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Theorem List for Metamath Proof Explorer - 5801-5900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcoeq0 5801 A composition of two relations is empty iff there is no overlap between the range of the second and the domain of the first. Useful in combination with coundi 5793 and coundir 5794 to prune meaningless terms in the result. (Contributed by Stefan O'Rear, 8-Oct-2014.)
((𝐴𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅)
 
Theoremcoiun 5802* Composition with an indexed union. (Contributed by NM, 21-Dec-2008.)
(𝐴 𝑥𝐶 𝐵) = 𝑥𝐶 (𝐴𝐵)
 
Theoremcocnvcnv1 5803 A composition is not affected by a double converse of its first argument. (Contributed by NM, 8-Oct-2007.)
(𝐴𝐵) = (𝐴𝐵)
 
Theoremcocnvcnv2 5804 A composition is not affected by a double converse of its second argument. (Contributed by NM, 8-Oct-2007.)
(𝐴𝐵) = (𝐴𝐵)
 
Theoremcores2 5805 Absorption of a reverse (preimage) restriction of the second member of a class composition. (Contributed by NM, 11-Dec-2006.)
(dom 𝐴𝐶 → (𝐴(𝐵𝐶)) = (𝐴𝐵))
 
Theoremco02 5806 Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.)
(𝐴 ∘ ∅) = ∅
 
Theoremco01 5807 Composition with the empty set. (Contributed by NM, 24-Apr-2004.)
(∅ ∘ 𝐴) = ∅
 
Theoremcoi1 5808 Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)
(Rel 𝐴 → (𝐴 ∘ I ) = 𝐴)
 
Theoremcoi2 5809 Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)
(Rel 𝐴 → ( I ∘ 𝐴) = 𝐴)
 
Theoremcoires1 5810 Composition with a restricted identity relation. (Contributed by FL, 19-Jun-2011.) (Revised by Stefan O'Rear, 7-Mar-2015.)
(𝐴 ∘ ( I ↾ 𝐵)) = (𝐴𝐵)
 
Theoremcoass 5811 Associative law for class composition. Theorem 27 of [Suppes] p. 64. Also Exercise 21 of [Enderton] p. 53. Interestingly, this law holds for any classes whatsoever, not just functions or even relations. (Contributed by NM, 27-Jan-1997.)
((𝐴𝐵) ∘ 𝐶) = (𝐴 ∘ (𝐵𝐶))
 
Theoremrelcnvtr 5812 A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.)
(Rel 𝑅 → ((𝑅𝑅) ⊆ 𝑅 ↔ (𝑅𝑅) ⊆ 𝑅))
 
Theoremrelssdmrn 5813 A relation is included in the Cartesian product of its domain and range. Exercise 4.12(t) of [Mendelson] p. 235. (Contributed by NM, 3-Aug-1994.)
(Rel 𝐴𝐴 ⊆ (dom 𝐴 × ran 𝐴))
 
Theoremcnvssrndm 5814 The converse is a subset of the cartesian product of range and domain. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐴 ⊆ (ran 𝐴 × dom 𝐴)
 
Theoremcossxp 5815 Composition as a subset of the Cartesian product of factors. (Contributed by Mario Carneiro, 12-Jan-2017.)
(𝐴𝐵) ⊆ (dom 𝐵 × ran 𝐴)
 
Theoremrelrelss 5816 Two ways to describe the structure of a two-place operation. (Contributed by NM, 17-Dec-2008.)
((Rel 𝐴 ∧ Rel dom 𝐴) ↔ 𝐴 ⊆ ((V × V) × V))
 
Theoremunielrel 5817 The membership relation for a relation is inherited by class union. (Contributed by NM, 17-Sep-2006.)
((Rel 𝑅𝐴𝑅) → 𝐴 𝑅)
 
Theoremrelfld 5818 The double union of a relation is its field. (Contributed by NM, 17-Sep-2006.)
(Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))
 
Theoremrelresfld 5819 Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.)
(Rel 𝑅 → (𝑅 𝑅) = 𝑅)
 
Theoremrelcoi2 5820 Composition with the identity relation restricted to a relation's field. (Contributed by FL, 2-May-2011.)
(Rel 𝑅 → (( I ↾ 𝑅) ∘ 𝑅) = 𝑅)
 
Theoremrelcoi1 5821 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 5822 The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.)
𝐴 = (dom 𝐴 ∪ ran 𝐴)
 
Theoremrelcnvfld 5823 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 5824 Alternate definition of domain df-dm 5272 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.)
dom 𝐴 = (𝐴𝐴)
 
Theoremunixp 5825 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 5826 A Cartesian product is empty iff its union is empty. (Contributed by NM, 20-Sep-2006.)
((𝐴 × 𝐵) = ∅ ↔ (𝐴 × 𝐵) = ∅)
 
Theoremunixpid 5827 Field of a square Cartesian product. (Contributed by FL, 10-Oct-2009.)
(𝐴 × 𝐴) = 𝐴
 
Theoremressn 5828 Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
(𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵}))
 
Theoremcnviin 5829* The converse of an intersection is the intersection of the converse. (Contributed by FL, 15-Oct-2012.)
(𝐴 ≠ ∅ → 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵)
 
Theoremcnvpo 5830 The converse of a partial order relation is a partial order relation. (Contributed by NM, 15-Jun-2005.)
(𝑅 Po 𝐴𝑅 Po 𝐴)
 
Theoremcnvso 5831 The converse of a strict order relation is a strict order relation. (Contributed by NM, 15-Jun-2005.)
(𝑅 Or 𝐴𝑅 Or 𝐴)
 
Theoremxpco 5832 Composition of two Cartesian products. (Contributed by Thierry Arnoux, 17-Nov-2017.)
(𝐵 ≠ ∅ → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶))
 
Theoremxpcoid 5833 Composition of two square Cartesian products. (Contributed by Thierry Arnoux, 14-Jan-2018.)
((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (𝐴 × 𝐴)
 
Theoremelsnxp 5834* 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 5835* Obsolete proof of elsnxp 5834 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 5836 The predecessors symbol.
class Pred(𝑅, 𝐴, 𝑋)
 
Definitiondf-pred 5837 Define the predecessor class of a relationship. This is the class of all elements 𝑦 of 𝐴 such that 𝑦𝑅𝑋 (see elpred 5850) . (Contributed by Scott Fenton, 29-Jan-2011.)
Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
 
Theorempredeq123 5838 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 5839 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝑅 = 𝑆 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐴, 𝑋))
 
Theorempredeq2 5840 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝐴 = 𝐵 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))
 
Theorempredeq3 5841 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝑋 = 𝑌 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))
 
Theoremnfpred 5842 Bound-variable hypothesis builder for the predecessor class. (Contributed by Scott Fenton, 9-Jun-2018.)
𝑥𝑅    &   𝑥𝐴    &   𝑥𝑋       𝑥Pred(𝑅, 𝐴, 𝑋)
 
Theorempredpredss 5843 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 5844 The predecessor class of 𝐴 is a subset of 𝐴. (Contributed by Scott Fenton, 2-Feb-2011.)
Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐴
 
Theoremsspred 5845 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 5846* 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 5847* 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 5848* 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 5849 Membership in a predecessor class - implicative version. (Contributed by Scott Fenton, 9-May-2012.)
𝑋 ∈ V       (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋)
 
Theoremelpred 5850 Membership in a predecessor class. (Contributed by Scott Fenton, 4-Feb-2011.)
𝑌 ∈ V       (𝑋𝐷 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌𝐴𝑌𝑅𝑋)))
 
Theoremelpredg 5851 Membership in a predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.)
((𝑋𝐵𝑌𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅𝑋))
 
Theorempredasetex 5852 The predecessor class exists when 𝐴 does. (Contributed by Scott Fenton, 8-Feb-2011.)
𝐴 ∈ V       Pred(𝑅, 𝐴, 𝑋) ∈ V
 
Theoremdffr4 5853* Alternate definition of well-founded relation. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 Pred(𝑅, 𝑥, 𝑦) = ∅))
 
Theorempredel 5854 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 5855 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 5856 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 5857 Closed form of elpredim 5849. (Contributed by Scott Fenton, 13-Apr-2011.) (Revised by NM, 5-Apr-2016.)
((𝑋𝑉𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)) → 𝑌𝑅𝑋)
 
Theoremsetlikespec 5858 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 5859 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 5860 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 5861 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 5862 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 5863 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 5864* 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 5865 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 5866 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 5867 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 5868* 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 5869* 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 5870* 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 5871* 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 5872* 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 5873* 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 5874* 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 5875* 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 5876* 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 5877* 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 5878* 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 5879 Extend the definition of a wff to include the ordinal predicate.
wff Ord 𝐴
 
Syntaxcon0 5880 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 5881 Extend the definition of a wff to include the limit ordinal predicate.
wff Lim 𝐴
 
Syntaxcsuc 5882 Extend class notation to include the successor function.
class suc 𝐴
 
Definitiondf-ord 5883 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 5884 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 5885 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 5938, dflim3 7208, and dflim4 for alternate definitions. (Contributed by NM, 22-Apr-1994.)
(Lim 𝐴 ↔ (Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴))
 
Definitiondf-suc 5886 Define the successor of a class. When applied to an ordinal number, the successor means the same thing as "plus 1" (see oa1suc 7776). 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 5957), so that the successor of any ordinal class is still an ordinal class (ordsuc 7175), 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 5887 Equality theorem for the ordinal predicate. (Contributed by NM, 17-Sep-1993.)
(𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵))
 
Theoremelong 5888 An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
(𝐴𝑉 → (𝐴 ∈ On ↔ Ord 𝐴))
 
Theoremelon 5889 An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
𝐴 ∈ V       (𝐴 ∈ On ↔ Ord 𝐴)
 
Theoremeloni 5890 An ordinal number has the ordinal property. (Contributed by NM, 5-Jun-1994.)
(𝐴 ∈ On → Ord 𝐴)
 
Theoremelon2 5891 An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004.)
(𝐴 ∈ On ↔ (Ord 𝐴𝐴 ∈ V))
 
Theoremlimeq 5892 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 5893 Epsilon well-orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.)
(Ord 𝐴 → E We 𝐴)
 
Theoremordtr 5894 An ordinal class is transitive. (Contributed by NM, 3-Apr-1994.)
(Ord 𝐴 → Tr 𝐴)
 
Theoremordfr 5895 Epsilon is well-founded on an ordinal class. (Contributed by NM, 22-Apr-1994.)
(Ord 𝐴 → E Fr 𝐴)
 
Theoremordelss 5896 An element of an ordinal class is a subset of it. (Contributed by NM, 30-May-1994.)
((Ord 𝐴𝐵𝐴) → 𝐵𝐴)
 
Theoremtrssord 5897 A transitive subclass of an ordinal class is ordinal. (Contributed by NM, 29-May-1994.)
((Tr 𝐴𝐴𝐵 ∧ Ord 𝐵) → Ord 𝐴)
 
Theoremordirr 5898 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 5899 A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998.)
((Ord 𝐴𝐵𝐴) → 𝐴𝐵)
 
Theoremordn2lp 5900 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))
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