P. 83 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 8201-8300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fival 8201*	The set of all the finite intersections of the elements of 𝐴. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) = {𝑦 ∣ ∃𝑥 ∈ (𝒫 𝐴 ∩ Fin)𝑦 = ∩ 𝑥})
Theorem	elfi 8202*	Specific properties of an element of (fi‘𝐵). (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ (𝒫 𝐵 ∩ Fin)𝐴 = ∩ 𝑥))
Theorem	elfi2 8203*	The empty intersection need not be considered in the set of finite intersections. (Contributed by Mario Carneiro, 21-Mar-2015.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ (fi‘𝐵) ↔ ∃𝑥 ∈ ((𝒫 𝐵 ∩ Fin) ∖ {∅})𝐴 = ∩ 𝑥))
Theorem	elfir 8204	Sufficient condition for an element of (fi‘𝐵). (Contributed by Mario Carneiro, 24-Nov-2013.)
⊢ ((𝐵 ∈ 𝑉 ∧ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ∩ 𝐴 ∈ (fi‘𝐵))
Theorem	intrnfi 8205	Sufficient condition for the intersection of the range of a function to be in the set of finite intersections. (Contributed by Mario Carneiro, 30-Aug-2015.)
⊢ ((𝐵 ∈ 𝑉 ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ∩ ran 𝐹 ∈ (fi‘𝐵))
Theorem	iinfi 8206*	An indexed intersection of elements of 𝐶 is an element of the finite intersections of 𝐶. (Contributed by Mario Carneiro, 30-Aug-2015.)
⊢ ((𝐶 ∈ 𝑉 ∧ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ (fi‘𝐶))
Theorem	inelfi 8207	The intersection of two sets is a finite intersection. (Contributed by Thierry Arnoux, 6-Jan-2017.)
⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ∩ 𝐵) ∈ (fi‘𝑋))
Theorem	ssfii 8208	Any element of a set 𝐴 is the intersection of a finite subset of 𝐴. (Contributed by FL, 27-Apr-2008.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (fi‘𝐴))
Theorem	fi0 8209	The set of finite intersections of the empty set. (Contributed by Mario Carneiro, 30-Aug-2015.)
⊢ (fi‘∅) = ∅
Theorem	fieq0 8210	If 𝐴 is not empty, the class of all the finite intersections of 𝐴 is not empty either. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 = ∅ ↔ (fi‘𝐴) = ∅))
Theorem	fiin 8211	The elements of (fi‘𝐶) are closed under finite intersection. (Contributed by Mario Carneiro, 24-Nov-2013.)
⊢ ((𝐴 ∈ (fi‘𝐶) ∧ 𝐵 ∈ (fi‘𝐶)) → (𝐴 ∩ 𝐵) ∈ (fi‘𝐶))
Theorem	dffi2 8212*	The set of finite intersections is the smallest set that contains 𝐴 and is closed under pairwise intersection. (Contributed by Mario Carneiro, 24-Nov-2013.)
⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) = ∩ {𝑧 ∣ (𝐴 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ∈ 𝑧)})
Theorem	fiss 8213	Subset relationship for function fi. (Contributed by Jeff Hankins, 7-Oct-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (fi‘𝐴) ⊆ (fi‘𝐵))
Theorem	inficl 8214*	A set which is closed under pairwise intersection is closed under finite intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴 ↔ (fi‘𝐴) = 𝐴))
Theorem	fipwuni 8215	The set of finite intersections of a set is contained in the powerset of the union of the elements of 𝐴. (Contributed by Mario Carneiro, 24-Nov-2013.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
⊢ (fi‘𝐴) ⊆ 𝒫 ∪ 𝐴
Theorem	fisn 8216	A singleton is closed under finite intersections. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ (fi‘{𝐴}) = {𝐴}
Theorem	fiuni 8217	The union of the finite intersections of a set is simply the union of the set itself. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 = ∪ (fi‘𝐴))
Theorem	fipwss 8218	If a set is a family of subsets of some base set, then so is its finite intersection. (Contributed by Stefan O'Rear, 2-Aug-2015.)
⊢ (𝐴 ⊆ 𝒫 𝑋 → (fi‘𝐴) ⊆ 𝒫 𝑋)
Theorem	elfiun 8219*	A finite intersection of elements taken from a union of collections. (Contributed by Jeff Hankins, 15-Nov-2009.) (Proof shortened by Mario Carneiro, 26-Nov-2013.)
⊢ ((𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐾) → (𝐴 ∈ (fi‘(𝐵 ∪ 𝐶)) ↔ (𝐴 ∈ (fi‘𝐵) ∨ 𝐴 ∈ (fi‘𝐶) ∨ ∃𝑥 ∈ (fi‘𝐵)∃𝑦 ∈ (fi‘𝐶)𝐴 = (𝑥 ∩ 𝑦))))
Theorem	dffi3 8220*	The set of finite intersections can be "constructed" inductively by iterating binary intersection ω-many times. (Contributed by Mario Carneiro, 21-Mar-2015.)
⊢ 𝑅 = (𝑢 ∈ V ↦ ran (𝑦 ∈ 𝑢, 𝑧 ∈ 𝑢 ↦ (𝑦 ∩ 𝑧)))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) = ∪ (rec(𝑅, 𝐴) “ ω))
Theorem	fifo 8221*	Describe a surjection from nonempty finite sets to finite intersections. (Contributed by Mario Carneiro, 18-May-2015.)
⊢ 𝐹 = (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ ∩ 𝑦)    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴))
2.4.30  Hall's marriage theorem
Theorem	marypha1lem 8222*	Core induction for Philip Hall's marriage theorem. (Contributed by Stefan O'Rear, 19-Feb-2015.)
⊢ (𝐴 ∈ Fin → (𝑏 ∈ Fin → ∀𝑐 ∈ 𝒫 (𝐴 × 𝑏)(∀𝑑 ∈ 𝒫 𝐴𝑑 ≼ (𝑐 “ 𝑑) → ∃𝑒 ∈ 𝒫 𝑐𝑒:𝐴–1-1→V)))
Theorem	marypha1 8223*	(Philip) Hall's marriage theorem, sufficiency: a finite relation contains an injection if there is no subset of its domain which would be forced to violate the pigeonhole principle. (Contributed by Stefan O'Rear, 20-Feb-2015.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → 𝐶 ⊆ (𝐴 × 𝐵))    &   ⊢ ((𝜑 ∧ 𝑑 ⊆ 𝐴) → 𝑑 ≼ (𝐶 “ 𝑑))    ⇒   ⊢ (𝜑 → ∃𝑓 ∈ 𝒫 𝐶𝑓:𝐴–1-1→𝐵)
Theorem	marypha2lem1 8224*	Lemma for marypha2 8228. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)
⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥))    ⇒   ⊢ 𝑇 ⊆ (𝐴 × ∪ ran 𝐹)
Theorem	marypha2lem2 8225*	Lemma for marypha2 8228. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)
⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥))    ⇒   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))}
Theorem	marypha2lem3 8226*	Lemma for marypha2 8228. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)
⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥))    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐺 ⊆ 𝑇 ↔ ∀𝑥 ∈ 𝐴 (𝐺‘𝑥) ∈ (𝐹‘𝑥)))
Theorem	marypha2lem4 8227*	Lemma for marypha2 8228. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)
⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥))    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝑇 “ 𝑋) = ∪ (𝐹 “ 𝑋))
Theorem	marypha2 8228*	Version of marypha1 8223 using a functional family of sets instead of a relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐹:𝐴⟶Fin)    &   ⊢ ((𝜑 ∧ 𝑑 ⊆ 𝐴) → 𝑑 ≼ ∪ (𝐹 “ 𝑑))    ⇒   ⊢ (𝜑 → ∃𝑔(𝑔:𝐴–1-1→V ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝐹‘𝑥)))
2.4.31  Supremum and infimum
Syntax	csup 8229	Extend class notation to include supremum of class 𝐴. Here 𝑅 is ordinarily a relation that strictly orders class 𝐵. For example, 𝑅 could be 'less than' and 𝐵 could be the set of real numbers.
class sup(𝐴, 𝐵, 𝑅)
Syntax	cinf 8230	Extend class notation to include infimum of class 𝐴. Here 𝑅 is ordinarily a relation that strictly orders class 𝐵. For example, 𝑅 could be 'less than' and 𝐵 could be the set of real numbers.
class inf(𝐴, 𝐵, 𝑅)
Definition	df-sup 8231*	Define the supremum of class 𝐴. It is meaningful when 𝑅 is a relation that strictly orders 𝐵 and when the supremum exists. For example, 𝑅 could be 'less than', 𝐵 could be the set of real numbers, and 𝐴 could be the set of all positive reals whose square is less than 2; in this case the supremum is defined as the square root of 2 per sqrtval 13825. See dfsup2 8233 for alternate definition not requiring dummy variables. (Contributed by NM, 22-May-1999.)
⊢ sup(𝐴, 𝐵, 𝑅) = ∪ {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧))}
Definition	df-inf 8232	Define the infimum of class 𝐴. It is meaningful when 𝑅 is a relation that strictly orders 𝐵 and when the infimum exists. For example, 𝑅 could be 'less than', 𝐵 could be the set of real numbers, and 𝐴 could be the set of all positive reals; in this case the infimum is 0. The infimum is defined as the supremum using the converse ordering relation. In the given example, 0 is the supremum of all reals (greatest real number) for which all positive reals are greater. (Contributed by AV, 2-Sep-2020.)
⊢ inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅)
Theorem	dfsup2 8233	Quantifier free definition of supremum. (Contributed by Scott Fenton, 19-Feb-2013.)
⊢ sup(𝐵, 𝐴, 𝑅) = ∪ (𝐴 ∖ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵)))))
Theorem	supeq1 8234	Equality theorem for supremum. (Contributed by NM, 22-May-1999.)
⊢ (𝐵 = 𝐶 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅))
Theorem	supeq1d 8235	Equality deduction for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅))
Theorem	supeq1i 8236	Equality inference for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ 𝐵 = 𝐶    ⇒   ⊢ sup(𝐵, 𝐴, 𝑅) = sup(𝐶, 𝐴, 𝑅)
Theorem	supeq2 8237	Equality theorem for supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝐵 = 𝐶 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐶, 𝑅))
Theorem	supeq3 8238	Equality theorem for supremum. (Contributed by Scott Fenton, 13-Jun-2018.)
⊢ (𝑅 = 𝑆 → sup(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, 𝑆))
Theorem	supeq123d 8239	Equality deduction for supremum. (Contributed by Stefan O'Rear, 20-Jan-2015.)
⊢ (𝜑 → 𝐴 = 𝐷)    &   ⊢ (𝜑 → 𝐵 = 𝐸)    &   ⊢ (𝜑 → 𝐶 = 𝐹)    ⇒   ⊢ (𝜑 → sup(𝐴, 𝐵, 𝐶) = sup(𝐷, 𝐸, 𝐹))
Theorem	nfsup 8240	Hypothesis builder for supremum. (Contributed by Mario Carneiro, 20-Mar-2014.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝑅    ⇒   ⊢ Ⅎ𝑥sup(𝐴, 𝐵, 𝑅)
Theorem	supmo 8241*	Any class 𝐵 has at most one supremum in 𝐴 (where 𝑅 is interpreted as 'less than'). (Contributed by NM, 5-May-1999.) (Revised by Mario Carneiro, 24-Dec-2016.)
⊢ (𝜑 → 𝑅 Or 𝐴)    ⇒   ⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
Theorem	supexd 8242	A supremum is a set. (Contributed by NM, 22-May-1999.) (Revised by Mario Carneiro, 24-Dec-2016.)
⊢ (𝜑 → 𝑅 Or 𝐴)    ⇒   ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ V)
Theorem	supeu 8243*	A supremum is unique. Similar to Theorem I.26 of [Apostol] p. 24 (but for suprema in general). (Contributed by NM, 12-Oct-2004.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))    ⇒   ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
Theorem	supval2 8244*	Alternate expression for the supremum. (Contributed by Mario Carneiro, 24-Dec-2016.) (Revised by Thierry Arnoux, 24-Sep-2017.)
⊢ (𝜑 → 𝑅 Or 𝐴)    ⇒   ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))))
Theorem	eqsup 8245*	Sufficient condition for an element to be equal to the supremum. (Contributed by Mario Carneiro, 21-Apr-2015.)
⊢ (𝜑 → 𝑅 Or 𝐴)    ⇒   ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝐶𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝐶 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → sup(𝐵, 𝐴, 𝑅) = 𝐶))
Theorem	eqsupd 8246*	Sufficient condition for an element to be equal to the supremum. (Contributed by Mario Carneiro, 21-Apr-2015.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝐶𝑅𝑦)    &   ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝐶)) → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)    ⇒   ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶)
Theorem	supcl 8247*	A supremum belongs to its base class (closure law). See also supub 8248 and suplub 8249. (Contributed by NM, 12-Oct-2004.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))    ⇒   ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
Theorem	supub 8248*	A supremum is an upper bound. See also supcl 8247 and suplub 8249. This proof demonstrates how to expand an iota-based definition (df-iota 5768) using riotacl2 6524. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))    ⇒   ⊢ (𝜑 → (𝐶 ∈ 𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
Theorem	suplub 8249*	A supremum is the least upper bound. See also supcl 8247 and supub 8248. (Contributed by NM, 13-Oct-2004.) (Revised by Mario Carneiro, 24-Dec-2016.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))    ⇒   ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ 𝐶𝑅sup(𝐵, 𝐴, 𝑅)) → ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧))
Theorem	suplub2 8250*	Bidirectional form of suplub 8249. (Contributed by Mario Carneiro, 6-Sep-2014.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    ⇒   ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) ↔ ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧))
Theorem	supnub 8251*	An upper bound is not less than the supremum. (Contributed by NM, 13-Oct-2004.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))    ⇒   ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐵 ¬ 𝐶𝑅𝑧) → ¬ 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
Theorem	supex 8252	A supremum is a set. (Contributed by NM, 22-May-1999.)
⊢ 𝑅 Or 𝐴    ⇒   ⊢ sup(𝐵, 𝐴, 𝑅) ∈ V
Theorem	sup00 8253	The supremum under an empty base set is always the empty set. (Contributed by AV, 4-Sep-2020.)
⊢ sup(𝐵, ∅, 𝑅) = ∅
Theorem	sup0riota 8254*	The supremum of an empty set is the smallest element of the base set. (Contributed by AV, 4-Sep-2020.)
⊢ (𝑅 Or 𝐴 → sup(∅, 𝐴, 𝑅) = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥))
Theorem	sup0 8255*	The supremum of an empty set under a base set which has a unique smallest element is the smallest element of the base set. (Contributed by AV, 4-Sep-2020.)
⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑋) ∧ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥) → sup(∅, 𝐴, 𝑅) = 𝑋)
Theorem	supmax 8256*	The greatest element of a set is its supremum. Note that the converse is not true; the supremum might not be an element of the set considered. (Contributed by Jeff Hoffman, 17-Jun-2008.) (Proof shortened by OpenAI, 30-Mar-2020.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝐶𝑅𝑦)    ⇒   ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶)
Theorem	fisup2g 8257*	A finite set satisfies the conditions to have a supremum. (Contributed by Mario Carneiro, 28-Apr-2015.)
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → ∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
Theorem	fisupcl 8258	A nonempty finite set contains its supremum. (Contributed by Jeff Madsen, 9-May-2011.)
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵)
Theorem	supgtoreq 8259	The supremum of a finite set is greater than or equal to all the elements of the set. (Contributed by AV, 1-Oct-2019.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑆 = sup(𝐵, 𝐴, 𝑅))    ⇒   ⊢ (𝜑 → (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆))
Theorem	suppr 8260	The supremum of a pair. (Contributed by NM, 17-Jun-2007.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → sup({𝐵, 𝐶}, 𝐴, 𝑅) = if(𝐶𝑅𝐵, 𝐵, 𝐶))
Theorem	supsn 8261	The supremum of a singleton. (Contributed by NM, 2-Oct-2007.)
⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → sup({𝐵}, 𝐴, 𝑅) = 𝐵)
Theorem	supisolem 8262*	Lemma for supiso 8264. (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    ⇒   ⊢ ((𝜑 ∧ 𝐷 ∈ 𝐴) → ((∀𝑦 ∈ 𝐶 ¬ 𝐷𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝐷 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧)) ↔ (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ (𝐹‘𝐷)𝑆𝑤 ∧ ∀𝑤 ∈ 𝐵 (𝑤𝑆(𝐹‘𝐷) → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣))))
Theorem	supisoex 8263*	Lemma for supiso 8264. (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧)))    ⇒   ⊢ (𝜑 → ∃𝑢 ∈ 𝐵 (∀𝑤 ∈ (𝐹 “ 𝐶) ¬ 𝑢𝑆𝑤 ∧ ∀𝑤 ∈ 𝐵 (𝑤𝑆𝑢 → ∃𝑣 ∈ (𝐹 “ 𝐶)𝑤𝑆𝑣)))
Theorem	supiso 8264*	Image of a supremum under an isomorphism. (Contributed by Mario Carneiro, 24-Dec-2016.)
⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐶 𝑦𝑅𝑧)))    &   ⊢ (𝜑 → 𝑅 Or 𝐴)    ⇒   ⊢ (𝜑 → sup((𝐹 “ 𝐶), 𝐵, 𝑆) = (𝐹‘sup(𝐶, 𝐴, 𝑅)))
Theorem	infeq1 8265	Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
⊢ (𝐵 = 𝐶 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))
Theorem	infeq1d 8266	Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅))
Theorem	infeq1i 8267	Equality inference for infimum. (Contributed by AV, 2-Sep-2020.)
⊢ 𝐵 = 𝐶    ⇒   ⊢ inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅)
Theorem	infeq2 8268	Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
⊢ (𝐵 = 𝐶 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐶, 𝑅))
Theorem	infeq3 8269	Equality theorem for infimum. (Contributed by AV, 2-Sep-2020.)
⊢ (𝑅 = 𝑆 → inf(𝐴, 𝐵, 𝑅) = inf(𝐴, 𝐵, 𝑆))
Theorem	infeq123d 8270	Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.)
⊢ (𝜑 → 𝐴 = 𝐷)    &   ⊢ (𝜑 → 𝐵 = 𝐸)    &   ⊢ (𝜑 → 𝐶 = 𝐹)    ⇒   ⊢ (𝜑 → inf(𝐴, 𝐵, 𝐶) = inf(𝐷, 𝐸, 𝐹))
Theorem	nfinf 8271	Hypothesis builder for infimum. (Contributed by AV, 2-Sep-2020.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝑅    ⇒   ⊢ Ⅎ𝑥inf(𝐴, 𝐵, 𝑅)
Theorem	infexd 8272	An infimum is a set. (Contributed by AV, 2-Sep-2020.)
⊢ (𝜑 → 𝑅 Or 𝐴)    ⇒   ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ V)
Theorem	eqinf 8273*	Sufficient condition for an element to be equal to the infimum. (Contributed by AV, 2-Sep-2020.)
⊢ (𝜑 → 𝑅 Or 𝐴)    ⇒   ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝐶 ∧ ∀𝑦 ∈ 𝐴 (𝐶𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)) → inf(𝐵, 𝐴, 𝑅) = 𝐶))
Theorem	eqinfd 8274*	Sufficient condition for an element to be equal to the infimum. (Contributed by AV, 3-Sep-2020.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝑦𝑅𝐶)    &   ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝐶𝑅𝑦)) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)    ⇒   ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)
Theorem	infval 8275*	Alternate expression for the infimum. (Contributed by AV, 2-Sep-2020.)
⊢ (𝜑 → 𝑅 Or 𝐴)    ⇒   ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))))
Theorem	infcllem 8276*	Lemma for infcl 8277, inflb 8278, infglb 8279, etc. (Contributed by AV, 3-Sep-2020.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)))
Theorem	infcl 8277*	An infimum belongs to its base class (closure law). See also inflb 8278 and infglb 8279. (Contributed by AV, 3-Sep-2020.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))    ⇒   ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐴)
Theorem	inflb 8278*	An infimum is a lower bound. See also infcl 8277 and infglb 8279. (Contributed by AV, 3-Sep-2020.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))    ⇒   ⊢ (𝜑 → (𝐶 ∈ 𝐵 → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅)))
Theorem	infglb 8279*	An infimum is the greatest lower bound. See also infcl 8277 and inflb 8278. (Contributed by AV, 3-Sep-2020.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))    ⇒   ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ inf(𝐵, 𝐴, 𝑅)𝑅𝐶) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝐶))
Theorem	infglbb 8280*	Bidirectional form of infglb 8279. (Contributed by AV, 3-Sep-2020.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    ⇒   ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (inf(𝐵, 𝐴, 𝑅)𝑅𝐶 ↔ ∃𝑧 ∈ 𝐵 𝑧𝑅𝐶))
Theorem	infnlb 8281*	A lower bound is not greater than the infimum. (Contributed by AV, 3-Sep-2020.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))    ⇒   ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝐶) → ¬ inf(𝐵, 𝐴, 𝑅)𝑅𝐶))
Theorem	infex 8282	An infimum is a set. (Contributed by AV, 3-Sep-2020.)
⊢ 𝑅 Or 𝐴    ⇒   ⊢ inf(𝐵, 𝐴, 𝑅) ∈ V
Theorem	infmin 8283*	The smallest element of a set is its infimum. Note that the converse is not true; the infimum might not be an element of the set considered. (Contributed by AV, 3-Sep-2020.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝑦𝑅𝐶)    ⇒   ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)
Theorem	infmo 8284*	Any class 𝐵 has at most one infimum in 𝐴 (where 𝑅 is interpreted as 'less than'). (Contributed by AV, 6-Oct-2020.)
⊢ (𝜑 → 𝑅 Or 𝐴)    ⇒   ⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))
Theorem	infeu 8285*	An infimum is unique. (Contributed by AV, 6-Oct-2020.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))    ⇒   ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))
Theorem	fimin2g 8286*	A finite set has a minimum under a total order. (Contributed by AV, 6-Oct-2020.)
⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥)
Theorem	fiming 8287*	A finite set has a minimum under a total order. (Contributed by AV, 6-Oct-2020.)
⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → 𝑥𝑅𝑦))
Theorem	fiinfg 8288*	Lemma showing existence and closure of infimum of a finite set. (Contributed by AV, 6-Oct-2020.)
⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐴 𝑧𝑅𝑦)))
Theorem	fiinf2g 8289*	A finite set satisfies the conditions to have an infimum. (Contributed by AV, 6-Oct-2020.)
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → ∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))
Theorem	fiinfcl 8290	A nonempty finite set contains its infimum. (Contributed by AV, 3-Sep-2020.)
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → inf(𝐵, 𝐴, 𝑅) ∈ 𝐵)
Theorem	infltoreq 8291	The infimum of a finite set is less than or equal to all the elements of the set. (Contributed by AV, 4-Sep-2020.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑆 = inf(𝐵, 𝐴, 𝑅))    ⇒   ⊢ (𝜑 → (𝑆𝑅𝐶 ∨ 𝐶 = 𝑆))
Theorem	infpr 8292	The infimum of a pair. (Contributed by AV, 4-Sep-2020.)
⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → inf({𝐵, 𝐶}, 𝐴, 𝑅) = if(𝐵𝑅𝐶, 𝐵, 𝐶))
Theorem	infsn 8293	The infimum of a singleton. (Contributed by NM, 2-Oct-2007.)
⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → inf({𝐵}, 𝐴, 𝑅) = 𝐵)
Theorem	inf00 8294	The infimum regarding an empty bas set is always the empty set. (Contributed by AV, 4-Sep-2020.)
⊢ inf(𝐵, ∅, 𝑅) = ∅
Theorem	infempty 8295*	The infimum of an empty set under a base set which has a unique greatest element is the greatest element of the base set. (Contributed by AV, 4-Sep-2020.)
⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑋𝑅𝑦) ∧ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦) → inf(∅, 𝐴, 𝑅) = 𝑋)
Theorem	infiso 8296*	Image of an infimum under an isomorphism. (Contributed by AV, 4-Sep-2020.)
⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐶 𝑧𝑅𝑦)))    &   ⊢ (𝜑 → 𝑅 Or 𝐴)    ⇒   ⊢ (𝜑 → inf((𝐹 “ 𝐶), 𝐵, 𝑆) = (𝐹‘inf(𝐶, 𝐴, 𝑅)))
2.4.32  Ordinal isomorphism, Hartog's theorem
Syntax	coi 8297	Extend class definition to include the canonical order isomorphism to an ordinal.
class OrdIso(𝑅, 𝐴)
Definition	df-oi 8298*	Define the canonical order isomorphism from the well-order 𝑅 on 𝐴 to an ordinal. (Contributed by Mario Carneiro, 23-May-2015.)
⊢ OrdIso(𝑅, 𝐴) = if((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴), (recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) ↾ {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑥)𝑧𝑅𝑡}), ∅)
Theorem	dfoi 8299*	Rewrite df-oi 8298 with abbreviations. (Contributed by Mario Carneiro, 24-Jun-2015.)
⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}    &   ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣))    &   ⊢ 𝐹 = recs(𝐺)    ⇒   ⊢ OrdIso(𝑅, 𝐴) = if((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴), (𝐹 ↾ {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}), ∅)
Theorem	oieq1 8300	Equality theorem for ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.)
⊢ (𝑅 = 𝑆 → OrdIso(𝑅, 𝐴) = OrdIso(𝑆, 𝐴))