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Theorem List for Intuitionistic Logic Explorer - 4301-4400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremonuniss2 4301* 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 4302 The class of ordinal numbers is a limit ordinal. (Contributed by NM, 24-Mar-1995.)
Lim On

Theoremordunisuc2r 4303* 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 4304 An ordinal number is a subset of On. (Contributed by NM, 11-Aug-1994.)
𝐴 ∈ On       𝐴 ⊆ On

Theoremonsuci 4305 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 4306* 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 4307 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 4308 Lemma for decidability and ordinals. The set {𝑥 ∈ {∅} ∣ 𝜑} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4310 or weak linearity in ordsoexmid 4350) with a proposition 𝜑. Our lemma states that it is an ordinal number. (Contributed by Jim Kingdon, 28-Jan-2019.)
{𝑥 ∈ {∅} ∣ 𝜑} ∈ On

Theoremordtriexmidlem2 4309* Lemma for decidability and ordinals. The set {𝑥 ∈ {∅} ∣ 𝜑} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4310 or weak linearity in ordsoexmid 4350) with a proposition 𝜑. Our lemma helps connect that set to excluded middle. (Contributed by Jim Kingdon, 28-Jan-2019.)
({𝑥 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)

Theoremordtriexmid 4310* 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 4311* Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 31-Jul-2019.)
𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑦𝑥)       (𝜑 ∨ ¬ 𝜑)

Theorem2ordpr 4312 Version of 2on 6138 with the definition of 2𝑜 expanded and expressed in terms of Ord. (Contributed by Jim Kingdon, 29-Aug-2021.)
Ord {∅, {∅}}

Theoremontr2exmid 4313* An ordinal transitivity law which implies excluded middle. (Contributed by Jim Kingdon, 17-Sep-2021.)
𝑥 ∈ On ∀𝑦𝑧 ∈ On ((𝑥𝑦𝑦𝑧) → 𝑥𝑧)       (𝜑 ∨ ¬ 𝜑)

Theoremordtri2or2exmidlem 4314* A set which is 2𝑜 if 𝜑 or if ¬ 𝜑 is an ordinal. (Contributed by Jim Kingdon, 29-Aug-2021.)
{𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∈ On

Theoremonsucsssucexmid 4315* The converse of onsucsssucr 4298 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 4316* Lemma for onsucelsucexmid 4318. (Contributed by Jim Kingdon, 2-Aug-2019.)
∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}

Theoremonsucelsucexmidlem 4317* Lemma for onsucelsucexmid 4318. The set {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} appears as 𝐴 in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5598), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4308. (Contributed by Jim Kingdon, 2-Aug-2019.)
{𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ∈ On

Theoremonsucelsucexmid 4318* The converse of onsucelsucr 4297 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 4297 does hold, as seen at nnsucelsuc 6200. (Contributed by Jim Kingdon, 2-Aug-2019.)
𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦 → suc 𝑥 ∈ suc 𝑦)       (𝜑 ∨ ¬ 𝜑)

Theoremordsucunielexmid 4319* The converse of sucunielr 4299 (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 4320* Lemma for regexmid 4323. 𝐴 is inhabited. (Contributed by Jim Kingdon, 3-Sep-2019.)
𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}       𝑦 𝑦𝐴

Theoremregexmidlem1 4321* Lemma for regexmid 4323. 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 4322* Lemma for reg2exmid 4324. 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 4323* 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 4325. (Contributed by Jim Kingdon, 3-Sep-2019.)

(∃𝑦 𝑦𝑥 → ∃𝑦(𝑦𝑥 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝑥)))       (𝜑 ∨ ¬ 𝜑)

Theoremreg2exmid 4324* 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 4325* 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 4326* -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 4327* Set (epsilon) induction. Theorem 5.22 of [TakeutiZaring] p. 21. (Contributed by NM, 17-Sep-2003.)
(∀𝑥(𝑥𝐴𝑥𝐴) → 𝐴 = V)

Theoremsetind2 4328 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 4329 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 4325, we could redefine Ord 𝐴 (df-iord 4166) to also require E Fr 𝐴 (df-frind 4132) and in that case any theorem related to irreflexivity of ordinals could use ordirr 4330 (which under that definition would presumably not need ax-setind 4325 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 4330. To encourage ordirr 4330 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 4330 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 4325. If in the definition of ordinals df-iord 4166, we also required that membership be well-founded on any ordinal (see df-frind 4132), then we could prove ordirr 4330 without ax-setind 4325. (Contributed by NM, 2-Jan-1994.)
(Ord 𝐴 → ¬ 𝐴𝐴)

Theoremonirri 4331 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 4332 A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998.)
((Ord 𝐴𝐵𝐴) → 𝐴𝐵)

Theoremordn2lp 4333 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 4334 An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.)
(Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅)

Theoremorddif 4335 Ordinal derived from its successor. (Contributed by NM, 20-May-1998.)
(Ord 𝐴𝐴 = (suc 𝐴 ∖ {𝐴}))

Theoremelirrv 4336 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 4337 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 4338 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 4339 Alternate proof of Russell's Paradox ru 2828, simplified using (indirectly) the Axiom of Set Induction ax-setind 4325. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
{𝑥𝑥𝑥} ∉ V

Theoremonprc 4340 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 4275), 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 4341 The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.)
suc On = On

Theoremen2lp 4342 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 4343 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 4344 Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4325 (via the preleq 4343 step). See df-op 3440 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 4345 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 4346 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 4347* At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 3999 can also be summarized as "at least two sets exist", the difference is that dtruarb 3999 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 4348* 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 4347. (Contributed by Jim Kingdon, 29-Dec-2018.)
¬ ∀𝑥 𝑥 = 𝑦

Theoremeunex 4349 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 4350 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 4351 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 4352 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 4353* 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 4354 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 4355 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 4171 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if 𝐴 ∈ On then both suc 𝐴 = 𝐴 (onunisuci 4232) and {𝑥 ∈ On ∣ 𝑥𝐴} = 𝐴 (onuniss2 4301).

Constructively (𝒫 𝐴 ∩ On) and suc 𝐴 cannot be shown to be equivalent (as proved at ordpwsucexmid 4358). (Contributed by Jim Kingdon, 21-Jul-2019.)

(Ord 𝐴 → suc 𝐴 ⊆ (𝒫 𝐴 ∩ On))

Theoremonnmin 4356 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 4357 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 4358* The subset in ordpwsucss 4355 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 4359* Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 29-Aug-2021.)
𝑥 ∈ On ∀𝑦 ∈ On (𝑥𝑦𝑦𝑥)       (𝜑 ∨ ¬ 𝜑)

Theoremonintexmid 4360* 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 4361 The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.)
E Fr 𝐴

Theoremordfr 4362 Epsilon is well-founded on an ordinal class. (Contributed by NM, 22-Apr-1994.)
(Ord 𝐴 → E Fr 𝐴)

Theoremordwe 4363 Epsilon well-orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.)
(Ord 𝐴 → E We 𝐴)

Theoremwetriext 4364* 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 4365 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 4366* Lemma for reg3exmid 4367. Our counterexample 𝐴 satisfies We. (Contributed by Jim Kingdon, 3-Oct-2021.)
𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}        E We 𝐴

Theoremreg3exmid 4367* If any inhabited set satisfying df-wetr 4134 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 4368* 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 4369* 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 4370* 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 4371* Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 ∈ On → (∀𝑦𝑥 𝜓𝜑))       (𝑥 ∈ On → 𝜑)

Theoremtfis2 4372* Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 ∈ On → (∀𝑦𝑥 𝜓𝜑))       (𝑥 ∈ On → 𝜑)

Theoremtfis3 4373* 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 4374* 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 4375* 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 4376* 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 4377 Extend class notation to include the class of natural numbers.
class ω

Definitiondf-iom 4378* 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 4168. 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 4379 instead for naming consistency with set.mm. (New usage is discouraged.)

ω = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)}

Theoremdfom3 4379* Alias for df-iom 4378. Use it instead of df-iom 4378 for naming consistency with set.mm. (Contributed by NM, 6-Aug-1994.)
ω = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)}

Theoremomex 4380 The existence of omega (the class of natural numbers). Axiom 7 of [TakeutiZaring] p. 43. (Contributed by NM, 6-Aug-1994.)
ω ∈ V

2.6.3  Peano's postulates

Theorempeano1 4381 Zero is a natural number. One of Peano's five postulates for arithmetic. Proposition 7.30(1) of [TakeutiZaring] p. 42. (Contributed by NM, 15-May-1994.)
∅ ∈ ω

Theorempeano2 4382 The successor of any natural number is a natural number. One of Peano's five postulates for arithmetic. Proposition 7.30(2) of [TakeutiZaring] p. 42. (Contributed by NM, 3-Sep-2003.)
(𝐴 ∈ ω → suc 𝐴 ∈ ω)

Theorempeano3 4383 The successor of any natural number is not zero. One of Peano's five postulates for arithmetic. Proposition 7.30(3) of [TakeutiZaring] p. 42. (Contributed by NM, 3-Sep-2003.)
(𝐴 ∈ ω → suc 𝐴 ≠ ∅)

Theorempeano4 4384 Two natural numbers are equal iff their successors are equal, i.e. the successor function is one-to-one. One of Peano's five postulates for arithmetic. Proposition 7.30(4) of [TakeutiZaring] p. 43. (Contributed by NM, 3-Sep-2003.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))

Theorempeano5 4385* The induction postulate: any class containing zero and closed under the successor operation contains all natural numbers. One of Peano's five postulates for arithmetic. Proposition 7.30(5) of [TakeutiZaring] p. 43. The more traditional statement of mathematical induction as a theorem schema, with a basis and an induction step, is derived from this theorem as theorem findes 4390. (Contributed by NM, 18-Feb-2004.)
((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → ω ⊆ 𝐴)

2.6.4  Finite induction (for finite ordinals)

Theoremfind 4386* The Principle of Finite Induction (mathematical induction). Corollary 7.31 of [TakeutiZaring] p. 43. The simpler hypothesis shown here was suggested in an email from "Colin" on 1-Oct-2001. The hypothesis states that 𝐴 is a set of natural numbers, zero belongs to 𝐴, and given any member of 𝐴 the member's successor also belongs to 𝐴. The conclusion is that every natural number is in 𝐴. (Contributed by NM, 22-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
(𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴)       𝐴 = ω

Theoremfinds 4387* Principle of Finite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. This is Metamath 100 proof #74. (Contributed by NM, 14-Apr-1995.)
(𝑥 = ∅ → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = suc 𝑦 → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   𝜓    &   (𝑦 ∈ ω → (𝜒𝜃))       (𝐴 ∈ ω → 𝜏)

Theoremfinds2 4388* Principle of Finite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 29-Nov-2002.)
(𝑥 = ∅ → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = suc 𝑦 → (𝜑𝜃))    &   (𝜏𝜓)    &   (𝑦 ∈ ω → (𝜏 → (𝜒𝜃)))       (𝑥 ∈ ω → (𝜏𝜑))

Theoremfinds1 4389* Principle of Finite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last two are the basis and the induction step. Theorem Schema 22 of [Suppes] p. 136. (Contributed by NM, 22-Mar-2006.)
(𝑥 = ∅ → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = suc 𝑦 → (𝜑𝜃))    &   𝜓    &   (𝑦 ∈ ω → (𝜒𝜃))       (𝑥 ∈ ω → 𝜑)

Theoremfindes 4390 Finite induction with explicit substitution. The first hypothesis is the basis and the second is the induction step. Theorem Schema 22 of [Suppes] p. 136. This is an alternative for Metamath 100 proof #74. (Contributed by Raph Levien, 9-Jul-2003.)
[∅ / 𝑥]𝜑    &   (𝑥 ∈ ω → (𝜑[suc 𝑥 / 𝑥]𝜑))       (𝑥 ∈ ω → 𝜑)

2.6.5  The Natural Numbers (continued)

Theoremnn0suc 4391* A natural number is either 0 or a successor. Similar theorems for arbitrary sets or real numbers will not be provable (without the law of the excluded middle), but equality of natural numbers is decidable. (Contributed by NM, 27-May-1998.)
(𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))

Theoremelnn 4392 A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.)
((𝐴𝐵𝐵 ∈ ω) → 𝐴 ∈ ω)

Theoremordom 4393 Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.)
Ord ω

Theoremomelon2 4394 Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.)
(ω ∈ V → ω ∈ On)

Theoremomelon 4395 Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.)
ω ∈ On

Theoremnnon 4396 A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
(𝐴 ∈ ω → 𝐴 ∈ On)

Theoremnnoni 4397 A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
𝐴 ∈ ω       𝐴 ∈ On

Theoremnnord 4398 A natural number is ordinal. (Contributed by NM, 17-Oct-1995.)
(𝐴 ∈ ω → Ord 𝐴)

Theoremomsson 4399 Omega is a subset of On. (Contributed by NM, 13-Jun-1994.)
ω ⊆ On

Theoremlimom 4400 Omega is a limit ordinal. Theorem 2.8 of [BellMachover] p. 473. (Contributed by NM, 26-Mar-1995.) (Proof rewritten by Jim Kingdon, 5-Jan-2019.)
Lim ω

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