P. 43 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 4201-4300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	sucunielr 4201	Successor and union. The converse (where B is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4216. (Contributed by Jim Kingdon, 2-Aug-2019.)
⊢ (suc A ∈ B → A ∈ ∪ B)
Theorem	unon 4202	The class of all ordinal numbers is its own union. Exercise 11 of [TakeutiZaring] p. 40. (Contributed by NM, 12-Nov-2003.)
⊢ ∪ On = On
Theorem	onuniss2 4203*	The union of the ordinal subsets of an ordinal number is that number. (Contributed by Jim Kingdon, 2-Aug-2019.)
⊢ (A ∈ On → ∪ {x ∈ On ∣ x ⊆ A} = A)
Theorem	limon 4204	The class of ordinal numbers is a limit ordinal. (Contributed by NM, 24-Mar-1995.)
⊢ Lim On
Theorem	ordunisuc2r 4205*	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 → (∀x ∈ A suc x ∈ A → A = ∪ A))
Theorem	onssi 4206	An ordinal number is a subset of On. (Contributed by NM, 11-Aug-1994.)
⊢ A ∈ On    ⇒   ⊢ A ⊆ On
Theorem	onsuci 4207	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
Theorem	ordtriexmidlem 4208	Lemma for decidability and ordinals. The set {x ∈ {∅} ∣ φ} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4210 or weak linearity in ordsoexmid 4240) with a proposition φ. Our lemma states that it is an ordinal number. (Contributed by Jim Kingdon, 28-Jan-2019.)
⊢ {x ∈ {∅} ∣ φ} ∈ On
Theorem	ordtriexmidlem2 4209*	Lemma for decidability and ordinals. The set {x ∈ {∅} ∣ φ} is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4210 or weak linearity in ordsoexmid 4240) with a proposition φ. Our lemma helps connect that set to excluded middle. (Contributed by Jim Kingdon, 28-Jan-2019.)
⊢ ({x ∈ {∅} ∣ φ} = ∅ → ¬ φ)
Theorem	ordtriexmid 4210*	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 (x ∈ y ∨ x = y ∨ y ∈ x)    ⇒   ⊢ (φ ∨ ¬ φ)
Theorem	ordtri2orexmid 4211*	Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 31-Jul-2019.)
⊢ ∀x ∈ On ∀y ∈ On (x ∈ y ∨ y ⊆ x)    ⇒   ⊢ (φ ∨ ¬ φ)
Theorem	onsucsssucexmid 4212*	The converse of onsucsssucr 4200 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 4213*	Lemma for onsucelsucexmid 4215. (Contributed by Jim Kingdon, 2-Aug-2019.)
⊢ ∅ ∈ {x ∈ {∅, {∅}} ∣ (x = ∅ ∨ φ)}
Theorem	onsucelsucexmidlem 4214*	Lemma for onsucelsucexmid 4215. The set {x ∈ {∅, {∅}} ∣ (x = ∅ ∨ φ)} appears as A in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5446), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4208. (Contributed by Jim Kingdon, 2-Aug-2019.)
⊢ {x ∈ {∅, {∅}} ∣ (x = ∅ ∨ φ)} ∈ On
Theorem	onsucelsucexmid 4215*	The converse of onsucelsucr 4199 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 4199 does hold, as seen at nnsucelsuc 6009. (Contributed by Jim Kingdon, 2-Aug-2019.)
⊢ ∀x ∈ On ∀y ∈ On (x ∈ y → suc x ∈ suc y)    ⇒   ⊢ (φ ∨ ¬ φ)
Theorem	ordsucunielexmid 4216*	The converse of sucunielr 4201 (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 4217*	Lemma for regexmid 4219. A is inhabited. (Contributed by Jim Kingdon, 3-Sep-2019.)
⊢ A = {x ∈ {∅, {∅}} ∣ (x = {∅} ∨ (x = ∅ ∧ φ))}    ⇒   ⊢ ∃y y ∈ A
Theorem	regexmidlem1 4218*	Lemma for regexmid 4219. 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	regexmid 4219*	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 4220. (Contributed by Jim Kingdon, 3-Sep-2019.)
⊢ (∃y y ∈ x → ∃y(y ∈ x ∧ ∀z(z ∈ y → ¬ z ∈ x)))    ⇒   ⊢ (φ ∨ ¬ φ)
2.5.2  Introduce the Axiom of Set Induction
Axiom	ax-setind 4220*	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. (Contributed by Jim Kingdon, 19-Oct-2018.)
⊢ (∀𝑎(∀y ∈ 𝑎 [y / 𝑎]φ → φ) → ∀𝑎φ)
Theorem	setindel 4221*	∈-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 4222*	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 4223	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 4224	No class is a member of itself. Exercise 6 of [TakeutiZaring] p. 22. (Contributed by NM, 7-Aug-1994.) (Proof rewritten by Mario Carneiro and Jim Kingdon, 26-Nov-2018.)
⊢ ¬ A ∈ A
Theorem	ordirr 4225	Epsilon irreflexivity of ordinals: no ordinal class is a member of itself. Theorem 2.2(i) of [BellMachover] p. 469, generalized to classes. (Contributed by NM, 2-Jan-1994.)
⊢ (Ord A → ¬ A ∈ A)
Theorem	elirrv 4226	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 4227	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 4228	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 4229	Alternate proof of Russell's Paradox ru 2757, simplified using (indirectly) the Axiom of Set Induction ax-setind 4220. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ {x ∣ x ∉ x} ∉ V
Theorem	onprc 4230	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 4178), 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 4231	The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.)
⊢ suc On = On
Theorem	en2lp 4232	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 4233	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 4234	Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4220 (via the preleq 4233 step). See df-op 3376 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 4235	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 4236	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 4237*	At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 3933 can also be summarized as "at least two sets exist", the difference is that dtruarb 3933 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 4238*	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 4237. (Contributed by Jim Kingdon, 29-Dec-2018.)
⊢ ¬ ∀x x = y
Theorem	eunex 4239	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 4240	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 4241	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	nlimsucg 4242	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 4243	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 4074 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if A ∈ On then both ∪ suc A = A (onunisuci 4135) and ∪ {x ∈ On ∣ x ⊆ A} = A (onuniss2 4203). Constructively (𝒫 A ∩ On) and suc A cannot be shown to be equivalent (as proved at ordpwsucexmid 4246). (Contributed by Jim Kingdon, 21-Jul-2019.)
⊢ (Ord A → suc A ⊆ (𝒫 A ∩ On))
Theorem	onnmin 4244	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 4245	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 4246*	The subset in ordpwsucss 4243 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	onpsssuc 4247	An ordinal number is a proper subset of its successor. (Contributed by Stefan O'Rear, 18-Nov-2014.)
⊢ (A ∈ On → A ⊊ suc A)
2.5.3  Transfinite induction
Theorem	tfi 4248*	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 4249*	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 4250*	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 4251*	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 4252*	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 4253*	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 4254*	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 4255*	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 (i.e. finite ordinals)
Syntax	com 4256	Extend class notation to include the class of natural numbers.
class 𝜔
Definition	df-iom 4257*	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 4071. 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 4258 instead for naming consistency with set.mm. (New usage is discouraged.)
⊢ 𝜔 = ∩ {x ∣ (∅ ∈ x ∧ ∀y ∈ x suc y ∈ x)}
Theorem	dfom3 4258*	Alias for df-iom 4257. Use it instead of df-iom 4257 for naming consistency with set.mm. (Contributed by NM, 6-Aug-1994.)
⊢ 𝜔 = ∩ {x ∣ (∅ ∈ x ∧ ∀y ∈ x suc y ∈ x)}
Theorem	omex 4259	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 4260	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 4261	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 4262	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 4263	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 4264*	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 4269. (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 4265*	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 4266*	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 4267*	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 4268*	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 4269	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 4270*	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	elnn 4271	A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.)
⊢ ((A ∈ B ∧ B ∈ 𝜔) → A ∈ 𝜔)
Theorem	ordom 4272	Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.)
⊢ Ord 𝜔
Theorem	omelon2 4273	Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.)
⊢ (𝜔 ∈ V → 𝜔 ∈ On)
Theorem	omelon 4274	Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.)
⊢ 𝜔 ∈ On
Theorem	nnon 4275	A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
⊢ (A ∈ 𝜔 → A ∈ On)
Theorem	nnoni 4276	A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
⊢ A ∈ 𝜔    ⇒   ⊢ A ∈ On
Theorem	nnord 4277	A natural number is ordinal. (Contributed by NM, 17-Oct-1995.)
⊢ (A ∈ 𝜔 → Ord A)
Theorem	omsson 4278	Omega is a subset of On. (Contributed by NM, 13-Jun-1994.)
⊢ 𝜔 ⊆ On
Theorem	limom 4279	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 4280	A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.)
⊢ (A ∈ 𝜔 ↔ suc A ∈ 𝜔)
Theorem	nnsuc 4281*	A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.)
⊢ ((A ∈ 𝜔 ∧ A ≠ ∅) → ∃x ∈ 𝜔 A = suc x)
Theorem	nndceq0 4282	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 4283	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 4284	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 4285*	If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4219 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6015 or nntri3or 6011), 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)
2.6.6  Relations
Syntax	cxp 4286	Extend the definition of a class to include the cross product.
class (A × B)
Syntax	ccnv 4287	Extend the definition of a class to include the converse of a class.
class ◡A
Syntax	cdm 4288	Extend the definition of a class to include the domain of a class.
class dom A
Syntax	crn 4289	Extend the definition of a class to include the range of a class.
class ran A
Syntax	cres 4290	Extend the definition of a class to include the restriction of a class. (Read: The restriction of A to B.)
class (A ↾ B)
Syntax	cima 4291	Extend the definition of a class to include the image of a class. (Read: The image of B under A.)
class (A “ B)
Syntax	ccom 4292	Extend the definition of a class to include the composition of two classes. (Read: The composition of A and B.)
class (A ∘ B)
Syntax	wrel 4293	Extend the definition of a wff to include the relation predicate. (Read: A is a relation.)
wff Rel A
Definition	df-xp 4294*	Define the cross product of two classes. Definition 9.11 of [Quine] p. 64. For example, ( { 1 , 5 } × { 2 , 7 } ) = ( { ⟨ 1 , 2 ⟩, ⟨ 1 , 7 ⟩ } ∪ { ⟨ 5 , 2 ⟩, ⟨ 5 , 7 ⟩ } ) . Another example is that the set of rational numbers are defined in using the cross-product ( Z × N ) ; the left- and right-hand sides of the cross-product represent the top (integer) and bottom (natural) numbers of a fraction. (Contributed by NM, 4-Jul-1994.)
⊢ (A × B) = {⟨x, y⟩ ∣ (x ∈ A ∧ y ∈ B)}
Definition	df-rel 4295	Define the relation predicate. Definition 6.4(1) of [TakeutiZaring] p. 23. For alternate definitions, see dfrel2 4714 and dfrel3 4721. (Contributed by NM, 1-Aug-1994.)
⊢ (Rel A ↔ A ⊆ (V × V))
Definition	df-cnv 4296*	Define the converse of a class. Definition 9.12 of [Quine] p. 64. The converse of a binary relation swaps its arguments, i.e., if A ∈ V and B ∈ V then (A◡𝑅B ↔ B𝑅A), as proven in brcnv 4461 (see df-br 3756 and df-rel 4295 for more on relations). For example, ◡ { ⟨ 2 , 6 ⟩, ⟨ 3 , 9 ⟩ } = { ⟨ 6 , 2 ⟩, ⟨ 9 , 3 ⟩ } . We use Quine's breve accent (smile) notation. Like Quine, we use it as a prefix, which eliminates the need for parentheses. Many authors use the postfix superscript "to the minus one." "Converse" is Quine's terminology; some authors call it "inverse," especially when the argument is a function. (Contributed by NM, 4-Jul-1994.)
⊢ ◡A = {⟨x, y⟩ ∣ yAx}
Definition	df-co 4297*	Define the composition of two classes. Definition 6.6(3) of [TakeutiZaring] p. 24. Note that Definition 7 of [Suppes] p. 63 reverses A and B, uses a slash instead of ∘, and calls the operation "relative product." (Contributed by NM, 4-Jul-1994.)
⊢ (A ∘ B) = {⟨x, y⟩ ∣ ∃z(xBz ∧ zAy)}
Definition	df-dm 4298*	Define the domain of a class. Definition 3 of [Suppes] p. 59. For example, F = { ⟨ 2 , 6 ⟩, ⟨ 3 , 9 ⟩ } → dom F = { 2 , 3 } . Contrast with range (defined in df-rn 4299). For alternate definitions see dfdm2 4795, dfdm3 4465, and dfdm4 4470. The notation "dom " is used by Enderton; other authors sometimes use script D. (Contributed by NM, 1-Aug-1994.)
⊢ dom A = {x ∣ ∃y xAy}
Definition	df-rn 4299	Define the range of a class. For example, F = { ⟨ 2 , 6 ⟩, ⟨ 3 , 9 ⟩ } -> ran F = { 6 , 9 } . Contrast with domain (defined in df-dm 4298). For alternate definitions, see dfrn2 4466, dfrn3 4467, and dfrn4 4724. The notation "ran " is used by Enderton; other authors sometimes use script R or script W. (Contributed by NM, 1-Aug-1994.)
⊢ ran A = dom ◡A
Definition	df-res 4300	Define the restriction of a class. Definition 6.6(1) of [TakeutiZaring] p. 24. For example ( F = { ⟨ 2 , 6 ⟩, ⟨ 3 , 9 ⟩ } ∧ B = { 1 , 2 } ) -> ( F ↾ B ) = { ⟨ 2 , 6 ⟩ } . (Contributed by NM, 2-Aug-1994.)
⊢ (A ↾ B) = (A ∩ (B × V))