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Theorem List for Metamath Proof Explorer - 6201-6300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsetlikespec 6201 If 𝑅 is set-like in 𝐴, then all predecessor classes of elements of 𝐴 exist. (Contributed by Scott Fenton, 20-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
((𝑋𝐴𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑋) ∈ V)
 
Theorempredidm 6202 Idempotent law for the predecessor class. (Contributed by Scott Fenton, 29-Mar-2011.)
Pred(𝑅, Pred(𝑅, 𝐴, 𝑋), 𝑋) = Pred(𝑅, 𝐴, 𝑋)
 
Theorempredin 6203 Intersection law for predecessor classes. (Contributed by Scott Fenton, 29-Mar-2011.)
Pred(𝑅, (𝐴𝐵), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∩ Pred(𝑅, 𝐵, 𝑋))
 
Theorempredun 6204 Union law for predecessor classes. (Contributed by Scott Fenton, 29-Mar-2011.)
Pred(𝑅, (𝐴𝐵), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∪ Pred(𝑅, 𝐵, 𝑋))
 
Theorempreddif 6205 Difference law for predecessor classes. (Contributed by Scott Fenton, 14-Apr-2011.)
Pred(𝑅, (𝐴𝐵), 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∖ Pred(𝑅, 𝐵, 𝑋))
 
Theorempredep 6206 The predecessor under the membership relation is equivalent to an intersection. (Contributed by Scott Fenton, 27-Mar-2011.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝑋𝐵 → Pred( E , 𝐴, 𝑋) = (𝐴𝑋))
 
Theoremtrpred 6207 The class of predecessors of an element of a transitive class for the membership relation is that element. (Contributed by BJ, 12-Oct-2024.)
((Tr 𝐴𝑋𝐴) → Pred( E , 𝐴, 𝑋) = 𝑋)
 
Theorempreddowncl 6208* A property of classes that are downward closed under predecessor. (Contributed by Scott Fenton, 13-Apr-2011.)
((𝐵𝐴 ∧ ∀𝑥𝐵 Pred(𝑅, 𝐴, 𝑥) ⊆ 𝐵) → (𝑋𝐵 → Pred(𝑅, 𝐵, 𝑋) = Pred(𝑅, 𝐴, 𝑋)))
 
Theorempredpoirr 6209 Given a partial ordering, a class is not a member of its predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.)
(𝑅 Po 𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋))
 
Theorempredfrirr 6210 Given a well-founded relation, a class is not a member of its predecessor class. (Contributed by Scott Fenton, 22-Apr-2011.)
(𝑅 Fr 𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋))
 
Theorempred0 6211 The predecessor class over is always . (Contributed by Scott Fenton, 16-Apr-2011.) (Proof shortened by AV, 11-Jun-2021.)
Pred(𝑅, ∅, 𝑋) = ∅
 
2.3.12  Well-founded induction (variant)
 
Theoremfrpomin 6212* Every nonempty (possibly proper) subclass of a class 𝐴 with a well-founded set-like partial order 𝑅 has a minimal element. The additional condition of partial order over frmin 9390 enables avoiding the axiom of infinity. (Contributed by Scott Fenton, 11-Feb-2022.)
(((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵𝑦𝐵 ¬ 𝑦𝑅𝑥)
 
Theoremfrpomin2 6213* Every nonempty (possibly proper) subclass of a class 𝐴 with a well-founded set-like partial order 𝑅 has a minimal element. The additional condition of partial order over frmin 9390 enables avoiding the axiom of infinity. (Contributed by Scott Fenton, 11-Feb-2022.)
(((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑥𝐵 Pred(𝑅, 𝐵, 𝑥) = ∅)
 
Theoremfrpoind 6214* The principle of well-founded induction over a partial order. This theorem is a version of frind 9391 that does not require the axiom of infinity and can be used to prove wfi 6221 and tfi 7651. (Contributed by Scott Fenton, 11-Feb-2022.)
(((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴 ∧ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵))) → 𝐴 = 𝐵)
 
Theoremfrpoinsg 6215* Well-Founded Induction Schema (variant). If a property passes from all elements less than 𝑦 of a well-founded set-like partial order class 𝐴 to 𝑦 itself (induction hypothesis), then the property holds for all elements of 𝐴. (Contributed by Scott Fenton, 11-Feb-2022.)
(((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑦𝐴) → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑𝜑))       ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)
 
Theoremfrpoins2fg 6216* Well-Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 24-Aug-2022.)
(𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓𝜑))    &   𝑦𝜓    &   (𝑦 = 𝑧 → (𝜑𝜓))       ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)
 
Theoremfrpoins2g 6217* Well-Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 24-Aug-2022.)
(𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓𝜑))    &   (𝑦 = 𝑧 → (𝜑𝜓))       ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)
 
Theoremfrpoins3g 6218* Well-Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 19-Aug-2024.)
(𝑥𝐴 → (∀𝑦 ∈ Pred (𝑅, 𝐴, 𝑥)𝜓𝜑))    &   (𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 = 𝐵 → (𝜑𝜒))       (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝐵𝐴) → 𝜒)
 
2.3.13  Well-ordered induction
 
Theoremtz6.26 6219* All nonempty subclasses of a class having a well-ordered set-like relation have minimal elements for that relation. Proposition 6.26 of [TakeutiZaring] p. 31. (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
(((𝑅 We 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
 
Theoremtz6.26i 6220* All nonempty subclasses of a class having a well-ordered set-like relation 𝑅 have 𝑅-minimal elements. Proposition 6.26 of [TakeutiZaring] p. 31. (Contributed by Scott Fenton, 14-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
𝑅 We 𝐴    &   𝑅 Se 𝐴       ((𝐵𝐴𝐵 ≠ ∅) → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
 
Theoremwfi 6221* The Principle of Well-Ordered Induction. Theorem 6.27 of [TakeutiZaring] p. 32. This principle states that if 𝐵 is a subclass of a well-ordered class 𝐴 with the property that every element of 𝐵 whose inital segment is included in 𝐴 is itself equal to 𝐴. (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
(((𝑅 We 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴 ∧ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵))) → 𝐴 = 𝐵)
 
Theoremwfii 6222* The Principle of Well-Ordered Induction. Theorem 6.27 of [TakeutiZaring] p. 32. This principle states that if 𝐵 is a subclass of a well-ordered class 𝐴 with the property that every element of 𝐵 whose inital segment is included in 𝐴 is itself equal to 𝐴. (Contributed by Scott Fenton, 29-Jan-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
𝑅 We 𝐴    &   𝑅 Se 𝐴       ((𝐵𝐴 ∧ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵)) → 𝐴 = 𝐵)
 
Theoremwfisg 6223* Well-Ordered Induction Schema. If a property passes from all elements less than 𝑦 of a well-ordered class 𝐴 to 𝑦 itself (induction hypothesis), then the property holds for all elements of 𝐴. (Contributed by Scott Fenton, 11-Feb-2011.)
(𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑𝜑))       ((𝑅 We 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)
 
Theoremwfis 6224* Well-Ordered Induction Schema. If all elements less than a given set 𝑥 of the well-ordered class 𝐴 have a property (induction hypothesis), then all elements of 𝐴 have that property. (Contributed by Scott Fenton, 29-Jan-2011.)
𝑅 We 𝐴    &   𝑅 Se 𝐴    &   (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑𝜑))       (𝑦𝐴𝜑)
 
Theoremwfis2fg 6225* Well-Ordered Induction Schema, using implicit substitution. (Contributed by Scott Fenton, 11-Feb-2011.)
𝑦𝜓    &   (𝑦 = 𝑧 → (𝜑𝜓))    &   (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓𝜑))       ((𝑅 We 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)
 
Theoremwfis2f 6226* Well-Ordered Induction schema, using implicit substitution. (Contributed by Scott Fenton, 29-Jan-2011.)
𝑅 We 𝐴    &   𝑅 Se 𝐴    &   𝑦𝜓    &   (𝑦 = 𝑧 → (𝜑𝜓))    &   (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓𝜑))       (𝑦𝐴𝜑)
 
Theoremwfis2g 6227* Well-Ordered Induction Schema, using implicit substitution. (Contributed by Scott Fenton, 11-Feb-2011.)
(𝑦 = 𝑧 → (𝜑𝜓))    &   (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓𝜑))       ((𝑅 We 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)
 
Theoremwfis2 6228* Well-Ordered Induction schema, using implicit substitution. (Contributed by Scott Fenton, 29-Jan-2011.)
𝑅 We 𝐴    &   𝑅 Se 𝐴    &   (𝑦 = 𝑧 → (𝜑𝜓))    &   (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓𝜑))       (𝑦𝐴𝜑)
 
Theoremwfis3 6229* Well-Ordered Induction schema, using implicit substitution. (Contributed by Scott Fenton, 29-Jan-2011.)
𝑅 We 𝐴    &   𝑅 Se 𝐴    &   (𝑦 = 𝑧 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜑𝜒))    &   (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓𝜑))       (𝐵𝐴𝜒)
 
2.3.14  Ordinals
 
Syntaxword 6230 Extend the definition of a wff to include the ordinal predicate.
wff Ord 𝐴
 
Syntaxcon0 6231 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
 
Syntaxwlim 6232 Extend the definition of a wff to include the limit ordinal predicate.
wff Lim 𝐴
 
Syntaxcsuc 6233 Extend class notation to include the successor function.
class suc 𝐴
 
Definitiondf-ord 6234 Define the ordinal predicate, which is true for a class that is transitive and is well-ordered by the membership relation. Variant of definition of [BellMachover] p. 468.

Some sources will define a notation for ordinal order corresponding to < and but we just use and respectively.

(Contributed by NM, 17-Sep-1993.)

(Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
 
Definitiondf-on 6235 Define the class of all ordinal numbers. Definition 7.11 of [TakeutiZaring] p. 38. (Contributed by NM, 5-Jun-1994.)
On = {𝑥 ∣ Ord 𝑥}
 
Definitiondf-lim 6236 Define the limit ordinal predicate, which is true for a nonempty ordinal that 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. See dflim2 6287, dflim3 7645, and dflim4 for alternate definitions. (Contributed by NM, 22-Apr-1994.)
(Lim 𝐴 ↔ (Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴))
 
Definitiondf-suc 6237 Define the successor of a class. When applied to an ordinal number, the successor means the same thing as "plus 1" (see oa1suc 8279). Definition 7.22 of [TakeutiZaring] p. 41, who use "+ 1" to denote this function. Ordinal natural numbers defined using this successor function and 0 as the empty set are also called von Neumann ordinals; 0 is the empty set {}, 1 is {0, {0}}, 2 is {1, {1}}, and so on. 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 6306), so that the successor of any ordinal class is still an ordinal class (ordsuc 7612), simplifying certain proofs. 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})
 
Theoremordeq 6238 Equality theorem for the ordinal predicate. (Contributed by NM, 17-Sep-1993.)
(𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵))
 
Theoremelong 6239 An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
(𝐴𝑉 → (𝐴 ∈ On ↔ Ord 𝐴))
 
Theoremelon 6240 An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
𝐴 ∈ V       (𝐴 ∈ On ↔ Ord 𝐴)
 
Theoremeloni 6241 An ordinal number has the ordinal property. (Contributed by NM, 5-Jun-1994.)
(𝐴 ∈ On → Ord 𝐴)
 
Theoremelon2 6242 An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004.)
(𝐴 ∈ On ↔ (Ord 𝐴𝐴 ∈ V))
 
Theoremlimeq 6243 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))
 
Theoremordwe 6244 Membership well-orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.)
(Ord 𝐴 → E We 𝐴)
 
Theoremordtr 6245 An ordinal class is transitive. (Contributed by NM, 3-Apr-1994.)
(Ord 𝐴 → Tr 𝐴)
 
Theoremordfr 6246 Membership is well-founded on an ordinal class. In other words, an ordinal class is well-founded. (Contributed by NM, 22-Apr-1994.)
(Ord 𝐴 → E Fr 𝐴)
 
Theoremordelss 6247 An element of an ordinal class is a subset of it. (Contributed by NM, 30-May-1994.)
((Ord 𝐴𝐵𝐴) → 𝐵𝐴)
 
Theoremtrssord 6248 A transitive subclass of an ordinal class is ordinal. (Contributed by NM, 29-May-1994.)
((Tr 𝐴𝐴𝐵 ∧ Ord 𝐵) → Ord 𝐴)
 
Theoremordirr 6249 No ordinal class is a member of itself. In other words, the membership relation is irreflexive on ordinal classes. Theorem 2.2(i) of [BellMachover] p. 469, generalized to classes. We prove this without invoking the Axiom of Regularity. (Contributed by NM, 2-Jan-1994.)
(Ord 𝐴 → ¬ 𝐴𝐴)
 
Theoremnordeq 6250 A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998.)
((Ord 𝐴𝐵𝐴) → 𝐴𝐵)
 
Theoremordn2lp 6251 An ordinal class cannot be an element of one of its members. Variant of first part of Theorem 2.2(vii) of [BellMachover] p. 469. (Contributed by NM, 3-Apr-1994.)
(Ord 𝐴 → ¬ (𝐴𝐵𝐵𝐴))
 
Theoremtz7.5 6252* A nonempty subclass of an ordinal class has a minimal element. Proposition 7.5 of [TakeutiZaring] p. 36. (Contributed by NM, 18-Feb-2004.) (Revised by David Abernethy, 16-Mar-2011.)
((Ord 𝐴𝐵𝐴𝐵 ≠ ∅) → ∃𝑥𝐵 (𝐵𝑥) = ∅)
 
Theoremordelord 6253 An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. (Contributed by NM, 23-Apr-1994.)
((Ord 𝐴𝐵𝐴) → Ord 𝐵)
 
Theoremtron 6254 The class of all ordinal numbers is transitive. (Contributed by NM, 4-May-2009.)
Tr On
 
Theoremordelon 6255 An element of an ordinal class is an ordinal number. (Contributed by NM, 26-Oct-2003.)
((Ord 𝐴𝐵𝐴) → 𝐵 ∈ On)
 
Theoremonelon 6256 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 ∧ 𝐵𝐴) → 𝐵 ∈ On)
 
Theoremtz7.7 6257 A transitive class belongs to an ordinal class iff it is strictly included in it. Proposition 7.7 of [TakeutiZaring] p. 37. (Contributed by NM, 5-May-1994.)
((Ord 𝐴 ∧ Tr 𝐵) → (𝐵𝐴 ↔ (𝐵𝐴𝐵𝐴)))
 
Theoremordelssne 6258 For ordinal classes, membership is equivalent to strict inclusion. Corollary 7.8 of [TakeutiZaring] p. 37. (Contributed by NM, 25-Nov-1995.)
((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵)))
 
Theoremordelpss 6259 For ordinal classes, membership is equivalent to strict inclusion. Corollary 7.8 of [TakeutiZaring] p. 37. (Contributed by NM, 17-Jun-1998.)
((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐴𝐵))
 
Theoremordsseleq 6260 For ordinal classes, inclusion is equivalent to membership or equality. (Contributed by NM, 25-Nov-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
 
Theoremordin 6261 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 (𝐴𝐵))
 
Theoremonin 6262 The intersection of two ordinal numbers is an ordinal number. (Contributed by NM, 7-Apr-1995.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵) ∈ On)
 
Theoremordtri3or 6263 A trichotomy law for ordinals. Proposition 7.10 of [TakeutiZaring] p. 38. (Contributed by NM, 10-May-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))
 
Theoremordtri1 6264 A trichotomy law for ordinals. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
 
Theoremontri1 6265 A trichotomy law for ordinal numbers. (Contributed by NM, 6-Nov-2003.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
 
Theoremordtri2 6266 A trichotomy law for ordinals. (Contributed by NM, 25-Nov-1995.)
((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
 
Theoremordtri3 6267 A trichotomy law for ordinals. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (Proof shortened by JJ, 24-Sep-2021.)
((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ ¬ (𝐴𝐵𝐵𝐴)))
 
Theoremordtri4 6268 A trichotomy law for ordinals. (Contributed by NM, 1-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐵)))
 
Theoremorddisj 6269 An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.)
(Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅)
 
Theoremonfr 6270 The ordinal class is well-founded. This proof does not require the axiom of regularity. This lemma is used in ordon 7580 (through epweon 7579) in order to eliminate the need for the axiom of regularity. (Contributed by NM, 17-May-1994.)
E Fr On
 
Theoremonelpss 6271 Relationship between membership and proper subset of an ordinal number. (Contributed by NM, 15-Sep-1995.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵)))
 
Theoremonsseleq 6272 Relationship between subset and membership of an ordinal number. (Contributed by NM, 15-Sep-1995.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
 
Theoremonelss 6273 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 → (𝐵𝐴𝐵𝐴))
 
Theoremordtr1 6274 Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.)
(Ord 𝐶 → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
 
Theoremordtr2 6275 Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
((Ord 𝐴 ∧ Ord 𝐶) → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
 
Theoremordtr3 6276 Transitive law for ordinal classes. (Contributed by Mario Carneiro, 30-Dec-2014.) (Proof shortened by JJ, 24-Sep-2021.)
((Ord 𝐵 ∧ Ord 𝐶) → (𝐴𝐵 → (𝐴𝐶𝐶𝐵)))
 
Theoremontr1 6277 Transitive law for ordinal numbers. Theorem 7M(b) of [Enderton] p. 192. (Contributed by NM, 11-Aug-1994.)
(𝐶 ∈ On → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
 
Theoremontr2 6278 Transitive law for ordinal numbers. Exercise 3 of [TakeutiZaring] p. 40. (Contributed by NM, 6-Nov-2003.)
((𝐴 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
 
Theoremordunidif 6279 The union of an ordinal stays the same if a subset equal to one of its elements is removed. (Contributed by NM, 10-Dec-2004.)
((Ord 𝐴𝐵𝐴) → (𝐴𝐵) = 𝐴)
 
Theoremordintdif 6280 If 𝐵 is smaller than 𝐴, then it equals the intersection of the difference. Exercise 11 in [TakeutiZaring] p. 44. (Contributed by Andrew Salmon, 14-Nov-2011.)
((Ord 𝐴 ∧ Ord 𝐵 ∧ (𝐴𝐵) ≠ ∅) → 𝐵 = (𝐴𝐵))
 
Theoremonintss 6281* 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 → (𝜑𝜓))       (𝐴 ∈ On → (𝜓 {𝑥 ∈ On ∣ 𝜑} ⊆ 𝐴))
 
Theoremoneqmini 6282* A way to show that an ordinal number equals the minimum of a collection of ordinal numbers: it must be in the collection, and it must not be larger than any member of the collection. (Contributed by NM, 14-Nov-2003.)
(𝐵 ⊆ On → ((𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵) → 𝐴 = 𝐵))
 
Theoremord0 6283 The empty set is an ordinal class. (Contributed by NM, 11-May-1994.)
Ord ∅
 
Theorem0elon 6284 The empty set is an ordinal number. Corollary 7N(b) of [Enderton] p. 193. (Contributed by NM, 17-Sep-1993.)
∅ ∈ On
 
Theoremord0eln0 6285 A nonempty ordinal contains the empty set. (Contributed by NM, 25-Nov-1995.)
(Ord 𝐴 → (∅ ∈ 𝐴𝐴 ≠ ∅))
 
Theoremon0eln0 6286 An ordinal number contains zero iff it is nonzero. (Contributed by NM, 6-Dec-2004.)
(𝐴 ∈ On → (∅ ∈ 𝐴𝐴 ≠ ∅))
 
Theoremdflim2 6287 An alternate definition of a limit ordinal. (Contributed by NM, 4-Nov-2004.)
(Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴𝐴 = 𝐴))
 
Theoreminton 6288 The intersection of the class of ordinal numbers is the empty set. (Contributed by NM, 20-Oct-2003.)
On = ∅
 
Theoremnlim0 6289 The empty set is not a limit ordinal. (Contributed by NM, 24-Mar-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
¬ Lim ∅
 
Theoremlimord 6290 A limit ordinal is ordinal. (Contributed by NM, 4-May-1995.)
(Lim 𝐴 → Ord 𝐴)
 
Theoremlimuni 6291 A limit ordinal is its own supremum (union). (Contributed by NM, 4-May-1995.)
(Lim 𝐴𝐴 = 𝐴)
 
Theoremlimuni2 6292 The union of a limit ordinal is a limit ordinal. (Contributed by NM, 19-Sep-2006.)
(Lim 𝐴 → Lim 𝐴)
 
Theorem0ellim 6293 A limit ordinal contains the empty set. (Contributed by NM, 15-May-1994.)
(Lim 𝐴 → ∅ ∈ 𝐴)
 
Theoremlimelon 6294 A limit ordinal class that is also a set is an ordinal number. (Contributed by NM, 26-Apr-2004.)
((𝐴𝐵 ∧ Lim 𝐴) → 𝐴 ∈ On)
 
Theoremonn0 6295 The class of all ordinal numbers is not empty. (Contributed by NM, 17-Sep-1995.)
On ≠ ∅
 
Theoremsuceq 6296 Equality of successors. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
 
Theoremelsuci 6297 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 → (𝐴𝐵𝐴 = 𝐵))
 
Theoremelsucg 6298 Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-1995.)
(𝐴𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
 
Theoremelsuc2g 6299 Variant of membership in a successor, requiring that 𝐵 rather than 𝐴 be a set. (Contributed by NM, 28-Oct-2003.)
(𝐵𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
 
Theoremelsuc 6300 Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-2003.)
𝐴 ∈ V       (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))
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