P. 45 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4401-4500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nfso 4401	Bound-variable hypothesis builder for total orders. (Contributed by Stefan O'Rear, 20-Jan-2015.)
⊢ Ⅎ𝑥𝑅    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥 𝑅 Or 𝐴
Theorem	pocl 4402	Properties of partial order relation in class notation. (Contributed by NM, 27-Mar-1997.)
⊢ (𝑅 Po 𝐴 → ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷))))
Theorem	ispod 4403*	Sufficient conditions for a partial order. (Contributed by NM, 9-Jul-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))    ⇒   ⊢ (𝜑 → 𝑅 Po 𝐴)
Theorem	swopolem 4404*	Perform the substitutions into the strict weak ordering law. (Contributed by Mario Carneiro, 31-Dec-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) → (𝑋𝑅𝑌 → (𝑋𝑅𝑍 ∨ 𝑍𝑅𝑌)))
Theorem	swopo 4405*	A strict weak order is a partial order. (Contributed by Mario Carneiro, 9-Jul-2014.)
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))    ⇒   ⊢ (𝜑 → 𝑅 Po 𝐴)
Theorem	poirr 4406	A partial order relation is irreflexive. (Contributed by NM, 27-Mar-1997.)
⊢ ((𝑅 Po 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵)
Theorem	potr 4407	A partial order relation is a transitive relation. (Contributed by NM, 27-Mar-1997.)
⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷))
Theorem	po2nr 4408	A partial order relation has no 2-cycle loops. (Contributed by NM, 27-Mar-1997.)
⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵))
Theorem	po3nr 4409	A partial order relation has no 3-cycle loops. (Contributed by NM, 27-Mar-1997.)
⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷 ∧ 𝐷𝑅𝐵))
Theorem	po0 4410	Any relation is a partial ordering of the empty set. (Contributed by NM, 28-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ 𝑅 Po ∅
Theorem	pofun 4411*	A function preserves a partial order relation. (Contributed by Jeff Madsen, 18-Jun-2011.)
⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ 𝑋𝑅𝑌}    &   ⊢ (𝑥 = 𝑦 → 𝑋 = 𝑌)    ⇒   ⊢ ((𝑅 Po 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑋 ∈ 𝐵) → 𝑆 Po 𝐴)
Theorem	sopo 4412	A strict linear order is a strict partial order. (Contributed by NM, 28-Mar-1997.)
⊢ (𝑅 Or 𝐴 → 𝑅 Po 𝐴)
Theorem	soss 4413	Subset theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ (𝐴 ⊆ 𝐵 → (𝑅 Or 𝐵 → 𝑅 Or 𝐴))
Theorem	soeq1 4414	Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)
⊢ (𝑅 = 𝑆 → (𝑅 Or 𝐴 ↔ 𝑆 Or 𝐴))
Theorem	soeq2 4415	Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)
⊢ (𝐴 = 𝐵 → (𝑅 Or 𝐴 ↔ 𝑅 Or 𝐵))
Theorem	sonr 4416	A strict order relation is irreflexive. (Contributed by NM, 24-Nov-1995.)
⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵)
Theorem	sotr 4417	A strict order relation is a transitive relation. (Contributed by NM, 21-Jan-1996.)
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷))
Theorem	issod 4418*	An irreflexive, transitive, trichotomous relation is a linear ordering (in the sense of df-iso 4396). (Contributed by NM, 21-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)
⊢ (𝜑 → 𝑅 Po 𝐴)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))    ⇒   ⊢ (𝜑 → 𝑅 Or 𝐴)
Theorem	sowlin 4419	A strict order relation satisfies weak linearity. (Contributed by Jim Kingdon, 6-Oct-2018.)
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐵𝑅𝐶 → (𝐵𝑅𝐷 ∨ 𝐷𝑅𝐶)))
Theorem	so2nr 4420	A strict order relation has no 2-cycle loops. (Contributed by NM, 21-Jan-1996.)
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵))
Theorem	so3nr 4421	A strict order relation has no 3-cycle loops. (Contributed by NM, 21-Jan-1996.)
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷 ∧ 𝐷𝑅𝐵))
Theorem	sotricim 4422	One direction of sotritric 4423 holds for all weakly linear orders. (Contributed by Jim Kingdon, 28-Sep-2019.)
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵𝑅𝐶 → ¬ (𝐵 = 𝐶 ∨ 𝐶𝑅𝐵)))
Theorem	sotritric 4423	A trichotomy relationship, given a trichotomous order. (Contributed by Jim Kingdon, 28-Sep-2019.)
⊢ 𝑅 Or 𝐴    &   ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵))    ⇒   ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵𝑅𝐶 ↔ ¬ (𝐵 = 𝐶 ∨ 𝐶𝑅𝐵)))
Theorem	sotritrieq 4424	A trichotomy relationship, given a trichotomous order. (Contributed by Jim Kingdon, 13-Dec-2019.)
⊢ 𝑅 Or 𝐴    &   ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵))    ⇒   ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵 = 𝐶 ↔ ¬ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵)))
Theorem	so0 4425	Any relation is a strict ordering of the empty set. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ 𝑅 Or ∅
2.3.9  Founded and set-like relations
Syntax	wfrfor 4426	Extend wff notation to include the well-founded predicate.
wff FrFor 𝑅𝐴𝑆
Syntax	wfr 4427	Extend wff notation to include the well-founded predicate. Read: ' 𝑅 is a well-founded relation on 𝐴.'
wff 𝑅 Fr 𝐴
Syntax	wse 4428	Extend wff notation to include the set-like predicate. Read: ' 𝑅 is set-like on 𝐴.'
wff 𝑅 Se 𝐴
Syntax	wwe 4429	Extend wff notation to include the well-ordering predicate. Read: ' 𝑅 well-orders 𝐴.'
wff 𝑅 We 𝐴
Definition	df-frfor 4430*	Define the well-founded relation predicate where 𝐴 might be a proper class. By passing in 𝑆 we allow it potentially to be a proper class rather than a set. (Contributed by Jim Kingdon and Mario Carneiro, 22-Sep-2021.)
⊢ ( FrFor 𝑅𝐴𝑆 ↔ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆) → 𝐴 ⊆ 𝑆))
Definition	df-frind 4431*	Define the well-founded relation predicate. In the presence of excluded middle, there are a variety of equivalent ways to define this. In our case, this definition, in terms of an inductive principle, works better than one along the lines of "there is an element which is minimal when A is ordered by R". Because 𝑠 is constrained to be a set (not a proper class) here, sometimes it may be necessary to use FrFor directly rather than via Fr. (Contributed by Jim Kingdon and Mario Carneiro, 21-Sep-2021.)
⊢ (𝑅 Fr 𝐴 ↔ ∀𝑠 FrFor 𝑅𝐴𝑠)
Definition	df-se 4432*	Define the set-like predicate. (Contributed by Mario Carneiro, 19-Nov-2014.)
⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
Definition	df-wetr 4433*	Define the well-ordering predicate. It is unusual to define "well-ordering" in the absence of excluded middle, but we mean an ordering which is like the ordering which we have for ordinals (for example, it does not entail trichotomy because ordinals do not have that as seen at ordtriexmid 4621). Given excluded middle, well-ordering is usually defined to require trichotomy (and the definition of Fr is typically also different). (Contributed by Mario Carneiro and Jim Kingdon, 23-Sep-2021.)
⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
Theorem	seex 4434*	The 𝑅-preimage of an element of the base set in a set-like relation is a set. (Contributed by Mario Carneiro, 19-Nov-2014.)
⊢ ((𝑅 Se 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝑥 ∈ 𝐴 ∣ 𝑥𝑅𝐵} ∈ V)
Theorem	exse 4435	Any relation on a set is set-like on it. (Contributed by Mario Carneiro, 22-Jun-2015.)
⊢ (𝐴 ∈ 𝑉 → 𝑅 Se 𝐴)
Theorem	sess1 4436	Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
⊢ (𝑅 ⊆ 𝑆 → (𝑆 Se 𝐴 → 𝑅 Se 𝐴))
Theorem	sess2 4437	Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
⊢ (𝐴 ⊆ 𝐵 → (𝑅 Se 𝐵 → 𝑅 Se 𝐴))
Theorem	seeq1 4438	Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
⊢ (𝑅 = 𝑆 → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐴))
Theorem	seeq2 4439	Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
⊢ (𝐴 = 𝐵 → (𝑅 Se 𝐴 ↔ 𝑅 Se 𝐵))
Theorem	nfse 4440	Bound-variable hypothesis builder for set-like relations. (Contributed by Mario Carneiro, 24-Jun-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
⊢ Ⅎ𝑥𝑅    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥 𝑅 Se 𝐴
Theorem	epse 4441	The epsilon relation is set-like on any class. (This is the origin of the term "set-like": a set-like relation "acts like" the epsilon relation of sets and their elements.) (Contributed by Mario Carneiro, 22-Jun-2015.)
⊢ E Se 𝐴
Theorem	frforeq1 4442	Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
⊢ (𝑅 = 𝑆 → ( FrFor 𝑅𝐴𝑇 ↔ FrFor 𝑆𝐴𝑇))
Theorem	freq1 4443	Equality theorem for the well-founded predicate. (Contributed by NM, 9-Mar-1997.)
⊢ (𝑅 = 𝑆 → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐴))
Theorem	frforeq2 4444	Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
⊢ (𝐴 = 𝐵 → ( FrFor 𝑅𝐴𝑇 ↔ FrFor 𝑅𝐵𝑇))
Theorem	freq2 4445	Equality theorem for the well-founded predicate. (Contributed by NM, 3-Apr-1994.)
⊢ (𝐴 = 𝐵 → (𝑅 Fr 𝐴 ↔ 𝑅 Fr 𝐵))
Theorem	frforeq3 4446	Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
⊢ (𝑆 = 𝑇 → ( FrFor 𝑅𝐴𝑆 ↔ FrFor 𝑅𝐴𝑇))
Theorem	nffrfor 4447	Bound-variable hypothesis builder for well-founded relations. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
⊢ Ⅎ𝑥𝑅    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝑆    ⇒   ⊢ Ⅎ𝑥 FrFor 𝑅𝐴𝑆
Theorem	nffr 4448	Bound-variable hypothesis builder for well-founded relations. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
⊢ Ⅎ𝑥𝑅    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥 𝑅 Fr 𝐴
Theorem	frirrg 4449	A well-founded relation is irreflexive. This is the case where 𝐴 exists. (Contributed by Jim Kingdon, 21-Sep-2021.)
⊢ ((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵)
Theorem	fr0 4450	Any relation is well-founded on the empty set. (Contributed by NM, 17-Sep-1993.)
⊢ 𝑅 Fr ∅
Theorem	frind 4451*	Induction over a well-founded set. (Contributed by Jim Kingdon, 28-Sep-2021.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ ((𝜒 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → 𝜓) → 𝜑))    &   ⊢ (𝜒 → 𝑅 Fr 𝐴)    &   ⊢ (𝜒 → 𝐴 ∈ 𝑉)    ⇒   ⊢ ((𝜒 ∧ 𝑥 ∈ 𝐴) → 𝜑)
Theorem	efrirr 4452	Irreflexivity of the epsilon relation: a class founded by epsilon is not a member of itself. (Contributed by NM, 18-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)
⊢ ( E Fr 𝐴 → ¬ 𝐴 ∈ 𝐴)
Theorem	tz7.2 4453	Similar to Theorem 7.2 of [TakeutiZaring] p. 35, of except that the Axiom of Regularity is not required due to antecedent E Fr 𝐴. (Contributed by NM, 4-May-1994.)
⊢ ((Tr 𝐴 ∧ E Fr 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ 𝐴))
Theorem	nfwe 4454	Bound-variable hypothesis builder for well-orderings. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
⊢ Ⅎ𝑥𝑅    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥 𝑅 We 𝐴
Theorem	weeq1 4455	Equality theorem for the well-ordering predicate. (Contributed by NM, 9-Mar-1997.)
⊢ (𝑅 = 𝑆 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐴))
Theorem	weeq2 4456	Equality theorem for the well-ordering predicate. (Contributed by NM, 3-Apr-1994.)
⊢ (𝐴 = 𝐵 → (𝑅 We 𝐴 ↔ 𝑅 We 𝐵))
Theorem	wefr 4457	A well-ordering is well-founded. (Contributed by NM, 22-Apr-1994.)
⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴)
Theorem	wepo 4458	A well-ordering is a partial ordering. (Contributed by Jim Kingdon, 23-Sep-2021.)
⊢ ((𝑅 We 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝑅 Po 𝐴)
Theorem	wetrep 4459*	An epsilon well-ordering is a transitive relation. (Contributed by NM, 22-Apr-1994.)
⊢ (( E We 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧))
Theorem	we0 4460	Any relation is a well-ordering of the empty set. (Contributed by NM, 16-Mar-1997.)
⊢ 𝑅 We ∅
2.3.10  Ordinals
Syntax	word 4461	Extend the definition of a wff to include the ordinal predicate.
wff Ord 𝐴
Syntax	con0 4462	Extend the definition of a class to include the class of all ordinal numbers. (The 0 in the name prevents creating a file called con.html, which causes problems in Windows.)
class On
Syntax	wlim 4463	Extend the definition of a wff to include the limit ordinal predicate.
wff Lim 𝐴
Syntax	csuc 4464	Extend class notation to include the successor function.
class suc 𝐴
Definition	df-iord 4465*	Define the ordinal predicate, which is true for a class that is transitive and whose elements are transitive. Definition of ordinal in [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic". Some sources will define a notation for ordinal order corresponding to < and ≤ but we just use ∈ and ⊆ respectively. (Contributed by Jim Kingdon, 10-Oct-2018.) Use its alias dford3 4466 instead for naming consistency with set.mm. (New usage is discouraged.)
⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 Tr 𝑥))
Theorem	dford3 4466*	Alias for df-iord 4465. Use it instead of df-iord 4465 for naming consistency with set.mm. (Contributed by Jim Kingdon, 10-Oct-2018.)
⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 Tr 𝑥))
Definition	df-on 4467	Define the class of all ordinal numbers. Definition 7.11 of [TakeutiZaring] p. 38. (Contributed by NM, 5-Jun-1994.)
⊢ On = {𝑥 ∣ Ord 𝑥}
Definition	df-ilim 4468	Define the limit ordinal predicate, which is true for an ordinal that has the empty set as an element and is not a successor (i.e. that is the union of itself). Our definition combines the definition of Lim of [BellMachover] p. 471 and Exercise 1 of [TakeutiZaring] p. 42, and then changes 𝐴 ≠ ∅ to ∅ ∈ 𝐴 (which would be equivalent given the law of the excluded middle, but which is not for us). (Contributed by Jim Kingdon, 11-Nov-2018.) Use its alias dflim2 4469 instead for naming consistency with set.mm. (New usage is discouraged.)
⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴))
Theorem	dflim2 4469	Alias for df-ilim 4468. Use it instead of df-ilim 4468 for naming consistency with set.mm. (Contributed by NM, 4-Nov-2004.)
⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴))
Definition	df-suc 4470	Define the successor of a class. When applied to an ordinal number, the successor means the same thing as "plus 1". Definition 7.22 of [TakeutiZaring] p. 41, who use "+ 1" to denote this function. Our definition is a generalization to classes. Although it is not conventional to use it with proper classes, it has no effect on a proper class (sucprc 4511). Some authors denote the successor operation with a prime (apostrophe-like) symbol, such as Definition 6 of [Suppes] p. 134 and the definition of successor in [Mendelson] p. 246 (who uses the symbol "Suc" as a predicate to mean "is a successor ordinal"). The definition of successor of [Enderton] p. 68 denotes the operation with a plus-sign superscript. (Contributed by NM, 30-Aug-1993.)
⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
Theorem	ordeq 4471	Equality theorem for the ordinal predicate. (Contributed by NM, 17-Sep-1993.)
⊢ (𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵))
Theorem	elong 4472	An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ On ↔ Ord 𝐴))
Theorem	elon 4473	An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ On ↔ Ord 𝐴)
Theorem	eloni 4474	An ordinal number has the ordinal property. (Contributed by NM, 5-Jun-1994.)
⊢ (𝐴 ∈ On → Ord 𝐴)
Theorem	elon2 4475	An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004.)
⊢ (𝐴 ∈ On ↔ (Ord 𝐴 ∧ 𝐴 ∈ V))
Theorem	limeq 4476	Equality theorem for the limit predicate. (Contributed by NM, 22-Apr-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ (𝐴 = 𝐵 → (Lim 𝐴 ↔ Lim 𝐵))
Theorem	ordtr 4477	An ordinal class is transitive. (Contributed by NM, 3-Apr-1994.)
⊢ (Ord 𝐴 → Tr 𝐴)
Theorem	ordelss 4478	An element of an ordinal class is a subset of it. (Contributed by NM, 30-May-1994.)
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ 𝐴)
Theorem	trssord 4479	A transitive subclass of an ordinal class is ordinal. (Contributed by NM, 29-May-1994.)
⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → Ord 𝐴)
Theorem	ordelord 4480	An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. (Contributed by NM, 23-Apr-1994.)
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)
Theorem	tron 4481	The class of all ordinal numbers is transitive. (Contributed by NM, 4-May-2009.)
⊢ Tr On
Theorem	ordelon 4482	An element of an ordinal class is an ordinal number. (Contributed by NM, 26-Oct-2003.)
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On)
Theorem	onelon 4483	An element of an ordinal number is an ordinal number. Theorem 2.2(iii) of [BellMachover] p. 469. (Contributed by NM, 26-Oct-2003.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On)
Theorem	ordin 4484	The intersection of two ordinal classes is ordinal. Proposition 7.9 of [TakeutiZaring] p. 37. (Contributed by NM, 9-May-1994.)
⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵))
Theorem	onin 4485	The intersection of two ordinal numbers is an ordinal number. (Contributed by NM, 7-Apr-1995.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∩ 𝐵) ∈ On)
Theorem	onelss 4486	An element of an ordinal number is a subset of the number. (Contributed by NM, 5-Jun-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
Theorem	ordtr1 4487	Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.)
⊢ (Ord 𝐶 → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))
Theorem	ontr1 4488	Transitive law for ordinal numbers. Theorem 7M(b) of [Enderton] p. 192. (Contributed by NM, 11-Aug-1994.)
⊢ (𝐶 ∈ On → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))
Theorem	onintss 4489*	If a property is true for an ordinal number, then the minimum ordinal number for which it is true is smaller or equal. Theorem Schema 61 of [Suppes] p. 228. (Contributed by NM, 3-Oct-2003.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ On → (𝜓 → ∩ {𝑥 ∈ On ∣ 𝜑} ⊆ 𝐴))
Theorem	ord0 4490	The empty set is an ordinal class. (Contributed by NM, 11-May-1994.)
⊢ Ord ∅
Theorem	0elon 4491	The empty set is an ordinal number. Corollary 7N(b) of [Enderton] p. 193. (Contributed by NM, 17-Sep-1993.)
⊢ ∅ ∈ On
Theorem	inton 4492	The intersection of the class of ordinal numbers is the empty set. (Contributed by NM, 20-Oct-2003.)
⊢ ∩ On = ∅
Theorem	nlim0 4493	The empty set is not a limit ordinal. (Contributed by NM, 24-Mar-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ ¬ Lim ∅
Theorem	limord 4494	A limit ordinal is ordinal. (Contributed by NM, 4-May-1995.)
⊢ (Lim 𝐴 → Ord 𝐴)
Theorem	limuni 4495	A limit ordinal is its own supremum (union). (Contributed by NM, 4-May-1995.)
⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)
Theorem	limuni2 4496	The union of a limit ordinal is a limit ordinal. (Contributed by NM, 19-Sep-2006.)
⊢ (Lim 𝐴 → Lim ∪ 𝐴)
Theorem	0ellim 4497	A limit ordinal contains the empty set. (Contributed by NM, 15-May-1994.)
⊢ (Lim 𝐴 → ∅ ∈ 𝐴)
Theorem	limelon 4498	A limit ordinal class that is also a set is an ordinal number. (Contributed by NM, 26-Apr-2004.)
⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → 𝐴 ∈ On)
Theorem	onn0 4499	The class of all ordinal numbers is not empty. (Contributed by NM, 17-Sep-1995.)
⊢ On ≠ ∅
Theorem	onm 4500	The class of all ordinal numbers is inhabited. (Contributed by Jim Kingdon, 6-Mar-2019.)
⊢ ∃𝑥 𝑥 ∈ On