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Theorem List for Metamath Proof Explorer - 6701-6800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcaofid1 6701* Transfer a right absorption law to the function operation. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑋)    &   ((𝜑𝑥𝑆) → (𝑥𝑅𝐵) = 𝐶)       (𝜑 → (𝐹𝑓 𝑅(𝐴 × {𝐵})) = (𝐴 × {𝐶}))
 
Theoremcaofid2 6702* Transfer a right absorption law to the function operation. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑋)    &   ((𝜑𝑥𝑆) → (𝐵𝑅𝑥) = 𝐶)       (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 𝑅𝐹) = (𝐴 × {𝐶}))
 
Theoremcaofcom 6703* Transfer a commutative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐺:𝐴𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑅𝑦) = (𝑦𝑅𝑥))       (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝐺𝑓 𝑅𝐹))
 
Theoremcaofrss 6704* Transfer a relation subset law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐺:𝐴𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑅𝑦𝑥𝑇𝑦))       (𝜑 → (𝐹𝑟 𝑅𝐺𝐹𝑟 𝑇𝐺))
 
Theoremcaofass 6705* Transfer an associative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐺:𝐴𝑆)    &   (𝜑𝐻:𝐴𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧)))       (𝜑 → ((𝐹𝑓 𝑅𝐺) ∘𝑓 𝑇𝐻) = (𝐹𝑓 𝑂(𝐺𝑓 𝑃𝐻)))
 
Theoremcaoftrn 6706* Transfer a transitivity law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐺:𝐴𝑆)    &   (𝜑𝐻:𝐴𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝑅𝑦𝑦𝑇𝑧) → 𝑥𝑈𝑧))       (𝜑 → ((𝐹𝑟 𝑅𝐺𝐺𝑟 𝑇𝐻) → 𝐹𝑟 𝑈𝐻))
 
Theoremcaofdi 6707* Transfer a distributive law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐾)    &   (𝜑𝐺:𝐴𝑆)    &   (𝜑𝐻:𝐴𝑆)    &   ((𝜑 ∧ (𝑥𝐾𝑦𝑆𝑧𝑆)) → (𝑥𝑇(𝑦𝑅𝑧)) = ((𝑥𝑇𝑦)𝑂(𝑥𝑇𝑧)))       (𝜑 → (𝐹𝑓 𝑇(𝐺𝑓 𝑅𝐻)) = ((𝐹𝑓 𝑇𝐺) ∘𝑓 𝑂(𝐹𝑓 𝑇𝐻)))
 
Theoremcaofdir 6708* Transfer a reverse distributive law to the function operation. (Contributed by NM, 19-Oct-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐾)    &   (𝜑𝐺:𝐴𝑆)    &   (𝜑𝐻:𝐴𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝐾)) → ((𝑥𝑅𝑦)𝑇𝑧) = ((𝑥𝑇𝑧)𝑂(𝑦𝑇𝑧)))       (𝜑 → ((𝐺𝑓 𝑅𝐻) ∘𝑓 𝑇𝐹) = ((𝐺𝑓 𝑇𝐹) ∘𝑓 𝑂(𝐻𝑓 𝑇𝐹)))
 
Theoremcaonncan 6709* Transfer nncan 10061-shaped laws to vectors of numbers. (Contributed by Stefan O'Rear, 27-Mar-2015.)
(𝜑𝐼𝑉)    &   (𝜑𝐴:𝐼𝑆)    &   (𝜑𝐵:𝐼𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑀(𝑥𝑀𝑦)) = 𝑦)       (𝜑 → (𝐴𝑓 𝑀(𝐴𝑓 𝑀𝐵)) = 𝐵)
 
2.3.20  Proper subset relation
 
Syntaxcrpss 6710 Extend class notation to include the reified proper subset relation.
class []
 
Definitiondf-rpss 6711* Define a relation which corresponds to proper subsethood df-pss 3460 on sets. This allows us to use proper subsethood with general concepts that require relations, such as strict ordering, see sorpss 6716. (Contributed by Stefan O'Rear, 2-Nov-2014.)
[] = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
 
Theoremrelrpss 6712 The proper subset relation is a relation. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Rel []
 
Theorembrrpssg 6713 The proper subset relation on sets is the same as class proper subsethood. (Contributed by Stefan O'Rear, 2-Nov-2014.)
(𝐵𝑉 → (𝐴 [] 𝐵𝐴𝐵))
 
Theorembrrpss 6714 The proper subset relation on sets is the same as class proper subsethood. (Contributed by Stefan O'Rear, 2-Nov-2014.)
𝐵 ∈ V       (𝐴 [] 𝐵𝐴𝐵)
 
Theoremporpss 6715 Every class is partially ordered by proper subsets. (Contributed by Stefan O'Rear, 2-Nov-2014.)
[] Po 𝐴
 
Theoremsorpss 6716* Express strict ordering under proper subsets, i.e. the notion of a chain of sets. (Contributed by Stefan O'Rear, 2-Nov-2014.)
( [] Or 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥))
 
Theoremsorpssi 6717 Property of a chain of sets. (Contributed by Stefan O'Rear, 2-Nov-2014.)
(( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶𝐶𝐵))
 
Theoremsorpssun 6718 A chain of sets is closed under binary union. (Contributed by Mario Carneiro, 16-May-2015.)
(( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶) ∈ 𝐴)
 
Theoremsorpssin 6719 A chain of sets is closed under binary intersection. (Contributed by Mario Carneiro, 16-May-2015.)
(( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶) ∈ 𝐴)
 
Theoremsorpssuni 6720* In a chain of sets, a maximal element is the union of the chain. (Contributed by Stefan O'Rear, 2-Nov-2014.)
( [] Or 𝑌 → (∃𝑢𝑌𝑣𝑌 ¬ 𝑢𝑣 𝑌𝑌))
 
Theoremsorpssint 6721* In a chain of sets, a minimal element is the intersection of the chain. (Contributed by Stefan O'Rear, 2-Nov-2014.)
( [] Or 𝑌 → (∃𝑢𝑌𝑣𝑌 ¬ 𝑣𝑢 𝑌𝑌))
 
Theoremsorpsscmpl 6722* The componentwise complement of a chain of sets is also a chain of sets. (Contributed by Stefan O'Rear, 2-Nov-2014.)
( [] Or 𝑌 → [] Or {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌})
 
2.4  ZF Set Theory - add the Axiom of Union
 
2.4.1  Introduce the Axiom of Union
 
Axiomax-un 6723* Axiom of Union. An axiom of Zermelo-Fraenkel set theory. It states that a set 𝑦 exists that includes the union of a given set 𝑥 i.e. the collection of all members of the members of 𝑥. The variant axun2 6725 states that the union itself exists. A version with the standard abbreviation for union is uniex2 6726. A version using class notation is uniex 6727.

The union of a class df-uni 4271 should not be confused with the union of two classes df-un 3449. Their relationship is shown in unipr 4283. (Contributed by NM, 23-Dec-1993.)

𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)
 
Theoremzfun 6724* Axiom of Union expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)
𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)
 
Theoremaxun2 6725* A variant of the Axiom of Union ax-un 6723. For any set 𝑥, there exists a set 𝑦 whose members are exactly the members of the members of 𝑥 i.e. the union of 𝑥. Axiom Union of [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑧𝑤𝑤𝑥))
 
Theoremuniex2 6726* The Axiom of Union using the standard abbreviation for union. Given any set 𝑥, its union 𝑦 exists. (Contributed by NM, 4-Jun-2006.)
𝑦 𝑦 = 𝑥
 
Theoremuniex 6727 The Axiom of Union in class notation. This says that if 𝐴 is a set i.e. 𝐴 ∈ V (see isset 3084), then the union of 𝐴 is also a set. Same as Axiom 3 of [TakeutiZaring] p. 16. (Contributed by NM, 11-Aug-1993.)
𝐴 ∈ V        𝐴 ∈ V
 
Theoremvuniex 6728 The union of a setvar is a set. (Contributed by BJ, 3-May-2021.)
𝑥 ∈ V
 
Theoremuniexg 6729 The ZF Axiom of Union in class notation, in the form of a theorem instead of an inference. We use the antecedent 𝐴𝑉 instead of 𝐴 ∈ V to make the theorem more general and thus shorten some proofs; obviously the universal class constant V is one possible substitution for class variable 𝑉. (Contributed by NM, 25-Nov-1994.)
(𝐴𝑉 𝐴 ∈ V)
 
Theoremunex 6730 The union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 1-Jul-1994.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝐵) ∈ V
 
Theoremtpex 6731 An unordered triple of classes exists. (Contributed by NM, 10-Apr-1994.)
{𝐴, 𝐵, 𝐶} ∈ V
 
Theoremunexb 6732 Existence of union is equivalent to existence of its components. (Contributed by NM, 11-Jun-1998.)
((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴𝐵) ∈ V)
 
Theoremunexg 6733 A union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 18-Sep-2006.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
 
Theoremxpexg 6734 The Cartesian product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. See also xpexgALT 6927. (Contributed by NM, 14-Aug-1994.)
((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)
 
Theorem3xpexg 6735 The Cartesian product of three sets is a set. (Contributed by Alexander van der Vekens, 21-Feb-2018.)
(𝑉𝑊 → ((𝑉 × 𝑉) × 𝑉) ∈ V)
 
Theoremxpex 6736 The Cartesian product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. (Contributed by NM, 14-Aug-1994.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴 × 𝐵) ∈ V
 
Theoremsqxpexg 6737 The Cartesian square of a set is a set. (Contributed by AV, 13-Jan-2020.)
(𝐴𝑉 → (𝐴 × 𝐴) ∈ V)
 
Theoremsnnex 6738* The class of all singletons is a proper class. (Contributed by NM, 10-Oct-2008.) (Proof shortened by Eric Schmidt, 7-Dec-2008.)
{𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V
 
Theoremdifex2 6739 If the subtrahend of a class difference exists, then the minuend exists iff the difference exists. (Contributed by NM, 12-Nov-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
(𝐵𝐶 → (𝐴 ∈ V ↔ (𝐴𝐵) ∈ V))
 
Theoremdifsnexi 6740 If the difference of a class and a singleton is a set, the class itself is a set. (Contributed by AV, 15-Jan-2019.)
((𝑁 ∖ {𝐾}) ∈ V → 𝑁 ∈ V)
 
Theoremuniuni 6741* Expression for double union that moves union into a class builder. (Contributed by FL, 28-May-2007.)
𝐴 = {𝑥 ∣ ∃𝑦(𝑥 = 𝑦𝑦𝐴)}
 
Theoremuniexb 6742 The Axiom of Union and its converse. A class is a set iff its union is a set. (Contributed by NM, 11-Nov-2003.)
(𝐴 ∈ V ↔ 𝐴 ∈ V)
 
Theorempwexb 6743 The Axiom of Power Sets and its converse. A class is a set iff its power class is a set. (Contributed by NM, 11-Nov-2003.)
(𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
 
Theoremeldifpw 6744 Membership in a power class difference. (Contributed by NM, 25-Mar-2007.)
𝐶 ∈ V       ((𝐴 ∈ 𝒫 𝐵 ∧ ¬ 𝐶𝐵) → (𝐴𝐶) ∈ (𝒫 (𝐵𝐶) ∖ 𝒫 𝐵))
 
Theoremelpwun 6745 Membership in the power class of a union. (Contributed by NM, 26-Mar-2007.)
𝐶 ∈ V       (𝐴 ∈ 𝒫 (𝐵𝐶) ↔ (𝐴𝐶) ∈ 𝒫 𝐵)
 
Theoremiunpw 6746* An indexed union of a power class in terms of the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33. (Contributed by NM, 29-Nov-2003.)
𝐴 ∈ V       (∃𝑥𝐴 𝑥 = 𝐴 ↔ 𝒫 𝐴 = 𝑥𝐴 𝒫 𝑥)
 
Theoremfr3nr 6747 A well-founded relation has no 3-cycle loops. Special case of Proposition 6.23 of [TakeutiZaring] p. 30. (Contributed by NM, 10-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)
((𝑅 Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ¬ (𝐵𝑅𝐶𝐶𝑅𝐷𝐷𝑅𝐵))
 
Theoremepne3 6748 A set well-founded by epsilon contains no 3-cycle loops. (Contributed by NM, 19-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)
(( E Fr 𝐴 ∧ (𝐵𝐴𝐶𝐴𝐷𝐴)) → ¬ (𝐵𝐶𝐶𝐷𝐷𝐵))
 
Theoremdfwe2 6749* Alternate definition of well-ordering. Definition 6.24(2) of [TakeutiZaring] p. 30. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
(𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
 
2.4.2  Ordinals (continued)
 
Theoremordon 6750 The class of all ordinal numbers is ordinal. Proposition 7.12 of [TakeutiZaring] p. 38, but without using the Axiom of Regularity. (Contributed by NM, 17-May-1994.)
Ord On
 
Theoremepweon 6751 The epsilon relation well-orders the class of ordinal numbers. Proposition 4.8(g) of [Mendelson] p. 244. (Contributed by NM, 1-Nov-2003.)
E We On
 
Theoremonprc 6752 No set contains all ordinal numbers. Proposition 7.13 of [TakeutiZaring] p. 38, but without using the Axiom of Regularity. This is also known as the Burali-Forti paradox (remark in [Enderton] p. 194). In 1897, Cesare Burali-Forti noticed that since the "set" of all ordinal numbers is an ordinal class (ordon 6750), it must be both an element of the set of all ordinal numbers yet greater than every such element. ZF set theory resolves this paradox by not allowing the class of all ordinal numbers to be a set (so instead it is a proper class). Here we prove the denial of its existence. (Contributed by NM, 18-May-1994.)
¬ On ∈ V
 
Theoremssorduni 6753 The union of a class of ordinal numbers is ordinal. Proposition 7.19 of [TakeutiZaring] p. 40. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
(𝐴 ⊆ On → Ord 𝐴)
 
Theoremssonuni 6754 The union of a set of ordinal numbers is an ordinal number. Theorem 9 of [Suppes] p. 132. (Contributed by NM, 1-Nov-2003.)
(𝐴𝑉 → (𝐴 ⊆ On → 𝐴 ∈ On))
 
Theoremssonunii 6755 The union of a set of ordinal numbers is an ordinal number. Corollary 7N(d) of [Enderton] p. 193. (Contributed by NM, 20-Sep-2003.)
𝐴 ∈ V       (𝐴 ⊆ On → 𝐴 ∈ On)
 
Theoremordeleqon 6756 A way to express the ordinal property of a class in terms of the class of ordinal numbers. Corollary 7.14 of [TakeutiZaring] p. 38 and its converse. (Contributed by NM, 1-Jun-2003.)
(Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
 
Theoremordsson 6757 Any ordinal class is a subclass of the class of ordinal numbers. Corollary 7.15 of [TakeutiZaring] p. 38. (Contributed by NM, 18-May-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
(Ord 𝐴𝐴 ⊆ On)
 
Theoremonss 6758 An ordinal number is a subset of the class of ordinal numbers. (Contributed by NM, 5-Jun-1994.)
(𝐴 ∈ On → 𝐴 ⊆ On)
 
Theorempredon 6759 For an ordinal, the predecessor under E and On is an identity relationship. (Contributed by Scott Fenton, 27-Mar-2011.)
(𝐴 ∈ On → Pred( E , On, 𝐴) = 𝐴)
 
Theoremssonprc 6760 Two ways of saying a class of ordinals is unbounded. (Contributed by Mario Carneiro, 8-Jun-2013.)
(𝐴 ⊆ On → (𝐴 ∉ V ↔ 𝐴 = On))
 
Theoremonuni 6761 The union of an ordinal number is an ordinal number. (Contributed by NM, 29-Sep-2006.)
(𝐴 ∈ On → 𝐴 ∈ On)
 
Theoremorduni 6762 The union of an ordinal class is ordinal. (Contributed by NM, 12-Sep-2003.)
(Ord 𝐴 → Ord 𝐴)
 
Theoremonint 6763 The intersection (infimum) of a nonempty class of ordinal numbers belongs to the class. Compare Exercise 4 of [TakeutiZaring] p. 45. (Contributed by NM, 31-Jan-1997.)
((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → 𝐴𝐴)
 
Theoremonint0 6764 The intersection of a class of ordinal numbers is zero iff the class contains zero. (Contributed by NM, 24-Apr-2004.)
(𝐴 ⊆ On → ( 𝐴 = ∅ ↔ ∅ ∈ 𝐴))
 
Theoremonssmin 6765* A nonempty class of ordinal numbers has the smallest member. Exercise 9 of [TakeutiZaring] p. 40. (Contributed by NM, 3-Oct-2003.)
((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∃𝑥𝐴𝑦𝐴 𝑥𝑦)
 
Theoremonminesb 6766 If a property is true for some ordinal number, it is true for a minimal ordinal number. This version uses explicit substitution. Theorem Schema 62 of [Suppes] p. 228. (Contributed by NM, 29-Sep-2003.)
(∃𝑥 ∈ On 𝜑[ {𝑥 ∈ On ∣ 𝜑} / 𝑥]𝜑)
 
Theoremonminsb 6767 If a property is true for some ordinal number, it is true for a minimal ordinal number. This version uses implicit substitution. Theorem Schema 62 of [Suppes] p. 228. (Contributed by NM, 3-Oct-2003.)
𝑥𝜓    &   (𝑥 = {𝑥 ∈ On ∣ 𝜑} → (𝜑𝜓))       (∃𝑥 ∈ On 𝜑𝜓)
 
Theoremoninton 6768 The intersection of a nonempty collection of ordinal numbers is an ordinal number. Compare Exercise 6 of [TakeutiZaring] p. 44. (Contributed by NM, 29-Jan-1997.)
((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → 𝐴 ∈ On)
 
Theoremonintrab 6769 The intersection of a class of ordinal numbers exists iff it is an ordinal number. (Contributed by NM, 6-Nov-2003.)
( {𝑥 ∈ On ∣ 𝜑} ∈ V ↔ {𝑥 ∈ On ∣ 𝜑} ∈ On)
 
Theoremonintrab2 6770 An existence condition equivalent to an intersection's being an ordinal number. (Contributed by NM, 6-Nov-2003.)
(∃𝑥 ∈ On 𝜑 {𝑥 ∈ On ∣ 𝜑} ∈ On)
 
Theoremonnmin 6771 No member of a set of ordinal numbers belongs to its minimum. (Contributed by NM, 2-Feb-1997.)
((𝐴 ⊆ On ∧ 𝐵𝐴) → ¬ 𝐵 𝐴)
 
Theoremonnminsb 6772* An ordinal number smaller than the minimum of a set of ordinal numbers does not have the property determining that set. 𝜓 is the wff resulting from the substitution of 𝐴 for 𝑥 in wff 𝜑. (Contributed by NM, 9-Nov-2003.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴 ∈ On → (𝐴 {𝑥 ∈ On ∣ 𝜑} → ¬ 𝜓))
 
Theoremoneqmin 6773* A way to show that an ordinal number equals the minimum of a nonempty 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 ∧ 𝐵 ≠ ∅) → (𝐴 = 𝐵 ↔ (𝐴𝐵 ∧ ∀𝑥𝐴 ¬ 𝑥𝐵)))
 
Theorembm2.5ii 6774* Problem 2.5(ii) of [BellMachover] p. 471. (Contributed by NM, 20-Sep-2003.)
𝐴 ∈ V       (𝐴 ⊆ On → 𝐴 = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥})
 
Theoremonminex 6775* If a wff is true for an ordinal number, there is the smallest ordinal number for which it is true. (Contributed by NM, 2-Feb-1997.) (Proof shortened by Mario Carneiro, 20-Nov-2016.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥 ∈ On 𝜑 → ∃𝑥 ∈ On (𝜑 ∧ ∀𝑦𝑥 ¬ 𝜓))
 
Theoremsucon 6776 The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.)
suc On = On
 
Theoremsucexb 6777 A successor exists iff its class argument exists. (Contributed by NM, 22-Jun-1998.)
(𝐴 ∈ V ↔ suc 𝐴 ∈ V)
 
Theoremsucexg 6778 The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.)
(𝐴𝑉 → suc 𝐴 ∈ V)
 
Theoremsucex 6779 The successor of a set is a set. (Contributed by NM, 30-Aug-1993.)
𝐴 ∈ V       suc 𝐴 ∈ V
 
Theoremonmindif2 6780 The minimum of a class of ordinal numbers is less than the minimum of that class with its minimum removed. (Contributed by NM, 20-Nov-2003.)
((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → 𝐴 (𝐴 ∖ { 𝐴}))
 
Theoremsuceloni 6781 The successor of an ordinal number is an ordinal number. Proposition 7.24 of [TakeutiZaring] p. 41. (Contributed by NM, 6-Jun-1994.)
(𝐴 ∈ On → suc 𝐴 ∈ On)
 
Theoremordsuc 6782 The successor of an ordinal class is ordinal. (Contributed by NM, 3-Apr-1995.)
(Ord 𝐴 ↔ Ord suc 𝐴)
 
Theoremordpwsuc 6783 The collection of ordinals in the power class of an ordinal is its successor. (Contributed by NM, 30-Jan-2005.)
(Ord 𝐴 → (𝒫 𝐴 ∩ On) = suc 𝐴)
 
Theoremonpwsuc 6784 The collection of ordinal numbers in the power set of an ordinal number is its successor. (Contributed by NM, 19-Oct-2004.)
(𝐴 ∈ On → (𝒫 𝐴 ∩ On) = suc 𝐴)
 
Theoremsucelon 6785 The successor of an ordinal number is an ordinal number. (Contributed by NM, 9-Sep-2003.)
(𝐴 ∈ On ↔ suc 𝐴 ∈ On)
 
Theoremordsucss 6786 The successor of an element of an ordinal class is a subset of it. (Contributed by NM, 21-Jun-1998.)
(Ord 𝐵 → (𝐴𝐵 → suc 𝐴𝐵))
 
Theoremonpsssuc 6787 An ordinal number is a proper subset of its successor. (Contributed by Stefan O'Rear, 18-Nov-2014.)
(𝐴 ∈ On → 𝐴 ⊊ suc 𝐴)
 
Theoremordelsuc 6788 A set belongs to an ordinal iff its successor is a subset of the ordinal. Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM, 29-Nov-2003.)
((𝐴𝐶 ∧ Ord 𝐵) → (𝐴𝐵 ↔ suc 𝐴𝐵))
 
Theoremonsucmin 6789* The successor of an ordinal number is the smallest larger ordinal number. (Contributed by NM, 28-Nov-2003.)
(𝐴 ∈ On → suc 𝐴 = {𝑥 ∈ On ∣ 𝐴𝑥})
 
Theoremordsucelsuc 6790 Membership is inherited by successors. Generalization of Exercise 9 of [TakeutiZaring] p. 42. (Contributed by NM, 22-Jun-1998.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
(Ord 𝐵 → (𝐴𝐵 ↔ suc 𝐴 ∈ suc 𝐵))
 
Theoremordsucsssuc 6791 The subclass relationship between two ordinal classes is inherited by their successors. (Contributed by NM, 4-Oct-2003.)
((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ suc 𝐴 ⊆ suc 𝐵))
 
Theoremordsucuniel 6792 Given an element 𝐴 of the union of an ordinal 𝐵, suc 𝐴 is an element of 𝐵 itself. (Contributed by Scott Fenton, 28-Mar-2012.) (Proof shortened by Mario Carneiro, 29-May-2015.)
(Ord 𝐵 → (𝐴 𝐵 ↔ suc 𝐴𝐵))
 
Theoremordsucun 6793 The successor of the maximum (i.e. union) of two ordinals is the maximum of their successors. (Contributed by NM, 28-Nov-2003.)
((Ord 𝐴 ∧ Ord 𝐵) → suc (𝐴𝐵) = (suc 𝐴 ∪ suc 𝐵))
 
Theoremordunpr 6794 The maximum of two ordinals is equal to one of them. (Contributed by Mario Carneiro, 25-Jun-2015.)
((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵𝐶) ∈ {𝐵, 𝐶})
 
Theoremordunel 6795 The maximum of two ordinals belongs to a third if each of them do. (Contributed by NM, 18-Sep-2006.) (Revised by Mario Carneiro, 25-Jun-2015.)
((Ord 𝐴𝐵𝐴𝐶𝐴) → (𝐵𝐶) ∈ 𝐴)
 
Theoremonsucuni 6796 A class of ordinal numbers is a subclass of the successor of its union. Similar to Proposition 7.26 of [TakeutiZaring] p. 41. (Contributed by NM, 19-Sep-2003.)
(𝐴 ⊆ On → 𝐴 ⊆ suc 𝐴)
 
Theoremordsucuni 6797 An ordinal class is a subclass of the successor of its union. (Contributed by NM, 12-Sep-2003.)
(Ord 𝐴𝐴 ⊆ suc 𝐴)
 
Theoremorduniorsuc 6798 An ordinal class is either its union or the successor of its union. If we adopt the view that zero is a limit ordinal, this means every ordinal class is either a limit or a successor. (Contributed by NM, 13-Sep-2003.)
(Ord 𝐴 → (𝐴 = 𝐴𝐴 = suc 𝐴))
 
Theoremunon 6799 The class of all ordinal numbers is its own union. Exercise 11 of [TakeutiZaring] p. 40. (Contributed by NM, 12-Nov-2003.)
On = On
 
Theoremordunisuc 6800 An ordinal class is equal to the union of its successor. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(Ord 𝐴 suc 𝐴 = 𝐴)
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