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Theorem List for Metamath Proof Explorer - 5701-5800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrelcoi2 5701 Composition with the identity relation restricted to a relation's field. (Contributed by FL, 2-May-2011.)
(Rel 𝑅 → (( I ↾ 𝑅) ∘ 𝑅) = 𝑅)
 
Theoremrelcoi1 5702 Composition with the identity relation restricted to a relation's field. (Contributed by FL, 8-May-2011.) (Proof shortened by OpenAI, 3-Jul-2020.)
(Rel 𝑅 → (𝑅 ∘ ( I ↾ 𝑅)) = 𝑅)
 
Theoremunidmrn 5703 The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.)
𝐴 = (dom 𝐴 ∪ ran 𝐴)
 
Theoremrelcnvfld 5704 if 𝑅 is a relation, its double union equals the double union of its converse. (Contributed by FL, 5-Jan-2009.)
(Rel 𝑅 𝑅 = 𝑅)
 
Theoremdfdm2 5705 Alternate definition of domain df-dm 5153 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.)
dom 𝐴 = (𝐴𝐴)
 
Theoremunixp 5706 The double class union of a nonempty Cartesian product is the union of it members. (Contributed by NM, 17-Sep-2006.)
((𝐴 × 𝐵) ≠ ∅ → (𝐴 × 𝐵) = (𝐴𝐵))
 
Theoremunixp0 5707 A Cartesian product is empty iff its union is empty. (Contributed by NM, 20-Sep-2006.)
((𝐴 × 𝐵) = ∅ ↔ (𝐴 × 𝐵) = ∅)
 
Theoremunixpid 5708 Field of a square Cartesian product. (Contributed by FL, 10-Oct-2009.)
(𝐴 × 𝐴) = 𝐴
 
Theoremressn 5709 Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
(𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵}))
 
Theoremcnviin 5710* The converse of an intersection is the intersection of the converse. (Contributed by FL, 15-Oct-2012.)
(𝐴 ≠ ∅ → 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵)
 
Theoremcnvpo 5711 The converse of a partial order relation is a partial order relation. (Contributed by NM, 15-Jun-2005.)
(𝑅 Po 𝐴𝑅 Po 𝐴)
 
Theoremcnvso 5712 The converse of a strict order relation is a strict order relation. (Contributed by NM, 15-Jun-2005.)
(𝑅 Or 𝐴𝑅 Or 𝐴)
 
Theoremxpco 5713 Composition of two Cartesian products. (Contributed by Thierry Arnoux, 17-Nov-2017.)
(𝐵 ≠ ∅ → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶))
 
Theoremxpcoid 5714 Composition of two square Cartesian products. (Contributed by Thierry Arnoux, 14-Jan-2018.)
((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (𝐴 × 𝐴)
 
Theoremelsnxp 5715* Elementhood to a cartesian product with a singleton. (Contributed by Thierry Arnoux, 10-Apr-2020.) (Proof shortened by JJ, 14-Jul-2021.)
(𝑋𝑉 → (𝑍 ∈ ({𝑋} × 𝐴) ↔ ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩))
 
TheoremelsnxpOLD 5716* Obsolete proof of elsnxp 5715 as of 14-Jul-2021. (Contributed by Thierry Arnoux, 10-Apr-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝑋𝑉 → (𝑍 ∈ ({𝑋} × 𝐴) ↔ ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩))
 
2.3.11  The Predecessor Class
 
Syntaxcpred 5717 The predecessors symbol.
class Pred(𝑅, 𝐴, 𝑋)
 
Definitiondf-pred 5718 Define the predecessor class of a relationship. This is the class of all elements 𝑦 of 𝐴 such that 𝑦𝑅𝑋 (see elpred 5731) . (Contributed by Scott Fenton, 29-Jan-2011.)
Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
 
Theorempredeq123 5719 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 13-Jun-2018.)
((𝑅 = 𝑆𝐴 = 𝐵𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐵, 𝑌))
 
Theorempredeq1 5720 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝑅 = 𝑆 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐴, 𝑋))
 
Theorempredeq2 5721 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝐴 = 𝐵 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))
 
Theorempredeq3 5722 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝑋 = 𝑌 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))
 
Theoremnfpred 5723 Bound-variable hypothesis builder for the predecessor class. (Contributed by Scott Fenton, 9-Jun-2018.)
𝑥𝑅    &   𝑥𝐴    &   𝑥𝑋       𝑥Pred(𝑅, 𝐴, 𝑋)
 
Theorempredpredss 5724 If 𝐴 is a subset of 𝐵, then their predecessor classes are also subsets. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝐴𝐵 → Pred(𝑅, 𝐴, 𝑋) ⊆ Pred(𝑅, 𝐵, 𝑋))
 
Theorempredss 5725 The predecessor class of 𝐴 is a subset of 𝐴. (Contributed by Scott Fenton, 2-Feb-2011.)
Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐴
 
Theoremsspred 5726 Another subset/predecessor class relationship. (Contributed by Scott Fenton, 6-Feb-2011.)
((𝐵𝐴 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))
 
Theoremdfpred2 5727* An alternate definition of predecessor class when 𝑋 is a set. (Contributed by Scott Fenton, 8-Feb-2011.)
𝑋 ∈ V       Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ {𝑦𝑦𝑅𝑋})
 
Theoremdfpred3 5728* An alternate definition of predecessor class when 𝑋 is a set. (Contributed by Scott Fenton, 13-Jun-2018.)
𝑋 ∈ V       Pred(𝑅, 𝐴, 𝑋) = {𝑦𝐴𝑦𝑅𝑋}
 
Theoremdfpred3g 5729* An alternate definition of predecessor class when 𝑋 is a set. (Contributed by Scott Fenton, 13-Jun-2018.)
(𝑋𝑉 → Pred(𝑅, 𝐴, 𝑋) = {𝑦𝐴𝑦𝑅𝑋})
 
Theoremelpredim 5730 Membership in a predecessor class - implicative version. (Contributed by Scott Fenton, 9-May-2012.)
𝑋 ∈ V       (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋)
 
Theoremelpred 5731 Membership in a predecessor class. (Contributed by Scott Fenton, 4-Feb-2011.)
𝑌 ∈ V       (𝑋𝐷 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌𝐴𝑌𝑅𝑋)))
 
Theoremelpredg 5732 Membership in a predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.)
((𝑋𝐵𝑌𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅𝑋))
 
Theorempredasetex 5733 The predecessor class exists when 𝐴 does. (Contributed by Scott Fenton, 8-Feb-2011.)
𝐴 ∈ V       Pred(𝑅, 𝐴, 𝑋) ∈ V
 
Theoremdffr4 5734* Alternate definition of well-founded relation. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 Pred(𝑅, 𝑥, 𝑦) = ∅))
 
Theorempredel 5735 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 5736 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 5737 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 5738 Closed form of elpredim 5730. (Contributed by Scott Fenton, 13-Apr-2011.) (Revised by NM, 5-Apr-2016.)
((𝑋𝑉𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)) → 𝑌𝑅𝑋)
 
Theoremsetlikespec 5739 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 5740 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 5741 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 5742 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 5743 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 5744 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 5745* 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 5746 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 5747 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 5748 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
 
Theoremtz6.26 5749* 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 5750* 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 5751* 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 5752* 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 5753* 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 5754* 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 5755* 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 5756* 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 5757* 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 5758* 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 5759* 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)
 
2.3.13  Ordinals
 
Syntaxword 5760 Extend the definition of a wff to include the ordinal predicate.
wff Ord 𝐴
 
Syntaxcon0 5761 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 5762 Extend the definition of a wff to include the limit ordinal predicate.
wff Lim 𝐴
 
Syntaxcsuc 5763 Extend class notation to include the successor function.
class suc 𝐴
 
Definitiondf-ord 5764 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 5765 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 5766 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 5819, dflim3 7089, and dflim4 for alternate definitions. (Contributed by NM, 22-Apr-1994.)
(Lim 𝐴 ↔ (Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴))
 
Definitiondf-suc 5767 Define the successor of a class. When applied to an ordinal number, the successor means the same thing as "plus 1" (see oa1suc 7656). 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 5838), so that the successor of any ordinal class is still an ordinal class (ordsuc 7056), 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 5768 Equality theorem for the ordinal predicate. (Contributed by NM, 17-Sep-1993.)
(𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵))
 
Theoremelong 5769 An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
(𝐴𝑉 → (𝐴 ∈ On ↔ Ord 𝐴))
 
Theoremelon 5770 An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
𝐴 ∈ V       (𝐴 ∈ On ↔ Ord 𝐴)
 
Theoremeloni 5771 An ordinal number has the ordinal property. (Contributed by NM, 5-Jun-1994.)
(𝐴 ∈ On → Ord 𝐴)
 
Theoremelon2 5772 An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004.)
(𝐴 ∈ On ↔ (Ord 𝐴𝐴 ∈ V))
 
Theoremlimeq 5773 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 5774 Epsilon well-orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.)
(Ord 𝐴 → E We 𝐴)
 
Theoremordtr 5775 An ordinal class is transitive. (Contributed by NM, 3-Apr-1994.)
(Ord 𝐴 → Tr 𝐴)
 
Theoremordfr 5776 Epsilon is well-founded on an ordinal class. (Contributed by NM, 22-Apr-1994.)
(Ord 𝐴 → E Fr 𝐴)
 
Theoremordelss 5777 An element of an ordinal class is a subset of it. (Contributed by NM, 30-May-1994.)
((Ord 𝐴𝐵𝐴) → 𝐵𝐴)
 
Theoremtrssord 5778 A transitive subclass of an ordinal class is ordinal. (Contributed by NM, 29-May-1994.)
((Tr 𝐴𝐴𝐵 ∧ Ord 𝐵) → Ord 𝐴)
 
Theoremordirr 5779 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 5780 A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998.)
((Ord 𝐴𝐵𝐴) → 𝐴𝐵)
 
Theoremordn2lp 5781 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 5782* 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 5783 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 5784 The class of all ordinal numbers is transitive. (Contributed by NM, 4-May-2009.)
Tr On
 
Theoremordelon 5785 An element of an ordinal class is an ordinal number. (Contributed by NM, 26-Oct-2003.)
((Ord 𝐴𝐵𝐴) → 𝐵 ∈ On)
 
Theoremonelon 5786 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 5787 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 5788 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 5789 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 5790 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 5791 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 5792 The intersection of two ordinal numbers is an ordinal number. (Contributed by NM, 7-Apr-1995.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵) ∈ On)
 
Theoremordtri3or 5793 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 5794 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 5795 A trichotomy law for ordinal numbers. (Contributed by NM, 6-Nov-2003.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
 
Theoremordtri2 5796 A trichotomy law for ordinals. (Contributed by NM, 25-Nov-1995.)
((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
 
Theoremordtri3 5797 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)))
 
Theoremordtri3OLD 5798 Obsolete proof of ordtri3 5797 as of 24-Sep-2021. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ ¬ (𝐴𝐵𝐵𝐴)))
 
Theoremordtri4 5799 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 5800 An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.)
(Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅)
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