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	po3nr 4401	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 4402	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 4403*	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 4404	A strict linear order is a strict partial order. (Contributed by NM, 28-Mar-1997.)
⊢ (𝑅 Or 𝐴 → 𝑅 Po 𝐴)
Theorem	soss 4405	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 4406	Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)
⊢ (𝑅 = 𝑆 → (𝑅 Or 𝐴 ↔ 𝑆 Or 𝐴))
Theorem	soeq2 4407	Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)
⊢ (𝐴 = 𝐵 → (𝑅 Or 𝐴 ↔ 𝑅 Or 𝐵))
Theorem	sonr 4408	A strict order relation is irreflexive. (Contributed by NM, 24-Nov-1995.)
⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵)
Theorem	sotr 4409	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 4410*	An irreflexive, transitive, trichotomous relation is a linear ordering (in the sense of df-iso 4388). (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 4411	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 4412	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 4413	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 4414	One direction of sotritric 4415 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 4415	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 4416	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 4417	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 4418	Extend wff notation to include the well-founded predicate.
wff FrFor 𝑅𝐴𝑆
Syntax	wfr 4419	Extend wff notation to include the well-founded predicate. Read: ' 𝑅 is a well-founded relation on 𝐴.'
wff 𝑅 Fr 𝐴
Syntax	wse 4420	Extend wff notation to include the set-like predicate. Read: ' 𝑅 is set-like on 𝐴.'
wff 𝑅 Se 𝐴
Syntax	wwe 4421	Extend wff notation to include the well-ordering predicate. Read: ' 𝑅 well-orders 𝐴.'
wff 𝑅 We 𝐴
Definition	df-frfor 4422*	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 4423*	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 4424*	Define the set-like predicate. (Contributed by Mario Carneiro, 19-Nov-2014.)
⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
Definition	df-wetr 4425*	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 4613). 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 4426*	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 4427	Any relation on a set is set-like on it. (Contributed by Mario Carneiro, 22-Jun-2015.)
⊢ (𝐴 ∈ 𝑉 → 𝑅 Se 𝐴)
Theorem	sess1 4428	Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
⊢ (𝑅 ⊆ 𝑆 → (𝑆 Se 𝐴 → 𝑅 Se 𝐴))
Theorem	sess2 4429	Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
⊢ (𝐴 ⊆ 𝐵 → (𝑅 Se 𝐵 → 𝑅 Se 𝐴))
Theorem	seeq1 4430	Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
⊢ (𝑅 = 𝑆 → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐴))
Theorem	seeq2 4431	Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
⊢ (𝐴 = 𝐵 → (𝑅 Se 𝐴 ↔ 𝑅 Se 𝐵))
Theorem	nfse 4432	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 4433	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 4434	Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
⊢ (𝑅 = 𝑆 → ( FrFor 𝑅𝐴𝑇 ↔ FrFor 𝑆𝐴𝑇))
Theorem	freq1 4435	Equality theorem for the well-founded predicate. (Contributed by NM, 9-Mar-1997.)
⊢ (𝑅 = 𝑆 → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐴))
Theorem	frforeq2 4436	Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
⊢ (𝐴 = 𝐵 → ( FrFor 𝑅𝐴𝑇 ↔ FrFor 𝑅𝐵𝑇))
Theorem	freq2 4437	Equality theorem for the well-founded predicate. (Contributed by NM, 3-Apr-1994.)
⊢ (𝐴 = 𝐵 → (𝑅 Fr 𝐴 ↔ 𝑅 Fr 𝐵))
Theorem	frforeq3 4438	Equality theorem for the well-founded predicate. (Contributed by Jim Kingdon, 22-Sep-2021.)
⊢ (𝑆 = 𝑇 → ( FrFor 𝑅𝐴𝑆 ↔ FrFor 𝑅𝐴𝑇))
Theorem	nffrfor 4439	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 4440	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 4441	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 4442	Any relation is well-founded on the empty set. (Contributed by NM, 17-Sep-1993.)
⊢ 𝑅 Fr ∅
Theorem	frind 4443*	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 4444	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 4445	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 4446	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 4447	Equality theorem for the well-ordering predicate. (Contributed by NM, 9-Mar-1997.)
⊢ (𝑅 = 𝑆 → (𝑅 We 𝐴 ↔ 𝑆 We 𝐴))
Theorem	weeq2 4448	Equality theorem for the well-ordering predicate. (Contributed by NM, 3-Apr-1994.)
⊢ (𝐴 = 𝐵 → (𝑅 We 𝐴 ↔ 𝑅 We 𝐵))
Theorem	wefr 4449	A well-ordering is well-founded. (Contributed by NM, 22-Apr-1994.)
⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴)
Theorem	wepo 4450	A well-ordering is a partial ordering. (Contributed by Jim Kingdon, 23-Sep-2021.)
⊢ ((𝑅 We 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝑅 Po 𝐴)
Theorem	wetrep 4451*	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 4452	Any relation is a well-ordering of the empty set. (Contributed by NM, 16-Mar-1997.)
⊢ 𝑅 We ∅
2.3.10  Ordinals
Syntax	word 4453	Extend the definition of a wff to include the ordinal predicate.
wff Ord 𝐴
Syntax	con0 4454	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 4455	Extend the definition of a wff to include the limit ordinal predicate.
wff Lim 𝐴
Syntax	csuc 4456	Extend class notation to include the successor function.
class suc 𝐴
Definition	df-iord 4457*	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 4458 instead for naming consistency with set.mm. (New usage is discouraged.)
⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 Tr 𝑥))
Theorem	dford3 4458*	Alias for df-iord 4457. Use it instead of df-iord 4457 for naming consistency with set.mm. (Contributed by Jim Kingdon, 10-Oct-2018.)
⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 Tr 𝑥))
Definition	df-on 4459	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 4460	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 4461 instead for naming consistency with set.mm. (New usage is discouraged.)
⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴))
Theorem	dflim2 4461	Alias for df-ilim 4460. Use it instead of df-ilim 4460 for naming consistency with set.mm. (Contributed by NM, 4-Nov-2004.)
⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ 𝐴 = ∪ 𝐴))
Definition	df-suc 4462	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 4503). 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 4463	Equality theorem for the ordinal predicate. (Contributed by NM, 17-Sep-1993.)
⊢ (𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵))
Theorem	elong 4464	An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ On ↔ Ord 𝐴))
Theorem	elon 4465	An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ On ↔ Ord 𝐴)
Theorem	eloni 4466	An ordinal number has the ordinal property. (Contributed by NM, 5-Jun-1994.)
⊢ (𝐴 ∈ On → Ord 𝐴)
Theorem	elon2 4467	An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004.)
⊢ (𝐴 ∈ On ↔ (Ord 𝐴 ∧ 𝐴 ∈ V))
Theorem	limeq 4468	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 4469	An ordinal class is transitive. (Contributed by NM, 3-Apr-1994.)
⊢ (Ord 𝐴 → Tr 𝐴)
Theorem	ordelss 4470	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 4471	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 4472	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 4473	The class of all ordinal numbers is transitive. (Contributed by NM, 4-May-2009.)
⊢ Tr On
Theorem	ordelon 4474	An element of an ordinal class is an ordinal number. (Contributed by NM, 26-Oct-2003.)
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On)
Theorem	onelon 4475	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 4476	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 4477	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 4478	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 4479	Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.)
⊢ (Ord 𝐶 → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))
Theorem	ontr1 4480	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 4481*	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 4482	The empty set is an ordinal class. (Contributed by NM, 11-May-1994.)
⊢ Ord ∅
Theorem	0elon 4483	The empty set is an ordinal number. Corollary 7N(b) of [Enderton] p. 193. (Contributed by NM, 17-Sep-1993.)
⊢ ∅ ∈ On
Theorem	inton 4484	The intersection of the class of ordinal numbers is the empty set. (Contributed by NM, 20-Oct-2003.)
⊢ ∩ On = ∅
Theorem	nlim0 4485	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 4486	A limit ordinal is ordinal. (Contributed by NM, 4-May-1995.)
⊢ (Lim 𝐴 → Ord 𝐴)
Theorem	limuni 4487	A limit ordinal is its own supremum (union). (Contributed by NM, 4-May-1995.)
⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)
Theorem	limuni2 4488	The union of a limit ordinal is a limit ordinal. (Contributed by NM, 19-Sep-2006.)
⊢ (Lim 𝐴 → Lim ∪ 𝐴)
Theorem	0ellim 4489	A limit ordinal contains the empty set. (Contributed by NM, 15-May-1994.)
⊢ (Lim 𝐴 → ∅ ∈ 𝐴)
Theorem	limelon 4490	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 4491	The class of all ordinal numbers is not empty. (Contributed by NM, 17-Sep-1995.)
⊢ On ≠ ∅
Theorem	onm 4492	The class of all ordinal numbers is inhabited. (Contributed by Jim Kingdon, 6-Mar-2019.)
⊢ ∃𝑥 𝑥 ∈ On
Theorem	suceq 4493	Equality of successors. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
⊢ (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
Theorem	elsuci 4494	Membership in a successor. This one-way implication does not require that either 𝐴 or 𝐵 be sets. (Contributed by NM, 6-Jun-1994.)
⊢ (𝐴 ∈ suc 𝐵 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))
Theorem	elsucg 4495	Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-1995.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
Theorem	elsuc2g 4496	Variant of membership in a successor, requiring that 𝐵 rather than 𝐴 be a set. (Contributed by NM, 28-Oct-2003.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵)))
Theorem	elsuc 4497	Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-2003.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ suc 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))
Theorem	elsuc2 4498	Membership in a successor. (Contributed by NM, 15-Sep-2003.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐵 ∈ suc 𝐴 ↔ (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴))
Theorem	nfsuc 4499	Bound-variable hypothesis builder for successor. (Contributed by NM, 15-Sep-2003.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥 suc 𝐴
Theorem	elelsuc 4500	Membership in a successor. (Contributed by NM, 20-Jun-1998.)
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ suc 𝐵)