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Theorem List for Metamath Proof Explorer - 7201-7300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcaofinvl 7201* Transfer a left inverse law to the function operation. (Contributed by NM, 22-Oct-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐵𝑊)    &   (𝜑𝑁:𝑆𝑆)    &   (𝜑𝐺 = (𝑣𝐴 ↦ (𝑁‘(𝐹𝑣))))    &   ((𝜑𝑥𝑆) → ((𝑁𝑥)𝑅𝑥) = 𝐵)       (𝜑 → (𝐺𝑓 𝑅𝐹) = (𝐴 × {𝐵}))

Theoremcaofid0l 7202* Transfer a left identity law to the function operation. (Contributed by NM, 21-Oct-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐵𝑊)    &   ((𝜑𝑥𝑆) → (𝐵𝑅𝑥) = 𝑥)       (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 𝑅𝐹) = 𝐹)

Theoremcaofid0r 7203* Transfer a right identity law to the function operation. (Contributed by NM, 21-Oct-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐵𝑊)    &   ((𝜑𝑥𝑆) → (𝑥𝑅𝐵) = 𝑥)       (𝜑 → (𝐹𝑓 𝑅(𝐴 × {𝐵})) = 𝐹)

Theoremcaofid1 7204* 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 7205* 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 7206* 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 7207* 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 7208* 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 7209* 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 7210* 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 7211* 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 7212* Transfer nncan 10652-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)

2.3.21  Proper subset relation

Syntaxcrpss 7213 Extend class notation to include the reified proper subset relation.
class []

Definitiondf-rpss 7214* Define a relation which corresponds to proper subsethood df-pss 3808 on sets. This allows us to use proper subsethood with general concepts that require relations, such as strict ordering, see sorpss 7219. (Contributed by Stefan O'Rear, 2-Nov-2014.)
[] = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}

Theoremrelrpss 7215 The proper subset relation is a relation. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Rel []

Theorembrrpssg 7216 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 7217 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 7218 Every class is partially ordered by proper subsets. (Contributed by Stefan O'Rear, 2-Nov-2014.)
[] Po 𝐴

Theoremsorpss 7219* 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 7220 Property of a chain of sets. (Contributed by Stefan O'Rear, 2-Nov-2014.)
(( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶𝐶𝐵))

Theoremsorpssun 7221 A chain of sets is closed under binary union. (Contributed by Mario Carneiro, 16-May-2015.)
(( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶) ∈ 𝐴)

Theoremsorpssin 7222 A chain of sets is closed under binary intersection. (Contributed by Mario Carneiro, 16-May-2015.)
(( [] Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶) ∈ 𝐴)

Theoremsorpssuni 7223* 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 7224* 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 7225* 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 7226* 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 7228 states that the union itself exists. A version with the standard abbreviation for union is uniex2 7229. A version using class notation is uniex 7230.

The union of a class df-uni 4672 should not be confused with the union of two classes df-un 3797. Their relationship is shown in unipr 4684. (Contributed by NM, 23-Dec-1993.)

𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)

Theoremzfun 7227* Axiom of Union expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)
𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)

Theoremaxun2 7228* A variant of the Axiom of Union ax-un 7226. 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 7229* 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 7230 The Axiom of Union in class notation. This says that if 𝐴 is a set i.e. 𝐴 ∈ V (see isset 3409), 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 7231 The union of a setvar is a set. (Contributed by BJ, 3-May-2021.)
𝑥 ∈ V

Theoremuniexg 7232 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 7233 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 7234 An unordered triple of classes exists. (Contributed by NM, 10-Apr-1994.)
{𝐴, 𝐵, 𝐶} ∈ V

Theoremunexb 7235 Existence of union is equivalent to existence of its components. (Contributed by NM, 11-Jun-1998.)
((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴𝐵) ∈ V)

Theoremunexg 7236 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 7237 The Cartesian product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. See also xpexgALT 7438. (Contributed by NM, 14-Aug-1994.)
((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)

Theoremxpexd 7238 The Cartesian product of two sets is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → (𝐴 × 𝐵) ∈ V)

Theorem3xpexg 7239 The Cartesian product of three sets is a set. (Contributed by Alexander van der Vekens, 21-Feb-2018.)
(𝑉𝑊 → ((𝑉 × 𝑉) × 𝑉) ∈ V)

Theoremxpex 7240 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 7241 The Cartesian square of a set is a set. (Contributed by AV, 13-Jan-2020.)
(𝐴𝑉 → (𝐴 × 𝐴) ∈ V)

Theoremabnexg 7242* Sufficient condition for a class abstraction to be a proper class. The class 𝐹 can be thought of as an expression in 𝑥 and the abstraction appearing in the statement as the class of values 𝐹 as 𝑥 varies through 𝐴. Assuming the antecedents, if that class is a set, then so is the "domain" 𝐴. The converse holds without antecedent, see abrexexg 7419. Note that the second antecedent 𝑥𝐴𝑥𝐹 cannot be translated to 𝐴𝐹 since 𝐹 may depend on 𝑥. In applications, one may take 𝐹 = {𝑥} or 𝐹 = 𝒫 𝑥 (see snnex 7244 and pwnex 7245 respectively, proved from abnex 7243, which is a consequence of abnexg 7242 with 𝐴 = V). (Contributed by BJ, 2-Dec-2021.)
(∀𝑥𝐴 (𝐹𝑉𝑥𝐹) → ({𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐹} ∈ 𝑊𝐴 ∈ V))

Theoremabnex 7243* Sufficient condition for a class abstraction to be a proper class. Lemma for snnex 7244 and pwnex 7245. See the comment of abnexg 7242. (Contributed by BJ, 2-May-2021.)
(∀𝑥(𝐹𝑉𝑥𝐹) → ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V)

Theoremsnnex 7244* The class of all singletons is a proper class. See also pwnex 7245. (Contributed by NM, 10-Oct-2008.) (Proof shortened by Eric Schmidt, 7-Dec-2008.) (Proof shortened by BJ, 5-Dec-2021.)
{𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V

Theorempwnex 7245* The class of all power sets is a proper class. See also snnex 7244. (Contributed by BJ, 2-May-2021.)
{𝑥 ∣ ∃𝑦 𝑥 = 𝒫 𝑦} ∉ V

Theoremdifex2 7246 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 7247 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 7248* Expression for double union that moves union into a class builder. (Contributed by FL, 28-May-2007.)
𝐴 = {𝑥 ∣ ∃𝑦(𝑥 = 𝑦𝑦𝐴)}

Theoremuniexr 7249 Converse of the Axiom of Union. Note that it does not require ax-un 7226. (Contributed by NM, 11-Nov-2003.)
( 𝐴𝑉𝐴 ∈ V)

Theoremuniexb 7250 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)

Theorempwexr 7251 Converse of the Axiom of Power Sets. Note that it does not require ax-pow 5077. (Contributed by NM, 11-Nov-2003.)
(𝒫 𝐴𝑉𝐴 ∈ V)

Theorempwexb 7252 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)

Theoremelpwpwel 7253 A class belongs to a double power class if and only if its union belongs to the power class. (Contributed by BJ, 22-Jan-2023.)
(𝐴 ∈ 𝒫 𝒫 𝐵 𝐴 ∈ 𝒫 𝐵)

Theoremeldifpw 7254 Membership in a power class difference. (Contributed by NM, 25-Mar-2007.)
𝐶 ∈ V       ((𝐴 ∈ 𝒫 𝐵 ∧ ¬ 𝐶𝐵) → (𝐴𝐶) ∈ (𝒫 (𝐵𝐶) ∖ 𝒫 𝐵))

Theoremelpwun 7255 Membership in the power class of a union. (Contributed by NM, 26-Mar-2007.)
𝐶 ∈ V       (𝐴 ∈ 𝒫 (𝐵𝐶) ↔ (𝐴𝐶) ∈ 𝒫 𝐵)

Theoremiunpw 7256* 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 7257 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 7258 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 7259* 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)

Theoremepweon 7260 The membership 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

Theoremordon 7261 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

Theoremonprc 7262 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 7261), 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 7263 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 7264 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 7265 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 7266 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 7267 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 7268 An ordinal number is a subset of the class of ordinal numbers. (Contributed by NM, 5-Jun-1994.)
(𝐴 ∈ On → 𝐴 ⊆ On)

Theorempredon 7269 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 7270 Two ways of saying a class of ordinals is unbounded. (Contributed by Mario Carneiro, 8-Jun-2013.)
(𝐴 ⊆ On → (𝐴 ∉ V ↔ 𝐴 = On))

Theoremonuni 7271 The union of an ordinal number is an ordinal number. (Contributed by NM, 29-Sep-2006.)
(𝐴 ∈ On → 𝐴 ∈ On)

Theoremorduni 7272 The union of an ordinal class is ordinal. (Contributed by NM, 12-Sep-2003.)
(Ord 𝐴 → Ord 𝐴)

Theoremonint 7273 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 7274 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 7275* 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 7276 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 7277 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 7278 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 7279 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 7280 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 7281 No member of a set of ordinal numbers belongs to its minimum. (Contributed by NM, 2-Feb-1997.)
((𝐴 ⊆ On ∧ 𝐵𝐴) → ¬ 𝐵 𝐴)

Theoremonnminsb 7282* 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 7283* 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)))

Theoremuniordint 7284* The union of a set of ordinals is equal to the intersection of its upper bounds. Problem 2.5(ii) of [BellMachover] p. 471. (Contributed by NM, 20-Sep-2003.)
𝐴 ∈ V       (𝐴 ⊆ On → 𝐴 = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥})

Theoremonminex 7285* If a wff is true for an ordinal number, then 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 7286 The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.)
suc On = On

Theoremsucexb 7287 A successor exists iff its class argument exists. (Contributed by NM, 22-Jun-1998.)
(𝐴 ∈ V ↔ suc 𝐴 ∈ V)

Theoremsucexg 7288 The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.)
(𝐴𝑉 → suc 𝐴 ∈ V)

Theoremsucex 7289 The successor of a set is a set. (Contributed by NM, 30-Aug-1993.)
𝐴 ∈ V       suc 𝐴 ∈ V

Theoremonmindif2 7290 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 7291 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 7292 The successor of an ordinal class is ordinal. (Contributed by NM, 3-Apr-1995.)
(Ord 𝐴 ↔ Ord suc 𝐴)

Theoremordpwsuc 7293 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 7294 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 7295 The successor of an ordinal number is an ordinal number. (Contributed by NM, 9-Sep-2003.)
(𝐴 ∈ On ↔ suc 𝐴 ∈ On)

Theoremordsucss 7296 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 7297 An ordinal number is a proper subset of its successor. (Contributed by Stefan O'Rear, 18-Nov-2014.)
(𝐴 ∈ On → 𝐴 ⊊ suc 𝐴)

Theoremordelsuc 7298 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 7299* 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 7300 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))

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