HomeHome Intuitionistic Logic Explorer
Theorem List (p. 45 of 133)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 4401-4500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremonun2 4401 The union of two ordinal numbers is an ordinal number. (Contributed by Jim Kingdon, 25-Jul-2019.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵) ∈ On)
 
Theoremonun2i 4402 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 4403 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 4404 An ordinal number is a subset of the class of ordinal numbers. (Contributed by NM, 5-Jun-1994.)
(𝐴 ∈ On → 𝐴 ⊆ On)
 
Theoremonuni 4405 The union of an ordinal number is an ordinal number. (Contributed by NM, 29-Sep-2006.)
(𝐴 ∈ On → 𝐴 ∈ On)
 
Theoremorduni 4406 The union of an ordinal class is ordinal. (Contributed by NM, 12-Sep-2003.)
(Ord 𝐴 → Ord 𝐴)
 
Theorembm2.5ii 4407* Problem 2.5(ii) of [BellMachover] p. 471. (Contributed by NM, 20-Sep-2003.)
𝐴 ∈ V       (𝐴 ⊆ On → 𝐴 = {𝑥 ∈ On ∣ ∀𝑦𝐴 𝑦𝑥})
 
Theoremsucexb 4408 A successor exists iff its class argument exists. (Contributed by NM, 22-Jun-1998.)
(𝐴 ∈ V ↔ suc 𝐴 ∈ V)
 
Theoremsucexg 4409 The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.)
(𝐴𝑉 → suc 𝐴 ∈ V)
 
Theoremsucex 4410 The successor of a set is a set. (Contributed by NM, 30-Aug-1993.)
𝐴 ∈ V       suc 𝐴 ∈ V
 
Theoremordsucim 4411 The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 8-Nov-2018.)
(Ord 𝐴 → Ord suc 𝐴)
 
Theoremsuceloni 4412 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 4413 The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 20-Nov-2018.)
(𝐴 ∈ V → (Ord 𝐴 ↔ Ord suc 𝐴))
 
Theoremsucelon 4414 The successor of an ordinal number is an ordinal number. (Contributed by NM, 9-Sep-2003.)
(𝐴 ∈ On ↔ suc 𝐴 ∈ On)
 
Theoremordsucss 4415 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 4416 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 4417 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 4418* 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 4419 Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4440. However, the converse does hold where 𝐵 is a natural number, as seen at nnsucelsuc 6380. (Contributed by Jim Kingdon, 17-Jul-2019.)
(𝐵 ∈ On → (suc 𝐴 ∈ suc 𝐵𝐴𝐵))
 
Theoremonsucsssucr 4420 The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4437. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.)
((𝐴 ∈ On ∧ Ord 𝐵) → (suc 𝐴 ⊆ suc 𝐵𝐴𝐵))
 
Theoremsucunielr 4421 Successor and union. The converse (where 𝐵 is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4441. (Contributed by Jim Kingdon, 2-Aug-2019.)
(suc 𝐴𝐵𝐴 𝐵)
 
Theoremunon 4422 The class of all ordinal numbers is its own union. Exercise 11 of [TakeutiZaring] p. 40. (Contributed by NM, 12-Nov-2003.)
On = On
 
Theoremonuniss2 4423* The union of the ordinal subsets of an ordinal number is that number. (Contributed by Jim Kingdon, 2-Aug-2019.)
(𝐴 ∈ On → {𝑥 ∈ On ∣ 𝑥𝐴} = 𝐴)
 
Theoremlimon 4424 The class of ordinal numbers is a limit ordinal. (Contributed by NM, 24-Mar-1995.)
Lim On
 
Theoremordunisuc2r 4425* An ordinal which contains the successor of each of its members is equal to its union. (Contributed by Jim Kingdon, 14-Nov-2018.)
(Ord 𝐴 → (∀𝑥𝐴 suc 𝑥𝐴𝐴 = 𝐴))
 
Theoremonssi 4426 An ordinal number is a subset of On. (Contributed by NM, 11-Aug-1994.)
𝐴 ∈ On       𝐴 ⊆ On
 
Theoremonsuci 4427 The successor of an ordinal number is an ordinal number. Corollary 7N(c) of [Enderton] p. 193. (Contributed by NM, 12-Jun-1994.)
𝐴 ∈ On       suc 𝐴 ∈ On
 
Theoremonintonm 4428* The intersection of an inhabited collection of ordinal numbers is an ordinal number. Compare Exercise 6 of [TakeutiZaring] p. 44. (Contributed by Mario Carneiro and Jim Kingdon, 30-Aug-2021.)
((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → 𝐴 ∈ On)
 
Theoremonintrab2im 4429 An existence condition which implies an intersection is an ordinal number. (Contributed by Jim Kingdon, 30-Aug-2021.)
(∃𝑥 ∈ On 𝜑 {𝑥 ∈ On ∣ 𝜑} ∈ On)
 
Theoremordtriexmidlem 4430 Lemma for decidability and ordinals. The set {𝑥 ∈ {∅} ∣ 𝜑} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4432 or weak linearity in ordsoexmid 4472) with a proposition 𝜑. Our lemma states that it is an ordinal number. (Contributed by Jim Kingdon, 28-Jan-2019.)
{𝑥 ∈ {∅} ∣ 𝜑} ∈ On
 
Theoremordtriexmidlem2 4431* Lemma for decidability and ordinals. The set {𝑥 ∈ {∅} ∣ 𝜑} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4432 or weak linearity in ordsoexmid 4472) with a proposition 𝜑. Our lemma helps connect that set to excluded middle. (Contributed by Jim Kingdon, 28-Jan-2019.)
({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
 
Theoremordtriexmid 4432* Ordinal trichotomy implies the law of the excluded middle (that is, decidability of an arbitrary proposition).

This theorem is stated in "Constructive ordinals", [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic".

(Contributed by Mario Carneiro and Jim Kingdon, 14-Nov-2018.)

𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑥 = 𝑦𝑦𝑥)       (𝜑 ∨ ¬ 𝜑)
 
Theoremordtri2orexmid 4433* Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 31-Jul-2019.)
𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑦𝑥)       (𝜑 ∨ ¬ 𝜑)
 
Theorem2ordpr 4434 Version of 2on 6315 with the definition of 2o expanded and expressed in terms of Ord. (Contributed by Jim Kingdon, 29-Aug-2021.)
Ord {∅, {∅}}
 
Theoremontr2exmid 4435* An ordinal transitivity law which implies excluded middle. (Contributed by Jim Kingdon, 17-Sep-2021.)
𝑥 ∈ On ∀𝑦𝑧 ∈ On ((𝑥𝑦𝑦𝑧) → 𝑥𝑧)       (𝜑 ∨ ¬ 𝜑)
 
Theoremordtri2or2exmidlem 4436* A set which is 2o if 𝜑 or if ¬ 𝜑 is an ordinal. (Contributed by Jim Kingdon, 29-Aug-2021.)
{𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∈ On
 
Theoremonsucsssucexmid 4437* The converse of onsucsssucr 4420 implies excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.)
𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦 → suc 𝑥 ⊆ suc 𝑦)       (𝜑 ∨ ¬ 𝜑)
 
Theoremonsucelsucexmidlem1 4438* Lemma for onsucelsucexmid 4440. (Contributed by Jim Kingdon, 2-Aug-2019.)
∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
 
Theoremonsucelsucexmidlem 4439* Lemma for onsucelsucexmid 4440. The set {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} appears as 𝐴 in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5758), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4430. (Contributed by Jim Kingdon, 2-Aug-2019.)
{𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ∈ On
 
Theoremonsucelsucexmid 4440* The converse of onsucelsucr 4419 implies excluded middle. On the other hand, if 𝑦 is constrained to be a natural number, instead of an arbitrary ordinal, then the converse of onsucelsucr 4419 does hold, as seen at nnsucelsuc 6380. (Contributed by Jim Kingdon, 2-Aug-2019.)
𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦 → suc 𝑥 ∈ suc 𝑦)       (𝜑 ∨ ¬ 𝜑)
 
Theoremordsucunielexmid 4441* The converse of sucunielr 4421 (where 𝐵 is an ordinal) implies excluded middle. (Contributed by Jim Kingdon, 2-Aug-2019.)
𝑥 ∈ On ∀𝑦 ∈ On (𝑥 𝑦 → suc 𝑥𝑦)       (𝜑 ∨ ¬ 𝜑)
 
2.5  IZF Set Theory - add the Axiom of Set Induction
 
2.5.1  The ZF Axiom of Foundation would imply Excluded Middle
 
Theoremregexmidlemm 4442* Lemma for regexmid 4445. 𝐴 is inhabited. (Contributed by Jim Kingdon, 3-Sep-2019.)
𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}       𝑦 𝑦𝐴
 
Theoremregexmidlem1 4443* Lemma for regexmid 4445. If 𝐴 has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2019.)
𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}       (∃𝑦(𝑦𝐴 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝐴)) → (𝜑 ∨ ¬ 𝜑))
 
Theoremreg2exmidlema 4444* Lemma for reg2exmid 4446. If 𝐴 has a minimal element (expressed by ), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.)
𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}       (∃𝑢𝐴𝑣𝐴 𝑢𝑣 → (𝜑 ∨ ¬ 𝜑))
 
Theoremregexmid 4445* The axiom of foundation implies excluded middle.

By foundation (or regularity), we mean the principle that every inhabited set has an element which is minimal (when arranged by ). The statement of foundation here is taken from Metamath Proof Explorer's ax-reg, and is identical (modulo one unnecessary quantifier) to the statement of foundation in Theorem "Foundation implies instances of EM" of [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic".

For this reason, IZF does not adopt foundation as an axiom and instead replaces it with ax-setind 4447. (Contributed by Jim Kingdon, 3-Sep-2019.)

(∃𝑦 𝑦𝑥 → ∃𝑦(𝑦𝑥 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝑥)))       (𝜑 ∨ ¬ 𝜑)
 
Theoremreg2exmid 4446* If any inhabited set has a minimal element (when expressed by ), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.)
𝑧(∃𝑤 𝑤𝑧 → ∃𝑥𝑧𝑦𝑧 𝑥𝑦)       (𝜑 ∨ ¬ 𝜑)
 
2.5.2  Introduce the Axiom of Set Induction
 
Axiomax-setind 4447* Axiom of -Induction (also known as set induction). An axiom of Intuitionistic Zermelo-Fraenkel set theory. Axiom 9 of [Crosilla] p. "Axioms of CZF and IZF". This replaces the Axiom of Foundation (also called Regularity) from Zermelo-Fraenkel set theory.

For more on axioms which might be adopted which are incompatible with this axiom (that is, Non-wellfounded Set Theory but in the absence of excluded middle), see Chapter 20 of [AczelRathjen], p. 183. (Contributed by Jim Kingdon, 19-Oct-2018.)

(∀𝑎(∀𝑦𝑎 [𝑦 / 𝑎]𝜑𝜑) → ∀𝑎𝜑)
 
Theoremsetindel 4448* -Induction in terms of membership in a class. (Contributed by Mario Carneiro and Jim Kingdon, 22-Oct-2018.)
(∀𝑥(∀𝑦(𝑦𝑥𝑦𝑆) → 𝑥𝑆) → 𝑆 = V)
 
Theoremsetind 4449* Set (epsilon) induction. Theorem 5.22 of [TakeutiZaring] p. 21. (Contributed by NM, 17-Sep-2003.)
(∀𝑥(𝑥𝐴𝑥𝐴) → 𝐴 = V)
 
Theoremsetind2 4450 Set (epsilon) induction, stated compactly. Given as a homework problem in 1992 by George Boolos (1940-1996). (Contributed by NM, 17-Sep-2003.)
(𝒫 𝐴𝐴𝐴 = V)
 
Theoremelirr 4451 No class is a member of itself. Exercise 6 of [TakeutiZaring] p. 22.

The reason that this theorem is marked as discouraged is a bit subtle. If we wanted to reduce usage of ax-setind 4447, we could redefine Ord 𝐴 (df-iord 4283) to also require E Fr 𝐴 (df-frind 4249) and in that case any theorem related to irreflexivity of ordinals could use ordirr 4452 (which under that definition would presumably not need ax-setind 4447 to prove it). But since ordinals have not yet been defined that way, we cannot rely on the "don't add additional axiom use" feature of the minimizer to get theorems to use ordirr 4452. To encourage ordirr 4452 when possible, we mark this theorem as discouraged.

(Contributed by NM, 7-Aug-1994.) (Proof rewritten by Mario Carneiro and Jim Kingdon, 26-Nov-2018.) (New usage is discouraged.)

¬ 𝐴𝐴
 
Theoremordirr 4452 Epsilon irreflexivity of ordinals: no ordinal class is a member of itself. Theorem 2.2(i) of [BellMachover] p. 469, generalized to classes. The present proof requires ax-setind 4447. If in the definition of ordinals df-iord 4283, we also required that membership be well-founded on any ordinal (see df-frind 4249), then we could prove ordirr 4452 without ax-setind 4447. (Contributed by NM, 2-Jan-1994.)
(Ord 𝐴 → ¬ 𝐴𝐴)
 
Theoremonirri 4453 An ordinal number is not a member of itself. Theorem 7M(c) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)
𝐴 ∈ On        ¬ 𝐴𝐴
 
Theoremnordeq 4454 A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998.)
((Ord 𝐴𝐵𝐴) → 𝐴𝐵)
 
Theoremordn2lp 4455 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))
 
Theoremorddisj 4456 An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.)
(Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅)
 
Theoremorddif 4457 Ordinal derived from its successor. (Contributed by NM, 20-May-1998.)
(Ord 𝐴𝐴 = (suc 𝐴 ∖ {𝐴}))
 
Theoremelirrv 4458 The membership relation is irreflexive: no set is a member of itself. Theorem 105 of [Suppes] p. 54. (Contributed by NM, 19-Aug-1993.)
¬ 𝑥𝑥
 
Theoremsucprcreg 4459 A class is equal to its successor iff it is a proper class (assuming the Axiom of Set Induction). (Contributed by NM, 9-Jul-2004.)
𝐴 ∈ V ↔ suc 𝐴 = 𝐴)
 
Theoremruv 4460 The Russell class is equal to the universe V. Exercise 5 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 4-Oct-2008.)
{𝑥𝑥𝑥} = V
 
TheoremruALT 4461 Alternate proof of Russell's Paradox ru 2903, simplified using (indirectly) the Axiom of Set Induction ax-setind 4447. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
{𝑥𝑥𝑥} ∉ V
 
Theoremonprc 4462 No set contains all ordinal numbers. Proposition 7.13 of [TakeutiZaring] p. 38. 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 4397), 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
 
Theoremsucon 4463 The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.)
suc On = On
 
Theoremen2lp 4464 No class has 2-cycle membership loops. Theorem 7X(b) of [Enderton] p. 206. (Contributed by NM, 16-Oct-1996.) (Proof rewritten by Mario Carneiro and Jim Kingdon, 27-Nov-2018.)
¬ (𝐴𝐵𝐵𝐴)
 
Theorempreleq 4465 Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V       (((𝐴𝐵𝐶𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶𝐵 = 𝐷))
 
Theoremopthreg 4466 Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4447 (via the preleq 4465 step). See df-op 3531 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 ∈ V       ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} ↔ (𝐴 = 𝐶𝐵 = 𝐷))
 
Theoremsuc11g 4467 The successor operation behaves like a one-to-one function (assuming the Axiom of Set Induction). Similar to Exercise 35 of [Enderton] p. 208 and its converse. (Contributed by NM, 25-Oct-2003.)
((𝐴𝑉𝐵𝑊) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
 
Theoremsuc11 4468 The successor operation behaves like a one-to-one function. Compare Exercise 16 of [Enderton] p. 194. (Contributed by NM, 3-Sep-2003.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
 
Theoremdtruex 4469* At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 4110 can also be summarized as "at least two sets exist", the difference is that dtruarb 4110 shows the existence of two sets which are not equal to each other, but this theorem says that given a specific 𝑦, we can construct a set 𝑥 which does not equal it. (Contributed by Jim Kingdon, 29-Dec-2018.)
𝑥 ¬ 𝑥 = 𝑦
 
Theoremdtru 4470* At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). If we assumed the law of the excluded middle this would be equivalent to dtruex 4469. (Contributed by Jim Kingdon, 29-Dec-2018.)
¬ ∀𝑥 𝑥 = 𝑦
 
Theoremeunex 4471 Existential uniqueness implies there is a value for which the wff argument is false. (Contributed by Jim Kingdon, 29-Dec-2018.)
(∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)
 
Theoremordsoexmid 4472 Weak linearity of ordinals implies the law of the excluded middle (that is, decidability of an arbitrary proposition). (Contributed by Mario Carneiro and Jim Kingdon, 29-Jan-2019.)
E Or On       (𝜑 ∨ ¬ 𝜑)
 
Theoremordsuc 4473 The successor of an ordinal class is ordinal. (Contributed by NM, 3-Apr-1995.) (Constructive proof by Mario Carneiro and Jim Kingdon, 20-Jul-2019.)
(Ord 𝐴 ↔ Ord suc 𝐴)
 
Theoremonsucuni2 4474 A successor ordinal is the successor of its union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = 𝐴)
 
Theorem0elsucexmid 4475* If the successor of any ordinal class contains the empty set, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2021.)
𝑥 ∈ On ∅ ∈ suc 𝑥       (𝜑 ∨ ¬ 𝜑)
 
Theoremnlimsucg 4476 A successor is not a limit ordinal. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴𝑉 → ¬ Lim suc 𝐴)
 
Theoremordpwsucss 4477 The collection of ordinals in the power class of an ordinal is a superset of its successor.

We can think of (𝒫 𝐴 ∩ On) as another possible definition of successor, which would be equivalent to df-suc 4288 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if 𝐴 ∈ On then both suc 𝐴 = 𝐴 (onunisuci 4349) and {𝑥 ∈ On ∣ 𝑥𝐴} = 𝐴 (onuniss2 4423).

Constructively (𝒫 𝐴 ∩ On) and suc 𝐴 cannot be shown to be equivalent (as proved at ordpwsucexmid 4480). (Contributed by Jim Kingdon, 21-Jul-2019.)

(Ord 𝐴 → suc 𝐴 ⊆ (𝒫 𝐴 ∩ On))
 
Theoremonnmin 4478 No member of a set of ordinal numbers belongs to its minimum. (Contributed by NM, 2-Feb-1997.) (Constructive proof by Mario Carneiro and Jim Kingdon, 21-Jul-2019.)
((𝐴 ⊆ On ∧ 𝐵𝐴) → ¬ 𝐵 𝐴)
 
Theoremssnel 4479 Relationship between subset and elementhood. In the context of ordinals this can be seen as an ordering law. (Contributed by Jim Kingdon, 22-Jul-2019.)
(𝐴𝐵 → ¬ 𝐵𝐴)
 
Theoremordpwsucexmid 4480* The subset in ordpwsucss 4477 cannot be equality. That is, strengthening it to equality implies excluded middle. (Contributed by Jim Kingdon, 30-Jul-2019.)
𝑥 ∈ On suc 𝑥 = (𝒫 𝑥 ∩ On)       (𝜑 ∨ ¬ 𝜑)
 
Theoremordtri2or2exmid 4481* Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 29-Aug-2021.)
𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑦𝑥)       (𝜑 ∨ ¬ 𝜑)
 
Theoremonintexmid 4482* If the intersection (infimum) of an inhabited class of ordinal numbers belongs to the class, excluded middle follows. The hypothesis would be provable given excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Aug-2021.)
((𝑦 ⊆ On ∧ ∃𝑥 𝑥𝑦) → 𝑦𝑦)       (𝜑 ∨ ¬ 𝜑)
 
Theoremzfregfr 4483 The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.)
E Fr 𝐴
 
Theoremordfr 4484 Epsilon is well-founded on an ordinal class. (Contributed by NM, 22-Apr-1994.)
(Ord 𝐴 → E Fr 𝐴)
 
Theoremordwe 4485 Epsilon well-orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.)
(Ord 𝐴 → E We 𝐴)
 
Theoremwetriext 4486* A trichotomous well-order is extensional. (Contributed by Jim Kingdon, 26-Sep-2021.)
(𝜑𝑅 We 𝐴)    &   (𝜑𝐴𝑉)    &   (𝜑 → ∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎))    &   (𝜑𝐵𝐴)    &   (𝜑𝐶𝐴)    &   (𝜑 → ∀𝑧𝐴 (𝑧𝑅𝐵𝑧𝑅𝐶))       (𝜑𝐵 = 𝐶)
 
Theoremwessep 4487 A subset of a set well-ordered by set membership is well-ordered by set membership. (Contributed by Jim Kingdon, 30-Sep-2021.)
(( E We 𝐴𝐵𝐴) → E We 𝐵)
 
Theoremreg3exmidlemwe 4488* Lemma for reg3exmid 4489. Our counterexample 𝐴 satisfies We. (Contributed by Jim Kingdon, 3-Oct-2021.)
𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}        E We 𝐴
 
Theoremreg3exmid 4489* If any inhabited set satisfying df-wetr 4251 for E has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Oct-2021.)
(( E We 𝑧 ∧ ∃𝑤 𝑤𝑧) → ∃𝑥𝑧𝑦𝑧 𝑥𝑦)       (𝜑 ∨ ¬ 𝜑)
 
Theoremdcextest 4490* If it is decidable whether {𝑥𝜑} is a set, then ¬ 𝜑 is decidable (where 𝑥 does not occur in 𝜑). From this fact, we can deduce (outside the formal system, since we cannot quantify over classes) that if it is decidable whether any class is a set, then "weak excluded middle" (that is, any negated proposition ¬ 𝜑 is decidable) holds. (Contributed by Jim Kingdon, 3-Jul-2022.)
DECID {𝑥𝜑} ∈ V       DECID ¬ 𝜑
 
2.5.3  Transfinite induction
 
Theoremtfi 4491* The Principle of Transfinite Induction. Theorem 7.17 of [TakeutiZaring] p. 39. This principle states that if 𝐴 is a class of ordinal numbers with the property that every ordinal number included in 𝐴 also belongs to 𝐴, then every ordinal number is in 𝐴.

(Contributed by NM, 18-Feb-2004.)

((𝐴 ⊆ On ∧ ∀𝑥 ∈ On (𝑥𝐴𝑥𝐴)) → 𝐴 = On)
 
Theoremtfis 4492* Transfinite Induction Schema. If all ordinal numbers less than a given number 𝑥 have a property (induction hypothesis), then all ordinal numbers have the property (conclusion). Exercise 25 of [Enderton] p. 200. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 20-Nov-2016.)
(𝑥 ∈ On → (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑))       (𝑥 ∈ On → 𝜑)
 
Theoremtfis2f 4493* Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 ∈ On → (∀𝑦𝑥 𝜓𝜑))       (𝑥 ∈ On → 𝜑)
 
Theoremtfis2 4494* Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 ∈ On → (∀𝑦𝑥 𝜓𝜑))       (𝑥 ∈ On → 𝜑)
 
Theoremtfis3 4495* Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 4-Nov-2003.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 = 𝐴 → (𝜑𝜒))    &   (𝑥 ∈ On → (∀𝑦𝑥 𝜓𝜑))       (𝐴 ∈ On → 𝜒)
 
Theoremtfisi 4496* A transfinite induction scheme in "implicit" form where the induction is done on an object derived from the object of interest. (Contributed by Stefan O'Rear, 24-Aug-2015.)
(𝜑𝐴𝑉)    &   (𝜑𝑇 ∈ On)    &   ((𝜑 ∧ (𝑅 ∈ On ∧ 𝑅𝑇) ∧ ∀𝑦(𝑆𝑅𝜒)) → 𝜓)    &   (𝑥 = 𝑦 → (𝜓𝜒))    &   (𝑥 = 𝐴 → (𝜓𝜃))    &   (𝑥 = 𝑦𝑅 = 𝑆)    &   (𝑥 = 𝐴𝑅 = 𝑇)       (𝜑𝜃)
 
2.6  IZF Set Theory - add the Axiom of Infinity
 
2.6.1  Introduce the Axiom of Infinity
 
Axiomax-iinf 4497* Axiom of Infinity. Axiom 5 of [Crosilla] p. "Axioms of CZF and IZF". (Contributed by Jim Kingdon, 16-Nov-2018.)
𝑥(∅ ∈ 𝑥 ∧ ∀𝑦(𝑦𝑥 → suc 𝑦𝑥))
 
Theoremzfinf2 4498* A standard version of the Axiom of Infinity, using definitions to abbreviate. Axiom Inf of [BellMachover] p. 472. (Contributed by NM, 30-Aug-1993.)
𝑥(∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)
 
2.6.2  The natural numbers (i.e. finite ordinals)
 
Syntaxcom 4499 Extend class notation to include the class of natural numbers.
class ω
 
Definitiondf-iom 4500* Define the class of natural numbers as the smallest inductive set, which is valid provided we assume the Axiom of Infinity. Definition 6.3 of [Eisenberg] p. 82.

Note: the natural numbers ω are a subset of the ordinal numbers df-on 4285. Later, when we define complex numbers, we will be able to also define a subset of the complex numbers with analogous properties and operations, but they will be different sets. (Contributed by NM, 6-Aug-1994.) Use its alias dfom3 4501 instead for naming consistency with set.mm. (New usage is discouraged.)

ω = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)}
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13239
  Copyright terms: Public domain < Previous  Next >