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Theorem List for Metamath Proof Explorer - 5501-5600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelpredg 5501 Membership in a predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.)
((𝑋𝐵𝑌𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅𝑋))
 
Theorempredasetex 5502 The predecessor class exists when 𝐴 does. (Contributed by Scott Fenton, 8-Feb-2011.)
𝐴 ∈ V       Pred(𝑅, 𝐴, 𝑋) ∈ V
 
Theoremdffr4 5503* Alternate definition of well-founded relation. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 Pred(𝑅, 𝑥, 𝑦) = ∅))
 
Theorempredel 5504 Membership in the predecessor class implies membership in the base class. (Contributed by Scott Fenton, 11-Feb-2011.)
(𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝐴)
 
Theorempredpo 5505 Property of the precessor class for partial orderings. (Contributed by Scott Fenton, 28-Apr-2012.)
((𝑅 Po 𝐴𝑋𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑋)))
 
Theorempredso 5506 Property of the predecessor class for strict orderings. (Contributed by Scott Fenton, 11-Feb-2011.)
((𝑅 Or 𝐴𝑋𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑋)))
 
Theorempredbrg 5507 Closed form of elpredim 5499. (Contributed by Scott Fenton, 13-Apr-2011.) (Revised by NM, 5-Apr-2016.)
((𝑋𝑉𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)) → 𝑌𝑅𝑋)
 
Theoremsetlikespec 5508 If 𝑅 is set-like in 𝐴, then all predecessors 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 5509 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 5510 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 5511 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 5512 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 5513 The predecessor under the epsilon relationship 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))
 
Theorempreddowncl 5514* 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 5515 Given a partial ordering, 𝑋 is not a member of its predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.)
(𝑅 Po 𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋))
 
Theorempredfrirr 5516 Given a well-founded relationship, 𝑋 is not a member of its predecessor class. (Contributed by Scott Fenton, 22-Apr-2011.)
(𝑅 Fr 𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋))
 
Theorempred0 5517 The predecessor class over is always . (Contributed by Scott Fenton, 16-Apr-2011.) (Proof shortened by AV, 11-Jun-2021.)
Pred(𝑅, ∅, 𝑋) = ∅
 
2.3.11  Well-founded induction
 
Theoremtz6.26 5518* All nonempty (possibly proper) subclasses of 𝐴, which has a well-founded relation 𝑅, have 𝑅-minimal elements. 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 5519* All nonempty (possibly proper) subclasses of 𝐴, which has a well-founded 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 5520* The Principle of Well-Founded 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 5521* The Principle of Well-Founded 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 5522* Well-Founded Induction Schema. If a property passes from all elements less than 𝑦 of a well-founded 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 5523* Well-Founded Induction Schema. If all elements less than a given set 𝑥 of the well-founded 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 5524* Well-Founded 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 5525* Well Founded 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 5526* Well-Founded 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 5527* Well Founded 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 5528* Well Founded 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)
 
2.3.12  Ordinals
 
Syntaxword 5529 Extend the definition of a wff to include the ordinal predicate.
wff Ord 𝐴
 
Syntaxcon0 5530 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 5531 Extend the definition of a wff to include the limit ordinal predicate.
wff Lim 𝐴
 
Syntaxcsuc 5532 Extend class notation to include the successor function.
class suc 𝐴
 
Definitiondf-ord 5533 Define the ordinal predicate, which is true for a class that is transitive and is well-ordered by the epsilon relation. Variant of definition of [BellMachover] p. 468. (Contributed by NM, 17-Sep-1993.)
(Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
 
Definitiondf-on 5534 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 5535 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 5586, dflim3 6815, and dflim4 for alternate definitions. (Contributed by NM, 22-Apr-1994.)
(Lim 𝐴 ↔ (Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴))
 
Definitiondf-suc 5536 Define the successor of a class. When applied to an ordinal number, the successor means the same thing as "plus 1" (see oa1suc 7374). 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 5605), so that the successor of any ordinal class is still an ordinal class (ordsuc 6782), 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 5537 Equality theorem for the ordinal predicate. (Contributed by NM, 17-Sep-1993.)
(𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵))
 
Theoremelong 5538 An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
(𝐴𝑉 → (𝐴 ∈ On ↔ Ord 𝐴))
 
Theoremelon 5539 An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
𝐴 ∈ V       (𝐴 ∈ On ↔ Ord 𝐴)
 
Theoremeloni 5540 An ordinal number has the ordinal property. (Contributed by NM, 5-Jun-1994.)
(𝐴 ∈ On → Ord 𝐴)
 
Theoremelon2 5541 An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004.)
(𝐴 ∈ On ↔ (Ord 𝐴𝐴 ∈ V))
 
Theoremlimeq 5542 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 5543 Epsilon well-orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.)
(Ord 𝐴 → E We 𝐴)
 
Theoremordtr 5544 An ordinal class is transitive. (Contributed by NM, 3-Apr-1994.)
(Ord 𝐴 → Tr 𝐴)
 
Theoremordfr 5545 Epsilon is well-founded on an ordinal class. (Contributed by NM, 22-Apr-1994.)
(Ord 𝐴 → E Fr 𝐴)
 
Theoremordelss 5546 An element of an ordinal class is a subset of it. (Contributed by NM, 30-May-1994.)
((Ord 𝐴𝐵𝐴) → 𝐵𝐴)
 
Theoremtrssord 5547 A transitive subclass of an ordinal class is ordinal. (Contributed by NM, 29-May-1994.)
((Tr 𝐴𝐴𝐵 ∧ Ord 𝐵) → Ord 𝐴)
 
Theoremordirr 5548 Epsilon irreflexivity of ordinals: no ordinal class is a member of itself. 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 5549 A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998.)
((Ord 𝐴𝐵𝐴) → 𝐴𝐵)
 
Theoremordn2lp 5550 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 5551* 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 5552 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 5553 The class of all ordinal numbers is transitive. (Contributed by NM, 4-May-2009.)
Tr On
 
Theoremordelon 5554 An element of an ordinal class is an ordinal number. (Contributed by NM, 26-Oct-2003.)
((Ord 𝐴𝐵𝐴) → 𝐵 ∈ On)
 
Theoremonelon 5555 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 5556 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 5557 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 5558 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 5559 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 5560 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 5561 The intersection of two ordinal numbers is an ordinal number. (Contributed by NM, 7-Apr-1995.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵) ∈ On)
 
Theoremordtri3or 5562 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 5563 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 5564 A trichotomy law for ordinal numbers. (Contributed by NM, 6-Nov-2003.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
 
Theoremordtri2 5565 A trichotomy law for ordinals. (Contributed by NM, 25-Nov-1995.)
((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
 
Theoremordtri3 5566 A trichotomy law for ordinals. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ ¬ (𝐴𝐵𝐵𝐴)))
 
Theoremordtri4 5567 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 5568 An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.)
(Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅)
 
Theoremonfr 5569 The ordinal class is well-founded. This lemma is needed for ordon 6750 in order to eliminate the need for the Axiom of Regularity. (Contributed by NM, 17-May-1994.)
E Fr On
 
Theoremonelpss 5570 Relationship between membership and proper subset of an ordinal number. (Contributed by NM, 15-Sep-1995.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵)))
 
Theoremonsseleq 5571 Relationship between subset and membership of an ordinal number. (Contributed by NM, 15-Sep-1995.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
 
Theoremonelss 5572 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 5573 Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.)
(Ord 𝐶 → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
 
Theoremordtr2 5574 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 5575 Transitive law for ordinal classes. (Contributed by Mario Carneiro, 30-Dec-2014.)
((Ord 𝐵 ∧ Ord 𝐶) → (𝐴𝐵 → (𝐴𝐶𝐶𝐵)))
 
Theoremontr1 5576 Transitive law for ordinal numbers. Theorem 7M(b) of [Enderton] p. 192. (Contributed by NM, 11-Aug-1994.)
(𝐶 ∈ On → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
 
Theoremontr2 5577 Transitive law for ordinal numbers. Exercise 3 of [TakeutiZaring] p. 40. (Contributed by NM, 6-Nov-2003.)
((𝐴 ∈ On ∧ 𝐶 ∈ On) → ((𝐴𝐵𝐵𝐶) → 𝐴𝐶))
 
Theoremordunidif 5578 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 5579 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 5580* 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 5581* 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 5582 The empty set is an ordinal class. (Contributed by NM, 11-May-1994.)
Ord ∅
 
Theorem0elon 5583 The empty set is an ordinal number. Corollary 7N(b) of [Enderton] p. 193. (Contributed by NM, 17-Sep-1993.)
∅ ∈ On
 
Theoremord0eln0 5584 A nonempty ordinal contains the empty set. (Contributed by NM, 25-Nov-1995.)
(Ord 𝐴 → (∅ ∈ 𝐴𝐴 ≠ ∅))
 
Theoremon0eln0 5585 An ordinal number contains zero iff it is nonzero. (Contributed by NM, 6-Dec-2004.)
(𝐴 ∈ On → (∅ ∈ 𝐴𝐴 ≠ ∅))
 
Theoremdflim2 5586 An alternate definition of a limit ordinal. (Contributed by NM, 4-Nov-2004.)
(Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴𝐴 = 𝐴))
 
Theoreminton 5587 The intersection of the class of ordinal numbers is the empty set. (Contributed by NM, 20-Oct-2003.)
On = ∅
 
Theoremnlim0 5588 The empty set is not a limit ordinal. (Contributed by NM, 24-Mar-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
¬ Lim ∅
 
Theoremlimord 5589 A limit ordinal is ordinal. (Contributed by NM, 4-May-1995.)
(Lim 𝐴 → Ord 𝐴)
 
Theoremlimuni 5590 A limit ordinal is its own supremum (union). (Contributed by NM, 4-May-1995.)
(Lim 𝐴𝐴 = 𝐴)
 
Theoremlimuni2 5591 The union of a limit ordinal is a limit ordinal. (Contributed by NM, 19-Sep-2006.)
(Lim 𝐴 → Lim 𝐴)
 
Theorem0ellim 5592 A limit ordinal contains the empty set. (Contributed by NM, 15-May-1994.)
(Lim 𝐴 → ∅ ∈ 𝐴)
 
Theoremlimelon 5593 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 5594 The class of all ordinal numbers is not empty. (Contributed by NM, 17-Sep-1995.)
On ≠ ∅
 
Theoremsuceq 5595 Equality of successors. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
 
Theoremelsuci 5596 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 5597 Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-1995.)
(𝐴𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
 
Theoremelsuc2g 5598 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 5599 Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-2003.)
𝐴 ∈ V       (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))
 
Theoremelsuc2 5600 Membership in a successor. (Contributed by NM, 15-Sep-2003.)
𝐴 ∈ V       (𝐵 ∈ suc 𝐴 ↔ (𝐵𝐴𝐵 = 𝐴))
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