P. 46 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4501-4600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	2ordpr 4501	Version of 2on 6393 with the definition of 2o expanded and expressed in terms of Ord. (Contributed by Jim Kingdon, 29-Aug-2021.)
⊢ Ord {∅, {∅}}
Theorem	ontr2exmid 4502*	An ordinal transitivity law which implies excluded middle. (Contributed by Jim Kingdon, 17-Sep-2021.)
⊢ ∀𝑥 ∈ On ∀𝑦∀𝑧 ∈ On ((𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧)    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
Theorem	ordtri2or2exmidlem 4503*	A set which is 2o if 𝜑 or ∅ if ¬ 𝜑 is an ordinal. (Contributed by Jim Kingdon, 29-Aug-2021.)
⊢ {𝑥 ∈ {∅, {∅}} ∣ 𝜑} ∈ On
Theorem	onsucsssucexmid 4504*	The converse of onsucsssucr 4486 implies excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.)
⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 → suc 𝑥 ⊆ suc 𝑦)    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
Theorem	onsucelsucexmidlem1 4505*	Lemma for onsucelsucexmid 4507. (Contributed by Jim Kingdon, 2-Aug-2019.)
⊢ ∅ ∈ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)}
Theorem	onsucelsucexmidlem 4506*	Lemma for onsucelsucexmid 4507. The set {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} appears as 𝐴 in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5833), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4496. (Contributed by Jim Kingdon, 2-Aug-2019.)
⊢ {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = ∅ ∨ 𝜑)} ∈ On
Theorem	onsucelsucexmid 4507*	The converse of onsucelsucr 4485 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 4485 does hold, as seen at nnsucelsuc 6459. (Contributed by Jim Kingdon, 2-Aug-2019.)
⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 → suc 𝑥 ∈ suc 𝑦)    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
Theorem	ordsucunielexmid 4508*	The converse of sucunielr 4487 (where 𝐵 is an ordinal) implies excluded middle. (Contributed by Jim Kingdon, 2-Aug-2019.)
⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ ∪ 𝑦 → suc 𝑥 ∈ 𝑦)    ⇒   ⊢ (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
Theorem	regexmidlemm 4509*	Lemma for regexmid 4512. 𝐴 is inhabited. (Contributed by Jim Kingdon, 3-Sep-2019.)
⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}    ⇒   ⊢ ∃𝑦 𝑦 ∈ 𝐴
Theorem	regexmidlem1 4510*	Lemma for regexmid 4512. If 𝐴 has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2019.)
⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}    ⇒   ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝐴)) → (𝜑 ∨ ¬ 𝜑))
Theorem	reg2exmidlema 4511*	Lemma for reg2exmid 4513. If 𝐴 has a minimal element (expressed by ⊆), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.)
⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}    ⇒   ⊢ (∃𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 𝑢 ⊆ 𝑣 → (𝜑 ∨ ¬ 𝜑))
Theorem	regexmid 4512*	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 4514. (Contributed by Jim Kingdon, 3-Sep-2019.)
⊢ (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)))    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
Theorem	reg2exmid 4513*	If any inhabited set has a minimal element (when expressed by ⊆), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.)
⊢ ∀𝑧(∃𝑤 𝑤 ∈ 𝑧 → ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 𝑥 ⊆ 𝑦)    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
2.5.2  Introduce the Axiom of Set Induction
Axiom	ax-setind 4514*	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(∀𝑦 ∈ 𝑎 [𝑦 / 𝑎]𝜑 → 𝜑) → ∀𝑎𝜑)
Theorem	setindel 4515*	∈-Induction in terms of membership in a class. (Contributed by Mario Carneiro and Jim Kingdon, 22-Oct-2018.)
⊢ (∀𝑥(∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆) → 𝑆 = V)
Theorem	setind 4516*	Set (epsilon) induction. Theorem 5.22 of [TakeutiZaring] p. 21. (Contributed by NM, 17-Sep-2003.)
⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → 𝐴 = V)
Theorem	setind2 4517	Set (epsilon) induction, stated compactly. Given as a homework problem in 1992 by George Boolos (1940-1996). (Contributed by NM, 17-Sep-2003.)
⊢ (𝒫 𝐴 ⊆ 𝐴 → 𝐴 = V)
Theorem	elirr 4518	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 4514, we could redefine Ord 𝐴 (df-iord 4344) to also require E Fr 𝐴 (df-frind 4310) and in that case any theorem related to irreflexivity of ordinals could use ordirr 4519 (which under that definition would presumably not need ax-setind 4514 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 4519. To encourage ordirr 4519 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.)
⊢ ¬ 𝐴 ∈ 𝐴
Theorem	ordirr 4519	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 4514. If in the definition of ordinals df-iord 4344, we also required that membership be well-founded on any ordinal (see df-frind 4310), then we could prove ordirr 4519 without ax-setind 4514. (Contributed by NM, 2-Jan-1994.)
⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴)
Theorem	onirri 4520	An ordinal number is not a member of itself. Theorem 7M(c) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)
⊢ 𝐴 ∈ On    ⇒   ⊢ ¬ 𝐴 ∈ 𝐴
Theorem	nordeq 4521	A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998.)
⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐴 ≠ 𝐵)
Theorem	ordn2lp 4522	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 → ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴))
Theorem	orddisj 4523	An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.)
⊢ (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅)
Theorem	orddif 4524	Ordinal derived from its successor. (Contributed by NM, 20-May-1998.)
⊢ (Ord 𝐴 → 𝐴 = (suc 𝐴 ∖ {𝐴}))
Theorem	elirrv 4525	The membership relation is irreflexive: no set is a member of itself. Theorem 105 of [Suppes] p. 54. (Contributed by NM, 19-Aug-1993.)
⊢ ¬ 𝑥 ∈ 𝑥
Theorem	sucprcreg 4526	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)
Theorem	ruv 4527	The Russell class is equal to the universe V. Exercise 5 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 4-Oct-2008.)
⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V
Theorem	ruALT 4528	Alternate proof of Russell's Paradox ru 2950, simplified using (indirectly) the Axiom of Set Induction ax-setind 4514. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V
Theorem	onprc 4529	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 4463), 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
Theorem	sucon 4530	The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.)
⊢ suc On = On
Theorem	en2lp 4531	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.)
⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴)
Theorem	preleq 4532	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    ⇒   ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	opthreg 4533	Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4514 (via the preleq 4532 step). See df-op 3585 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 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	suc11g 4534	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.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵))
Theorem	suc11 4535	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 𝐵 ↔ 𝐴 = 𝐵))
Theorem	dtruex 4536*	At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 4170 can also be summarized as "at least two sets exist", the difference is that dtruarb 4170 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
Theorem	dtru 4537*	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 4536. (Contributed by Jim Kingdon, 29-Dec-2018.)
⊢ ¬ ∀𝑥 𝑥 = 𝑦
Theorem	eunex 4538	Existential uniqueness implies there is a value for which the wff argument is false. (Contributed by Jim Kingdon, 29-Dec-2018.)
⊢ (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)
Theorem	ordsoexmid 4539	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)
Theorem	ordsuc 4540	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)
Theorem	onsucuni2 4541	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)
Theorem	0elsucexmid 4542*	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)
Theorem	nlimsucg 4543	A successor is not a limit ordinal. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ (𝐴 ∈ 𝑉 → ¬ Lim suc 𝐴)
Theorem	ordpwsucss 4544	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 4349 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if 𝐴 ∈ On then both ∪ suc 𝐴 = 𝐴 (onunisuci 4410) and ∪ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = 𝐴 (onuniss2 4489). Constructively (𝒫 𝐴 ∩ On) and suc 𝐴 cannot be shown to be equivalent (as proved at ordpwsucexmid 4547). (Contributed by Jim Kingdon, 21-Jul-2019.)
⊢ (Ord 𝐴 → suc 𝐴 ⊆ (𝒫 𝐴 ∩ On))
Theorem	onnmin 4545	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 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵 ∈ ∩ 𝐴)
Theorem	ssnel 4546	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.)
⊢ (𝐴 ⊆ 𝐵 → ¬ 𝐵 ∈ 𝐴)
Theorem	ordpwsucexmid 4547*	The subset in ordpwsucss 4544 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)
Theorem	ordtri2or2exmid 4548*	Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 29-Aug-2021.)
⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
Theorem	ontri2orexmidim 4549*	Ordinal trichotomy implies excluded middle. Closed form of ordtri2or2exmid 4548. (Contributed by Jim Kingdon, 26-Aug-2024.)
⊢ (∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) → DECID 𝜑)
Theorem	onintexmid 4550*	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 𝑥 ∈ 𝑦) → ∩ 𝑦 ∈ 𝑦)    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
Theorem	zfregfr 4551	The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.)
⊢ E Fr 𝐴
Theorem	ordfr 4552	Epsilon is well-founded on an ordinal class. (Contributed by NM, 22-Apr-1994.)
⊢ (Ord 𝐴 → E Fr 𝐴)
Theorem	ordwe 4553	Epsilon well-orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.)
⊢ (Ord 𝐴 → E We 𝐴)
Theorem	wetriext 4554*	A trichotomous well-order is extensional. (Contributed by Jim Kingdon, 26-Sep-2021.)
⊢ (𝜑 → 𝑅 We 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎))    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 (𝑧𝑅𝐵 ↔ 𝑧𝑅𝐶))    ⇒   ⊢ (𝜑 → 𝐵 = 𝐶)
Theorem	wessep 4555	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 𝐴 ∧ 𝐵 ⊆ 𝐴) → E We 𝐵)
Theorem	reg3exmidlemwe 4556*	Lemma for reg3exmid 4557. Our counterexample 𝐴 satisfies We. (Contributed by Jim Kingdon, 3-Oct-2021.)
⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))}    ⇒   ⊢ E We 𝐴
Theorem	reg3exmid 4557*	If any inhabited set satisfying df-wetr 4312 for E has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Oct-2021.)
⊢ (( E We 𝑧 ∧ ∃𝑤 𝑤 ∈ 𝑧) → ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 𝑥 ⊆ 𝑦)    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
Theorem	dcextest 4558*	If it is decidable whether {𝑥 ∣ 𝜑} 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 {𝑥 ∣ 𝜑} ∈ V    ⇒   ⊢ DECID ¬ 𝜑
2.5.3  Transfinite induction
Theorem	tfi 4559*	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 (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → 𝐴 = On)
Theorem	tfis 4560*	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 → (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑))    ⇒   ⊢ (𝑥 ∈ On → 𝜑)
Theorem	tfis2f 4561*	Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑))    ⇒   ⊢ (𝑥 ∈ On → 𝜑)
Theorem	tfis2 4562*	Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑))    ⇒   ⊢ (𝑥 ∈ On → 𝜑)
Theorem	tfis3 4563*	Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 4-Nov-2003.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑))    ⇒   ⊢ (𝐴 ∈ On → 𝜒)
Theorem	tfisi 4564*	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.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑇 ∈ On)    &   ⊢ ((𝜑 ∧ (𝑅 ∈ On ∧ 𝑅 ⊆ 𝑇) ∧ ∀𝑦(𝑆 ∈ 𝑅 → 𝜒)) → 𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃))    &   ⊢ (𝑥 = 𝑦 → 𝑅 = 𝑆)    &   ⊢ (𝑥 = 𝐴 → 𝑅 = 𝑇)    ⇒   ⊢ (𝜑 → 𝜃)
2.6  IZF Set Theory - add the Axiom of Infinity
2.6.1  Introduce the Axiom of Infinity
Axiom	ax-iinf 4565*	Axiom of Infinity. Axiom 5 of [Crosilla] p. "Axioms of CZF and IZF". (Contributed by Jim Kingdon, 16-Nov-2018.)
⊢ ∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥))
Theorem	zfinf2 4566*	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 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)
2.6.2  The natural numbers
Syntax	com 4567	Extend class notation to include the class of natural numbers.
class ω
Definition	df-iom 4568*	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 4346. Later, when we define complex numbers, we will be able to also define a subset of the complex numbers (df-inn 8858) with analogous properties and operations, but they will be different sets. We are unable to use the terms finite ordinal and natural number interchangeably, as shown at exmidonfin 7150. (Contributed by NM, 6-Aug-1994.) Use its alias dfom3 4569 instead for naming consistency with set.mm. (New usage is discouraged.)
⊢ ω = ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)}
Theorem	dfom3 4569*	Alias for df-iom 4568. Use it instead of df-iom 4568 for naming consistency with set.mm. (Contributed by NM, 6-Aug-1994.)
⊢ ω = ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)}
Theorem	omex 4570	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
Theorem	peano1 4571	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
Theorem	peano2 4572	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)
Theorem	peano3 4573	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 ≠ ∅)
Theorem	peano4 4574	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 𝐵 ↔ 𝐴 = 𝐵))
Theorem	peano5 4575*	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 4580. (Contributed by NM, 18-Feb-2004.)
⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴)
2.6.4  Finite induction (for finite ordinals)
Theorem	find 4576*	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 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴)    ⇒   ⊢ 𝐴 = ω
Theorem	finds 4577*	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 = ∅ → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ 𝜓    &   ⊢ (𝑦 ∈ ω → (𝜒 → 𝜃))    ⇒   ⊢ (𝐴 ∈ ω → 𝜏)
Theorem	finds2 4578*	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 = ∅ → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))    &   ⊢ (𝜏 → 𝜓)    &   ⊢ (𝑦 ∈ ω → (𝜏 → (𝜒 → 𝜃)))    ⇒   ⊢ (𝑥 ∈ ω → (𝜏 → 𝜑))
Theorem	finds1 4579*	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 = ∅ → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃))    &   ⊢ 𝜓    &   ⊢ (𝑦 ∈ ω → (𝜒 → 𝜃))    ⇒   ⊢ (𝑥 ∈ ω → 𝜑)
Theorem	findes 4580	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)
Theorem	nn0suc 4581*	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))
Theorem	elomssom 4582	A natural number ordinal is, as a set, included in the set of natural number ordinals. (Contributed by NM, 21-Jun-1998.) Extract this result from the previous proof of elnn 4583. (Revised by BJ, 7-Aug-2024.)
⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω)
Theorem	elnn 4583	A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.)
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω)
Theorem	ordom 4584	Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.)
⊢ Ord ω
Theorem	omelon2 4585	Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.)
⊢ (ω ∈ V → ω ∈ On)
Theorem	omelon 4586	Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.)
⊢ ω ∈ On
Theorem	nnon 4587	A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
Theorem	nnoni 4588	A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
⊢ 𝐴 ∈ ω    ⇒   ⊢ 𝐴 ∈ On
Theorem	nnord 4589	A natural number is ordinal. (Contributed by NM, 17-Oct-1995.)
⊢ (𝐴 ∈ ω → Ord 𝐴)
Theorem	omsson 4590	Omega is a subset of On. (Contributed by NM, 13-Jun-1994.)
⊢ ω ⊆ On
Theorem	limom 4591	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
Theorem	peano2b 4592	A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.)
⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
Theorem	nnsuc 4593*	A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.)
⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥)
Theorem	nnsucpred 4594	The successor of the precedessor of a nonzero natural number. (Contributed by Jim Kingdon, 31-Jul-2022.)
⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → suc ∪ 𝐴 = 𝐴)
Theorem	nndceq0 4595	A natural number is either zero or nonzero. Decidable equality for natural numbers is a special case of the law of the excluded middle which holds in most constructive set theories including ours. (Contributed by Jim Kingdon, 5-Jan-2019.)
⊢ (𝐴 ∈ ω → DECID 𝐴 = ∅)
Theorem	0elnn 4596	A natural number is either the empty set or has the empty set as an element. (Contributed by Jim Kingdon, 23-Aug-2019.)
⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
Theorem	nn0eln0 4597	A natural number is nonempty iff it contains the empty set. Although in constructive mathematics it is generally more natural to work with inhabited sets and ignore the whole concept of nonempty sets, in the specific case of natural numbers this theorem may be helpful in converting proofs which were written assuming excluded middle. (Contributed by Jim Kingdon, 28-Aug-2019.)
⊢ (𝐴 ∈ ω → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅))
Theorem	nnregexmid 4598*	If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4512 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6467 or nntri3or 6461), sets of natural numbers are a different animal. (Contributed by Jim Kingdon, 6-Sep-2019.)
⊢ ((𝑥 ⊆ ω ∧ ∃𝑦 𝑦 ∈ 𝑥) → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)))    ⇒   ⊢ (𝜑 ∨ ¬ 𝜑)
Theorem	omsinds 4599*	Strong (or "total") induction principle over ω. (Contributed by Scott Fenton, 17-Jul-2015.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 ∈ ω → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑))    ⇒   ⊢ (𝐴 ∈ ω → 𝜒)
Theorem	nnpredcl 4600	The predecessor of a natural number is a natural number. This theorem is most interesting when the natural number is a successor (as seen in theorems like onsucuni2 4541) but also holds when it is ∅ by uni0 3816. (Contributed by Jim Kingdon, 31-Jul-2022.)
⊢ (𝐴 ∈ ω → ∪ 𝐴 ∈ ω)