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Theorem List for Intuitionistic Logic Explorer - 4201-4300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremonn0 4201 The class of all ordinal numbers is not empty. (Contributed by NM, 17-Sep-1995.)
On ≠ ∅
 
Theoremonm 4202 The class of all ordinal numbers is inhabited. (Contributed by Jim Kingdon, 6-Mar-2019.)
𝑥 𝑥 ∈ On
 
Theoremsuceq 4203 Equality of successors. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
 
Theoremelsuci 4204 Membership in a successor. This one-way implication does not require that either 𝐴 or 𝐵 be sets. (Contributed by NM, 6-Jun-1994.)
(𝐴 ∈ suc 𝐵 → (𝐴𝐵𝐴 = 𝐵))
 
Theoremelsucg 4205 Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-1995.)
(𝐴𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
 
Theoremelsuc2g 4206 Variant of membership in a successor, requiring that 𝐵 rather than 𝐴 be a set. (Contributed by NM, 28-Oct-2003.)
(𝐵𝑉 → (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
 
Theoremelsuc 4207 Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-2003.)
𝐴 ∈ V       (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵))
 
Theoremelsuc2 4208 Membership in a successor. (Contributed by NM, 15-Sep-2003.)
𝐴 ∈ V       (𝐵 ∈ suc 𝐴 ↔ (𝐵𝐴𝐵 = 𝐴))
 
Theoremnfsuc 4209 Bound-variable hypothesis builder for successor. (Contributed by NM, 15-Sep-2003.)
𝑥𝐴       𝑥 suc 𝐴
 
Theoremelelsuc 4210 Membership in a successor. (Contributed by NM, 20-Jun-1998.)
(𝐴𝐵𝐴 ∈ suc 𝐵)
 
Theoremsucel 4211* Membership of a successor in another class. (Contributed by NM, 29-Jun-2004.)
(suc 𝐴𝐵 ↔ ∃𝑥𝐵𝑦(𝑦𝑥 ↔ (𝑦𝐴𝑦 = 𝐴)))
 
Theoremsuc0 4212 The successor of the empty set. (Contributed by NM, 1-Feb-2005.)
suc ∅ = {∅}
 
Theoremsucprc 4213 A proper class is its own successor. (Contributed by NM, 3-Apr-1995.)
𝐴 ∈ V → suc 𝐴 = 𝐴)
 
Theoremunisuc 4214 A transitive class is equal to the union of its successor. Combines Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton] p. 73. (Contributed by NM, 30-Aug-1993.)
𝐴 ∈ V       (Tr 𝐴 suc 𝐴 = 𝐴)
 
Theoremunisucg 4215 A transitive class is equal to the union of its successor. Combines Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton] p. 73. (Contributed by Jim Kingdon, 18-Aug-2019.)
(𝐴𝑉 → (Tr 𝐴 suc 𝐴 = 𝐴))
 
Theoremsssucid 4216 A class is included in its own successor. Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized to arbitrary classes). (Contributed by NM, 31-May-1994.)
𝐴 ⊆ suc 𝐴
 
Theoremsucidg 4217 Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized). (Contributed by NM, 25-Mar-1995.) (Proof shortened by Scott Fenton, 20-Feb-2012.)
(𝐴𝑉𝐴 ∈ suc 𝐴)
 
Theoremsucid 4218 A set belongs to its successor. (Contributed by NM, 22-Jun-1994.) (Proof shortened by Alan Sare, 18-Feb-2012.) (Proof shortened by Scott Fenton, 20-Feb-2012.)
𝐴 ∈ V       𝐴 ∈ suc 𝐴
 
Theoremnsuceq0g 4219 No successor is empty. (Contributed by Jim Kingdon, 14-Oct-2018.)
(𝐴𝑉 → suc 𝐴 ≠ ∅)
 
Theoremeqelsuc 4220 A set belongs to the successor of an equal set. (Contributed by NM, 18-Aug-1994.)
𝐴 ∈ V       (𝐴 = 𝐵𝐴 ∈ suc 𝐵)
 
Theoremiunsuc 4221* Inductive definition for the indexed union at a successor. (Contributed by Mario Carneiro, 4-Feb-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
𝐴 ∈ V    &   (𝑥 = 𝐴𝐵 = 𝐶)        𝑥 ∈ suc 𝐴𝐵 = ( 𝑥𝐴 𝐵𝐶)
 
Theoremsuctr 4222 The successor of a transitive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.)
(Tr 𝐴 → Tr suc 𝐴)
 
Theoremtrsuc 4223 A set whose successor belongs to a transitive class also belongs. (Contributed by NM, 5-Sep-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
((Tr 𝐴 ∧ suc 𝐵𝐴) → 𝐵𝐴)
 
Theoremtrsucss 4224 A member of the successor of a transitive class is a subclass of it. (Contributed by NM, 4-Oct-2003.)
(Tr 𝐴 → (𝐵 ∈ suc 𝐴𝐵𝐴))
 
Theoremsucssel 4225 A set whose successor is a subset of another class is a member of that class. (Contributed by NM, 16-Sep-1995.)
(𝐴𝑉 → (suc 𝐴𝐵𝐴𝐵))
 
Theoremorduniss 4226 An ordinal class includes its union. (Contributed by NM, 13-Sep-2003.)
(Ord 𝐴 𝐴𝐴)
 
Theoremonordi 4227 An ordinal number is an ordinal class. (Contributed by NM, 11-Jun-1994.)
𝐴 ∈ On       Ord 𝐴
 
Theoremontrci 4228 An ordinal number is a transitive class. (Contributed by NM, 11-Jun-1994.)
𝐴 ∈ On       Tr 𝐴
 
Theoremoneli 4229 A member of an ordinal number is an ordinal number. Theorem 7M(a) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)
𝐴 ∈ On       (𝐵𝐴𝐵 ∈ On)
 
Theoremonelssi 4230 A member of an ordinal number is a subset of it. (Contributed by NM, 11-Aug-1994.)
𝐴 ∈ On       (𝐵𝐴𝐵𝐴)
 
Theoremonelini 4231 An element of an ordinal number equals the intersection with it. (Contributed by NM, 11-Jun-1994.)
𝐴 ∈ On       (𝐵𝐴𝐵 = (𝐵𝐴))
 
Theoremoneluni 4232 An ordinal number equals its union with any element. (Contributed by NM, 13-Jun-1994.)
𝐴 ∈ On       (𝐵𝐴 → (𝐴𝐵) = 𝐴)
 
Theoremonunisuci 4233 An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994.)
𝐴 ∈ On        suc 𝐴 = 𝐴
 
2.4  IZF Set Theory - add the Axiom of Union
 
2.4.1  Introduce the Axiom of Union
 
Axiomax-un 4234* Axiom of Union. An axiom of Intuitionistic 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 4236 states that the union itself exists. A version with the standard abbreviation for union is uniex2 4237. A version using class notation is uniex 4238.

This is Axiom 3 of [Crosilla] p. "Axioms of CZF and IZF", except (a) unnecessary quantifiers are removed, (b) Crosilla has a biconditional rather than an implication (but the two are equivalent by bm1.3ii 3935), and (c) the order of the conjuncts is swapped (which is equivalent by ancom 262).

The union of a class df-uni 3637 should not be confused with the union of two classes df-un 2992. Their relationship is shown in unipr 3650. (Contributed by NM, 23-Dec-1993.)

𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)
 
Theoremzfun 4235* Axiom of Union expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)
𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)
 
Theoremaxun2 4236* A variant of the Axiom of Union ax-un 4234. 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 4237* 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 4238 The Axiom of Union in class notation. This says that if 𝐴 is a set i.e. 𝐴 ∈ V (see isset 2619), 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
 
Theoremuniexg 4239 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 4240 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
 
Theoremunexb 4241 Existence of union is equivalent to existence of its components. (Contributed by NM, 11-Jun-1998.)
((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴𝐵) ∈ V)
 
Theoremunexg 4242 A union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 18-Sep-2006.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
 
Theoremtpexg 4243 An unordered triple of classes exists. (Contributed by NM, 10-Apr-1994.)
((𝐴𝑈𝐵𝑉𝐶𝑊) → {𝐴, 𝐵, 𝐶} ∈ V)
 
Theoremunisn3 4244* Union of a singleton in the form of a restricted class abstraction. (Contributed by NM, 3-Jul-2008.)
(𝐴𝐵 {𝑥𝐵𝑥 = 𝐴} = 𝐴)
 
Theoremsnnex 4245* 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
 
Theoremopeluu 4246 Each member of an ordered pair belongs to the union of the union of a class to which the ordered pair belongs. Lemma 3D of [Enderton] p. 41. (Contributed by NM, 31-Mar-1995.) (Revised by Mario Carneiro, 27-Feb-2016.)
𝐴 ∈ V    &   𝐵 ∈ V       (⟨𝐴, 𝐵⟩ ∈ 𝐶 → (𝐴 𝐶𝐵 𝐶))
 
Theoremuniuni 4247* Expression for double union that moves union into a class builder. (Contributed by FL, 28-May-2007.)
𝐴 = {𝑥 ∣ ∃𝑦(𝑥 = 𝑦𝑦𝐴)}
 
Theoremeusv1 4248* Two ways to express single-valuedness of a class expression 𝐴(𝑥). (Contributed by NM, 14-Oct-2010.)
(∃!𝑦𝑥 𝑦 = 𝐴 ↔ ∃𝑦𝑥 𝑦 = 𝐴)
 
Theoremeusvnf 4249* Even if 𝑥 is free in 𝐴, it is effectively bound when 𝐴(𝑥) is single-valued. (Contributed by NM, 14-Oct-2010.) (Revised by Mario Carneiro, 14-Oct-2016.)
(∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
 
Theoremeusvnfb 4250* Two ways to say that 𝐴(𝑥) is a set expression that does not depend on 𝑥. (Contributed by Mario Carneiro, 18-Nov-2016.)
(∃!𝑦𝑥 𝑦 = 𝐴 ↔ (𝑥𝐴𝐴 ∈ V))
 
Theoremeusv2i 4251* Two ways to express single-valuedness of a class expression 𝐴(𝑥). (Contributed by NM, 14-Oct-2010.) (Revised by Mario Carneiro, 18-Nov-2016.)
(∃!𝑦𝑥 𝑦 = 𝐴 → ∃!𝑦𝑥 𝑦 = 𝐴)
 
Theoremeusv2nf 4252* Two ways to express single-valuedness of a class expression 𝐴(𝑥). (Contributed by Mario Carneiro, 18-Nov-2016.)
𝐴 ∈ V       (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
 
Theoremeusv2 4253* Two ways to express single-valuedness of a class expression 𝐴(𝑥). (Contributed by NM, 15-Oct-2010.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
𝐴 ∈ V       (∃!𝑦𝑥 𝑦 = 𝐴 ↔ ∃!𝑦𝑥 𝑦 = 𝐴)
 
Theoremreusv1 4254* Two ways to express single-valuedness of a class expression 𝐶(𝑦). (Contributed by NM, 16-Dec-2012.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
(∃𝑦𝐵 𝜑 → (∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
 
Theoremreusv3i 4255* Two ways of expressing existential uniqueness via an indirect equality. (Contributed by NM, 23-Dec-2012.)
(𝑦 = 𝑧 → (𝜑𝜓))    &   (𝑦 = 𝑧𝐶 = 𝐷)       (∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) → ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷))
 
Theoremreusv3 4256* Two ways to express single-valuedness of a class expression 𝐶(𝑦). See reusv1 4254 for the connection to uniqueness. (Contributed by NM, 27-Dec-2012.)
(𝑦 = 𝑧 → (𝜑𝜓))    &   (𝑦 = 𝑧𝐶 = 𝐷)       (∃𝑦𝐵 (𝜑𝐶𝐴) → (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) ↔ ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))
 
Theoremalxfr 4257* Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 18-Feb-2007.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((∀𝑦 𝐴𝐵 ∧ ∀𝑥𝑦 𝑥 = 𝐴) → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
 
Theoremralxfrd 4258* Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 15-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
((𝜑𝑦𝐶) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐵 𝜓 ↔ ∀𝑦𝐶 𝜒))
 
Theoremrexxfrd 4259* Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by FL, 10-Apr-2007.) (Revised by Mario Carneiro, 15-Aug-2014.)
((𝜑𝑦𝐶) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐵 𝜓 ↔ ∃𝑦𝐶 𝜒))
 
Theoremralxfr2d 4260* Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by Mario Carneiro, 20-Aug-2014.)
((𝜑𝑦𝐶) → 𝐴𝑉)    &   (𝜑 → (𝑥𝐵 ↔ ∃𝑦𝐶 𝑥 = 𝐴))    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐵 𝜓 ↔ ∀𝑦𝐶 𝜒))
 
Theoremrexxfr2d 4261* Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by Mario Carneiro, 20-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
((𝜑𝑦𝐶) → 𝐴𝑉)    &   (𝜑 → (𝑥𝐵 ↔ ∃𝑦𝐶 𝑥 = 𝐴))    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐵 𝜓 ↔ ∃𝑦𝐶 𝜒))
 
Theoremralxfr 4262* Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.)
(𝑦𝐶𝐴𝐵)    &   (𝑥𝐵 → ∃𝑦𝐶 𝑥 = 𝐴)    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥𝐵 𝜑 ↔ ∀𝑦𝐶 𝜓)
 
TheoremralxfrALT 4263* Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. This proof does not use ralxfrd 4258. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑦𝐶𝐴𝐵)    &   (𝑥𝐵 → ∃𝑦𝐶 𝑥 = 𝐴)    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥𝐵 𝜑 ↔ ∀𝑦𝐶 𝜓)
 
Theoremrexxfr 4264* Transfer existence from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.)
(𝑦𝐶𝐴𝐵)    &   (𝑥𝐵 → ∃𝑦𝐶 𝑥 = 𝐴)    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∃𝑥𝐵 𝜑 ↔ ∃𝑦𝐶 𝜓)
 
Theoremrabxfrd 4265* Class builder membership after substituting an expression 𝐴 (containing 𝑦) for 𝑥 in the class expression 𝜒. (Contributed by NM, 16-Jan-2012.)
𝑦𝐵    &   𝑦𝐶    &   ((𝜑𝑦𝐷) → 𝐴𝐷)    &   (𝑥 = 𝐴 → (𝜓𝜒))    &   (𝑦 = 𝐵𝐴 = 𝐶)       ((𝜑𝐵𝐷) → (𝐶 ∈ {𝑥𝐷𝜓} ↔ 𝐵 ∈ {𝑦𝐷𝜒}))
 
Theoremrabxfr 4266* Class builder membership after substituting an expression 𝐴 (containing 𝑦) for 𝑥 in the class expression 𝜑. (Contributed by NM, 10-Jun-2005.)
𝑦𝐵    &   𝑦𝐶    &   (𝑦𝐷𝐴𝐷)    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵𝐴 = 𝐶)       (𝐵𝐷 → (𝐶 ∈ {𝑥𝐷𝜑} ↔ 𝐵 ∈ {𝑦𝐷𝜓}))
 
Theoremreuhypd 4267* A theorem useful for eliminating restricted existential uniqueness hypotheses. (Contributed by NM, 16-Jan-2012.)
((𝜑𝑥𝐶) → 𝐵𝐶)    &   ((𝜑𝑥𝐶𝑦𝐶) → (𝑥 = 𝐴𝑦 = 𝐵))       ((𝜑𝑥𝐶) → ∃!𝑦𝐶 𝑥 = 𝐴)
 
Theoremreuhyp 4268* A theorem useful for eliminating restricted existential uniqueness hypotheses. (Contributed by NM, 15-Nov-2004.)
(𝑥𝐶𝐵𝐶)    &   ((𝑥𝐶𝑦𝐶) → (𝑥 = 𝐴𝑦 = 𝐵))       (𝑥𝐶 → ∃!𝑦𝐶 𝑥 = 𝐴)
 
Theoremuniexb 4269 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 4270 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)
 
Theoremuniv 4271 The union of the universe is the universe. Exercise 4.12(c) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.)
V = V
 
Theoremeldifpw 4272 Membership in a power class difference. (Contributed by NM, 25-Mar-2007.)
𝐶 ∈ V       ((𝐴 ∈ 𝒫 𝐵 ∧ ¬ 𝐶𝐵) → (𝐴𝐶) ∈ (𝒫 (𝐵𝐶) ∖ 𝒫 𝐵))
 
Theoremop1stb 4273 Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (Contributed by NM, 25-Nov-2003.)
𝐴 ∈ V    &   𝐵 ∈ V        𝐴, 𝐵⟩ = 𝐴
 
Theoremop1stbg 4274 Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (Contributed by Jim Kingdon, 17-Dec-2018.)
((𝐴𝑉𝐵𝑊) → 𝐴, 𝐵⟩ = 𝐴)
 
Theoremiunpw 4275* 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)
 
2.4.2  Ordinals (continued)
 
Theoremordon 4276 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
 
Theoremssorduni 4277 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 4278 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 4279 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)
 
Theoremonun2 4280 The union of two ordinal numbers is an ordinal number. (Contributed by Jim Kingdon, 25-Jul-2019.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵) ∈ On)
 
Theoremonun2i 4281 The union of two ordinal numbers is an ordinal number. (Contributed by NM, 13-Jun-1994.) (Constructive proof by Jim Kingdon, 25-Jul-2019.)
𝐴 ∈ On    &   𝐵 ∈ On       (𝐴𝐵) ∈ On
 
Theoremordsson 4282 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.)
(Ord 𝐴𝐴 ⊆ On)
 
Theoremonss 4283 An ordinal number is a subset of the class of ordinal numbers. (Contributed by NM, 5-Jun-1994.)
(𝐴 ∈ On → 𝐴 ⊆ On)
 
Theoremonuni 4284 The union of an ordinal number is an ordinal number. (Contributed by NM, 29-Sep-2006.)
(𝐴 ∈ On → 𝐴 ∈ On)
 
Theoremorduni 4285 The union of an ordinal class is ordinal. (Contributed by NM, 12-Sep-2003.)
(Ord 𝐴 → Ord 𝐴)
 
Theorembm2.5ii 4286* Problem 2.5(ii) of [BellMachover] p. 471. (Contributed by NM, 20-Sep-2003.)
𝐴 ∈ V       (𝐴 ⊆ On → 𝐴 = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥})
 
Theoremsucexb 4287 A successor exists iff its class argument exists. (Contributed by NM, 22-Jun-1998.)
(𝐴 ∈ V ↔ suc 𝐴 ∈ V)
 
Theoremsucexg 4288 The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.)
(𝐴𝑉 → suc 𝐴 ∈ V)
 
Theoremsucex 4289 The successor of a set is a set. (Contributed by NM, 30-Aug-1993.)
𝐴 ∈ V       suc 𝐴 ∈ V
 
Theoremordsucim 4290 The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 8-Nov-2018.)
(Ord 𝐴 → Ord suc 𝐴)
 
Theoremsuceloni 4291 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)
 
Theoremordsucg 4292 The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 20-Nov-2018.)
(𝐴 ∈ V → (Ord 𝐴 ↔ Ord suc 𝐴))
 
Theoremsucelon 4293 The successor of an ordinal number is an ordinal number. (Contributed by NM, 9-Sep-2003.)
(𝐴 ∈ On ↔ suc 𝐴 ∈ On)
 
Theoremordsucss 4294 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))
 
Theoremordelsuc 4295 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))
 
Theoremonsucssi 4296 A set belongs to an ordinal number iff its successor is a subset of the ordinal number. Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM, 16-Sep-1995.)
𝐴 ∈ On    &   𝐵 ∈ On       (𝐴𝐵 ↔ suc 𝐴𝐵)
 
Theoremonsucmin 4297* 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})
 
Theoremonsucelsucr 4298 Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4319. However, the converse does hold where 𝐵 is a natural number, as seen at nnsucelsuc 6206. (Contributed by Jim Kingdon, 17-Jul-2019.)
(𝐵 ∈ On → (suc 𝐴 ∈ suc 𝐵𝐴𝐵))
 
Theoremonsucsssucr 4299 The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4316. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.)
((𝐴 ∈ On ∧ Ord 𝐵) → (suc 𝐴 ⊆ suc 𝐵𝐴𝐵))
 
Theoremsucunielr 4300 Successor and union. The converse (where 𝐵 is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4320. (Contributed by Jim Kingdon, 2-Aug-2019.)
(suc 𝐴𝐵𝐴 𝐵)
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