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Theorem List for Metamath Proof Explorer - 6101-6200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdmco 6101 The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.)
dom (𝐴𝐵) = (𝐵 “ dom 𝐴)
 
Theoremcoeq0 6102 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 6094 and coundir 6095 to prune meaningless terms in the result. (Contributed by Stefan O'Rear, 8-Oct-2014.)
((𝐴𝐵) = ∅ ↔ (dom 𝐴 ∩ ran 𝐵) = ∅)
 
Theoremcoiun 6103* Composition with an indexed union. (Contributed by NM, 21-Dec-2008.)
(𝐴 𝑥𝐶 𝐵) = 𝑥𝐶 (𝐴𝐵)
 
Theoremcocnvcnv1 6104 A composition is not affected by a double converse of its first argument. (Contributed by NM, 8-Oct-2007.)
(𝐴𝐵) = (𝐴𝐵)
 
Theoremcocnvcnv2 6105 A composition is not affected by a double converse of its second argument. (Contributed by NM, 8-Oct-2007.)
(𝐴𝐵) = (𝐴𝐵)
 
Theoremcores2 6106 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 6107 Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.)
(𝐴 ∘ ∅) = ∅
 
Theoremco01 6108 Composition with the empty set. (Contributed by NM, 24-Apr-2004.)
(∅ ∘ 𝐴) = ∅
 
Theoremcoi1 6109 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 6110 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 6111 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 6112 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))
 
Theoremrelcnvtrg 6113 General form of relcnvtr 6114. (Contributed by Peter Mazsa, 17-Oct-2023.)
((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → ((𝑅𝑆) ⊆ 𝑇 ↔ (𝑆𝑅) ⊆ 𝑇))
 
Theoremrelcnvtr 6114 A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.) (Proof shortened by Peter Mazsa, 17-Oct-2023.)
(Rel 𝑅 → ((𝑅𝑅) ⊆ 𝑅 ↔ (𝑅𝑅) ⊆ 𝑅))
 
Theoremrelssdmrn 6115 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 6116 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 6117 Composition as a subset of the Cartesian product of factors. (Contributed by Mario Carneiro, 12-Jan-2017.)
(𝐴𝐵) ⊆ (dom 𝐵 × ran 𝐴)
 
Theoremrelrelss 6118 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 6119 The membership relation for a relation is inherited by class union. (Contributed by NM, 17-Sep-2006.)
((Rel 𝑅𝐴𝑅) → 𝐴 𝑅)
 
Theoremrelfld 6120 The double union of a relation is its field. (Contributed by NM, 17-Sep-2006.)
(Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))
 
Theoremrelresfld 6121 Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.)
(Rel 𝑅 → (𝑅 𝑅) = 𝑅)
 
Theoremrelcoi2 6122 Composition with the identity relation restricted to a relation's field. (Contributed by FL, 2-May-2011.)
(Rel 𝑅 → (( I ↾ 𝑅) ∘ 𝑅) = 𝑅)
 
Theoremrelcoi1 6123 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 6124 The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.)
𝐴 = (dom 𝐴 ∪ ran 𝐴)
 
Theoremrelcnvfld 6125 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 6126 Alternate definition of domain df-dm 5559 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.)
dom 𝐴 = (𝐴𝐴)
 
Theoremunixp 6127 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 6128 A Cartesian product is empty iff its union is empty. (Contributed by NM, 20-Sep-2006.)
((𝐴 × 𝐵) = ∅ ↔ (𝐴 × 𝐵) = ∅)
 
Theoremunixpid 6129 Field of a Cartesian square. (Contributed by FL, 10-Oct-2009.)
(𝐴 × 𝐴) = 𝐴
 
Theoremressn 6130 Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
(𝐴 ↾ {𝐵}) = ({𝐵} × (𝐴 “ {𝐵}))
 
Theoremcnviin 6131* The converse of an intersection is the intersection of the converse. (Contributed by FL, 15-Oct-2012.)
(𝐴 ≠ ∅ → 𝑥𝐴 𝐵 = 𝑥𝐴 𝐵)
 
Theoremcnvpo 6132 The converse of a partial order relation is a partial order relation. (Contributed by NM, 15-Jun-2005.)
(𝑅 Po 𝐴𝑅 Po 𝐴)
 
Theoremcnvso 6133 The converse of a strict order relation is a strict order relation. (Contributed by NM, 15-Jun-2005.)
(𝑅 Or 𝐴𝑅 Or 𝐴)
 
Theoremxpco 6134 Composition of two Cartesian products. (Contributed by Thierry Arnoux, 17-Nov-2017.)
(𝐵 ≠ ∅ → ((𝐵 × 𝐶) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐶))
 
Theoremxpcoid 6135 Composition of two Cartesian squares. (Contributed by Thierry Arnoux, 14-Jan-2018.)
((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) = (𝐴 × 𝐴)
 
Theoremelsnxp 6136* Membership in 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⟩))
 
Theoremreu3op 6137* There is a unique ordered pair fulfilling a wff iff there are uniquely two sets fulfilling a corresponding wff. (Contributed by AV, 1-Jul-2023.)
(𝑝 = ⟨𝑎, 𝑏⟩ → (𝜓𝜒))       (∃!𝑝 ∈ (𝑋 × 𝑌)𝜓 ↔ (∃𝑎𝑋𝑏𝑌 𝜒 ∧ ∃𝑥𝑋𝑦𝑌𝑎𝑋𝑏𝑌 (𝜒 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
 
Theoremreuop 6138* There is a unique ordered pair fulfilling a wff iff there are uniquely two sets fulfilling a corresponding wff. (Contributed by AV, 23-Jun-2023.)
(𝑝 = ⟨𝑎, 𝑏⟩ → (𝜓𝜒))    &   (𝑝 = ⟨𝑥, 𝑦⟩ → (𝜓𝜃))       (∃!𝑝 ∈ (𝑋 × 𝑌)𝜓 ↔ ∃𝑎𝑋𝑏𝑌 (𝜒 ∧ ∀𝑥𝑋𝑦𝑌 (𝜃 → ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)))
 
Theoremopreu2reurex 6139* There is a unique ordered pair fulfilling a wff iff there are uniquely two sets fulfilling a corresponding wff. (Contributed by AV, 24-Jun-2023.) (Revised by AV, 1-Jul-2023.)
(𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑𝜒))       (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ (∃!𝑎𝐴𝑏𝐵 𝜒 ∧ ∃!𝑏𝐵𝑎𝐴 𝜒))
 
Theoremopreu2reu 6140* If there is a unique ordered pair fulfilling a wff, then there is a double restricted unique existential qualification fulfilling a corresponding wff. (Contributed by AV, 25-Jun-2023.) (Revised by AV, 2-Jul-2023.)
(𝑝 = ⟨𝑎, 𝑏⟩ → (𝜑𝜒))       (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 → ∃!𝑎𝐴 ∃!𝑏𝐵 𝜒)
 
2.3.11  The Predecessor Class
 
Syntaxcpred 6141 The predecessors symbol.
class Pred(𝑅, 𝐴, 𝑋)
 
Definitiondf-pred 6142 Define the predecessor class of a binary relation. This is the class of all elements 𝑦 of 𝐴 such that 𝑦𝑅𝑋 (see elpred 6155) . (Contributed by Scott Fenton, 29-Jan-2011.)
Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (𝑅 “ {𝑋}))
 
Theorempredeq123 6143 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 6144 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝑅 = 𝑆 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐴, 𝑋))
 
Theorempredeq2 6145 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝐴 = 𝐵 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋))
 
Theorempredeq3 6146 Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝑋 = 𝑌 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌))
 
Theoremnfpred 6147 Bound-variable hypothesis builder for the predecessor class. (Contributed by Scott Fenton, 9-Jun-2018.)
𝑥𝑅    &   𝑥𝐴    &   𝑥𝑋       𝑥Pred(𝑅, 𝐴, 𝑋)
 
Theorempredpredss 6148 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 6149 The predecessor class of 𝐴 is a subset of 𝐴. (Contributed by Scott Fenton, 2-Feb-2011.)
Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐴
 
Theoremsspred 6150 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 6151* 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 6152* 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 6153* 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 6154 Membership in a predecessor class - implicative version. (Contributed by Scott Fenton, 9-May-2012.)
𝑋 ∈ V       (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋)
 
Theoremelpred 6155 Membership in a predecessor class. (Contributed by Scott Fenton, 4-Feb-2011.)
𝑌 ∈ V       (𝑋𝐷 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌𝐴𝑌𝑅𝑋)))
 
Theoremelpredg 6156 Membership in a predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.)
((𝑋𝐵𝑌𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅𝑋))
 
Theorempredasetex 6157 The predecessor class exists when 𝐴 does. (Contributed by Scott Fenton, 8-Feb-2011.)
𝐴 ∈ V       Pred(𝑅, 𝐴, 𝑋) ∈ V
 
Theoremdffr4 6158* Alternate definition of well-founded relation. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥 Pred(𝑅, 𝑥, 𝑦) = ∅))
 
Theorempredel 6159 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 6160 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 6161 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 6162 Closed form of elpredim 6154. (Contributed by Scott Fenton, 13-Apr-2011.) (Revised by NM, 5-Apr-2016.)
((𝑋𝑉𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)) → 𝑌𝑅𝑋)
 
Theoremsetlikespec 6163 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 6164 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 6165 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 6166 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 6167 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 6168 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))
 
Theorempreddowncl 6169* 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 6170 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 6171 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 6172 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 6173* 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 6174* All nonempty subclasses of a class having a well-founded 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 6175* 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 6176* 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 6177* 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 6178* 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 6179* 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 6180* 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 6181* 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 6182* 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 6183* 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 6184 Extend the definition of a wff to include the ordinal predicate.
wff Ord 𝐴
 
Syntaxcon0 6185 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 6186 Extend the definition of a wff to include the limit ordinal predicate.
wff Lim 𝐴
 
Syntaxcsuc 6187 Extend class notation to include the successor function.
class suc 𝐴
 
Definitiondf-ord 6188 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. (Contributed by NM, 17-Sep-1993.)
(Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
 
Definitiondf-on 6189 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 6190 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 6241, dflim3 7550, and dflim4 for alternate definitions. (Contributed by NM, 22-Apr-1994.)
(Lim 𝐴 ↔ (Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴))
 
Definitiondf-suc 6191 Define the successor of a class. When applied to an ordinal number, the successor means the same thing as "plus 1" (see oa1suc 8147). 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 6260), so that the successor of any ordinal class is still an ordinal class (ordsuc 7517), 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 6192 Equality theorem for the ordinal predicate. (Contributed by NM, 17-Sep-1993.)
(𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵))
 
Theoremelong 6193 An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
(𝐴𝑉 → (𝐴 ∈ On ↔ Ord 𝐴))
 
Theoremelon 6194 An ordinal number is an ordinal set. (Contributed by NM, 5-Jun-1994.)
𝐴 ∈ V       (𝐴 ∈ On ↔ Ord 𝐴)
 
Theoremeloni 6195 An ordinal number has the ordinal property. (Contributed by NM, 5-Jun-1994.)
(𝐴 ∈ On → Ord 𝐴)
 
Theoremelon2 6196 An ordinal number is an ordinal set. (Contributed by NM, 8-Feb-2004.)
(𝐴 ∈ On ↔ (Ord 𝐴𝐴 ∈ V))
 
Theoremlimeq 6197 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 6198 Membership well-orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.)
(Ord 𝐴 → E We 𝐴)
 
Theoremordtr 6199 An ordinal class is transitive. (Contributed by NM, 3-Apr-1994.)
(Ord 𝐴 → Tr 𝐴)
 
Theoremordfr 6200 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)
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