → (har‘𝐴) = ∩ {𝑥 ∈ ran card ∣ 𝐴 ≺ 𝑥})
Theorem	harval3on 43535*	For any ordinal number 𝐴 let (har‘𝐴) denote the least cardinal that is greater than 𝐴. (Contributed by RP, 4-Nov-2023.)
⊢ (𝐴 ∈ On → (har‘𝐴) = ∩ {𝑥 ∈ ran card ∣ 𝐴 ≺ 𝑥})
Theorem	omssrncard 43536	All natural numbers are cardinals. (Contributed by RP, 1-Oct-2023.)
⊢ ω ⊆ ran card
Theorem	0iscard 43537	0 is a cardinal number. (Contributed by RP, 1-Oct-2023.)
⊢ ∅ ∈ ran card
Theorem	1iscard 43538	1 is a cardinal number. (Contributed by RP, 1-Oct-2023.)
⊢ 1o ∈ ran card
Theorem	omiscard 43539	ω is a cardinal number. (Contributed by RP, 1-Oct-2023.)
⊢ ω ∈ ran card
Theorem	sucomisnotcard 43540	ω +o 1o is not a cardinal number. (Contributed by RP, 1-Oct-2023.)
⊢ ¬ (ω +o 1o) ∈ ran card
Theorem	nna1iscard 43541	For any natural number, the add one operation is results in a cardinal number. (Contributed by RP, 1-Oct-2023.)
⊢ (𝑁 ∈ ω → (𝑁 +o 1o) ∈ ran card)
Theorem	har2o 43542	The least cardinal greater than 2 is 3. (Contributed by RP, 5-Nov-2023.)
⊢ (har‘2o) = 3o
Theorem	en2pr 43543*	A class is equinumerous to ordinal two iff it is a pair of distinct sets. (Contributed by RP, 11-Oct-2023.)
⊢ (𝐴 ≈ 2o ↔ ∃𝑥∃𝑦(𝐴 = {𝑥, 𝑦} ∧ 𝑥 ≠ 𝑦))
Theorem	pr2cv 43544	If an unordered pair is equinumerous to ordinal two, then both parts are sets. (Contributed by RP, 8-Oct-2023.)
⊢ ({𝐴, 𝐵} ≈ 2o → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Theorem	pr2el1 43545	If an unordered pair is equinumerous to ordinal two, then a part is a member. (Contributed by RP, 21-Oct-2023.)
⊢ ({𝐴, 𝐵} ≈ 2o → 𝐴 ∈ {𝐴, 𝐵})
Theorem	pr2cv1 43546	If an unordered pair is equinumerous to ordinal two, then a part is a set. (Contributed by RP, 21-Oct-2023.)
⊢ ({𝐴, 𝐵} ≈ 2o → 𝐴 ∈ V)
Theorem	pr2el2 43547	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 43548	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 43549	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 43550	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 43551	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 43552	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 43553	(ℵ‘1o) is the least uncountable ordinal. (Contributed by RP, 18-Nov-2023.)
⊢ (ℵ‘1o) = ∩ {𝑥 ∈ On ∣ ω ≺ 𝑥}
Theorem	alephiso2 43554	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 43555	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 43556*	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 43557*	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 43558	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 43559	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 43560	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 9284 where two textbook definitions are shown to be equivalent. This property is seen often with ordinal numbers (onin 6366, ordelinel 6438), chains of sets ordered by the proper subset relation (sorpssin 7710), various sets in the field of topology (inopn 22793, incld 22937, innei 23019, ... ) and "universal" classes like weak universes (wunin 10673, tskin 10719) and the class of all sets (inex1g 5277).
Theorem	fipjust 43561*	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 43562*	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 43563*	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 43564*	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 43565*	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 43566*	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 43567*	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 43568*	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 43569*	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 43570	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 43571*	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 43572*	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 43573*	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 43574*	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 43575*	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 43576*	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 43577	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 43578*	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 43579*	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 43580	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 43581	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 43582	Subclass theorem for converse. (Contributed by RP, 22-Oct-2020.)
⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ◡𝐴 ⊆ ◡𝐵))
Theorem	relnonrel 43583	The non-relation part of a relation is empty. (Contributed by RP, 22-Oct-2020.)
⊢ (Rel 𝐴 ↔ (𝐴 ∖ ◡◡𝐴) = ∅)
Theorem	cnvnonrel 43584	The converse of the non-relation part of a class is empty. (Contributed by RP, 18-Oct-2020.)
⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅
Theorem	brnonrel 43585	A non-relation cannot relate any two classes. (Contributed by RP, 23-Oct-2020.)
⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) → ¬ 𝑋(𝐴 ∖ ◡◡𝐴)𝑌)
Theorem	dmnonrel 43586	The domain of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
⊢ dom (𝐴 ∖ ◡◡𝐴) = ∅
Theorem	rnnonrel 43587	The range of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
⊢ ran (𝐴 ∖ ◡◡𝐴) = ∅
Theorem	resnonrel 43588	A restriction of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) = ∅
Theorem	imanonrel 43589	An image under the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
⊢ ((𝐴 ∖ ◡◡𝐴) “ 𝐵) = ∅
Theorem	cononrel1 43590	Composition with the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
⊢ ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ∅
Theorem	cononrel2 43591	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 6023 by Thierry Arnoux.
Theorem	elmapintab 43592*	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 43593	The function value of any class under a non-relation is empty. (Contributed by RP, 23-Oct-2020.)
⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) = ∅
Theorem	elinlem 43594	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 43595	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 43596*	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 43597	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 43598*	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 43599*	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 43600*	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 43601. (Contributed by RP, 26-Sep-2020.)
⊢ (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐵 ↔ ∀𝑥∀𝑦(𝑥𝐴𝑦 → (𝑥𝐵𝑥 ∧ 𝑦𝐵𝑦)))