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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | regexmidlemm 4201* | Lemma for regexmid 4203. A is inhabited. (Contributed by Jim Kingdon, 3-Sep-2019.) |
⊢ A = {x ∈ {∅, {∅}} ∣ (x = {∅} ∨ (x = ∅ ∧ φ))} ⇒ ⊢ ∃y y ∈ A | ||
Theorem | regexmidlem1 4202* | Lemma for regexmid 4203. 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 4203* |
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 4204. (Contributed by Jim Kingdon, 3-Sep-2019.) |
⊢ (∃y y ∈ x → ∃y(y ∈ x ∧ ∀z(z ∈ y → ¬ z ∈ x))) ⇒ ⊢ (φ ∨ ¬ φ) | ||
Axiom | ax-setind 4204* | 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 4205* | ∈-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 4206* | 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 4207 | 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 4208 | 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 4209 | 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 4210 | 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 4211 | 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 4212 | 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 4213 | Alternate proof of Russell's Paradox ru 2740, simplified using (indirectly) the Axiom of Set Induction ax-setind 4204. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ {x ∣ x ∉ x} ∉ V | ||
Theorem | onprc 4214 | 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 4162), 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 4215 | The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.) |
⊢ suc On = On | ||
Theorem | en2lp 4216 | 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 4217 | 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 4218 | Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4204 (via the preleq 4217 step). See df-op 3359 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 4219 | 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 4220 | 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 4221* | At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 3916 can also be summarized as "at least two sets exist", the difference is that dtruarb 3916 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 4222* | 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 4221. (Contributed by Jim Kingdon, 29-Dec-2018.) |
⊢ ¬ ∀x x = y | ||
Theorem | eunex 4223 | 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 4224 | 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 4225 | 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 4226 | 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 4227 |
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 4057 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if A ∈ On then both ∪ suc A = A (onunisuci 4119) and ∪ {x ∈ On ∣ x ⊆ A} = A (onuniss2 4187). Constructively (𝒫 A ∩ On) and suc A cannot be shown to be equivalent (as proved at ordpwsucexmid 4230). (Contributed by Jim Kingdon, 21-Jul-2019.) |
⊢ (Ord A → suc A ⊆ (𝒫 A ∩ On)) | ||
Theorem | onnmin 4228 | 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 4229 | 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 4230* | The subset in ordpwsucss 4227 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 4231 | An ordinal number is a proper subset of its successor. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
⊢ (A ∈ On → A ⊊ suc A) | ||
Theorem | tfi 4232* |
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 4233* | 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 4234* | 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 4235* | 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 4236* | 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 4237* | 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) | ||
Axiom | ax-iinf 4238* | 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 4239* | 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) | ||
Syntax | com 4240 | Extend class notation to include the class of natural numbers. |
class 𝜔 | ||
Definition | df-iom 4241* |
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 4054. 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 4242 instead for naming consistency with set.mm. (New usage is discouraged.) |
⊢ 𝜔 = ∩ {x ∣ (∅ ∈ x ∧ ∀y ∈ x suc y ∈ x)} | ||
Theorem | dfom3 4242* | Alias for df-iom 4241. Use it instead of df-iom 4241 for naming consistency with set.mm. (Contributed by NM, 6-Aug-1994.) |
⊢ 𝜔 = ∩ {x ∣ (∅ ∈ x ∧ ∀y ∈ x suc y ∈ x)} | ||
Theorem | omex 4243 | The existence of omega (the class of natural numbers). Axiom 7 of [TakeutiZaring] p. 43. (Contributed by NM, 6-Aug-1994.) |
⊢ 𝜔 ∈ V | ||
Theorem | peano1 4244 | 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 4245 | 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 4246 | 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 4247 | 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 4248* | 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 4253. (Contributed by NM, 18-Feb-2004.) |
⊢ ((∅ ∈ A ∧ ∀x ∈ 𝜔 (x ∈ A → suc x ∈ A)) → 𝜔 ⊆ A) | ||
Theorem | find 4249* | 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 4250* | 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 4251* | 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 4252* | 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 4253 | 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) | ||
Theorem | nn0suc 4254* | 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 4255 | A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.) |
⊢ ((A ∈ B ∧ B ∈ 𝜔) → A ∈ 𝜔) | ||
Theorem | ordom 4256 | Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.) |
⊢ Ord 𝜔 | ||
Theorem | omelon2 4257 | Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.) |
⊢ (𝜔 ∈ V → 𝜔 ∈ On) | ||
Theorem | omelon 4258 | Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.) |
⊢ 𝜔 ∈ On | ||
Theorem | nnon 4259 | A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.) |
⊢ (A ∈ 𝜔 → A ∈ On) | ||
Theorem | nnoni 4260 | A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.) |
⊢ A ∈ 𝜔 ⇒ ⊢ A ∈ On | ||
Theorem | nnord 4261 | A natural number is ordinal. (Contributed by NM, 17-Oct-1995.) |
⊢ (A ∈ 𝜔 → Ord A) | ||
Theorem | omsson 4262 | Omega is a subset of On. (Contributed by NM, 13-Jun-1994.) |
⊢ 𝜔 ⊆ On | ||
Theorem | limom 4263 | 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 4264 | A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.) |
⊢ (A ∈ 𝜔 ↔ suc A ∈ 𝜔) | ||
Theorem | nnsuc 4265* | A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.) |
⊢ ((A ∈ 𝜔 ∧ A ≠ ∅) → ∃x ∈ 𝜔 A = suc x) | ||
Theorem | nndceq0 4266 | 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 4267 | 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 4268 | 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 4269* | If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4203 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 5990 or nntri3or 5987), 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) | ||
Syntax | cxp 4270 | Extend the definition of a class to include the cross product. |
class (A × B) | ||
Syntax | ccnv 4271 | Extend the definition of a class to include the converse of a class. |
class ^{◡}A | ||
Syntax | cdm 4272 | Extend the definition of a class to include the domain of a class. |
class dom A | ||
Syntax | crn 4273 | Extend the definition of a class to include the range of a class. |
class ran A | ||
Syntax | cres 4274 | 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 4275 | 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 4276 | 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 4277 | Extend the definition of a wff to include the relation predicate. (Read: A is a relation.) |
wff Rel A | ||
Definition | df-xp 4278* | 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 4279 | Define the relation predicate. Definition 6.4(1) of [TakeutiZaring] p. 23. For alternate definitions, see dfrel2 4698 and dfrel3 4705. (Contributed by NM, 1-Aug-1994.) |
⊢ (Rel A ↔ A ⊆ (V × V)) | ||
Definition | df-cnv 4280* | 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 4445 (see df-br 3739 and df-rel 4279 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 4281* | 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 4282* | 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 4283). For alternate definitions see dfdm2 4779, dfdm3 4449, and dfdm4 4454. 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 4283 | Define the range of a class. For example, F = { ⟨ 2 , 6 ⟩, ⟨ 3 , 9 ⟩ } -> ran F = { 6 , 9 } . Contrast with domain (defined in df-dm 4282). For alternate definitions, see dfrn2 4450, dfrn3 4451, and dfrn4 4708. 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 4284 | 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)) | ||
Definition | df-ima 4285 | Define the image of a class (as restricted by another class). Definition 6.6(2) of [TakeutiZaring] p. 24. For example, ( F = { ⟨ 2 , 6 ⟩, ⟨ 3 , 9 ⟩ } /\ B = { 1 , 2 } ) -> ( F “ B ) = { 6 } . Contrast with restriction (df-res 4284) and range (df-rn 4283). For an alternate definition, see dfima2 4597. (Contributed by NM, 2-Aug-1994.) |
⊢ (A “ B) = ran (A ↾ B) | ||
Theorem | xpeq1 4286 | Equality theorem for cross product. (Contributed by NM, 4-Jul-1994.) |
⊢ (A = B → (A × 𝐶) = (B × 𝐶)) | ||
Theorem | xpeq2 4287 | Equality theorem for cross product. (Contributed by NM, 5-Jul-1994.) |
⊢ (A = B → (𝐶 × A) = (𝐶 × B)) | ||
Theorem | elxpi 4288* | Membership in a cross product. Uses fewer axioms than elxp 4289. (Contributed by NM, 4-Jul-1994.) |
⊢ (A ∈ (B × 𝐶) → ∃x∃y(A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ 𝐶))) | ||
Theorem | elxp 4289* | Membership in a cross product. (Contributed by NM, 4-Jul-1994.) |
⊢ (A ∈ (B × 𝐶) ↔ ∃x∃y(A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ 𝐶))) | ||
Theorem | elxp2 4290* | Membership in a cross product. (Contributed by NM, 23-Feb-2004.) |
⊢ (A ∈ (B × 𝐶) ↔ ∃x ∈ B ∃y ∈ 𝐶 A = ⟨x, y⟩) | ||
Theorem | xpeq12 4291 | Equality theorem for cross product. (Contributed by FL, 31-Aug-2009.) |
⊢ ((A = B ∧ 𝐶 = 𝐷) → (A × 𝐶) = (B × 𝐷)) | ||
Theorem | xpeq1i 4292 | Equality inference for cross product. (Contributed by NM, 21-Dec-2008.) |
⊢ A = B ⇒ ⊢ (A × 𝐶) = (B × 𝐶) | ||
Theorem | xpeq2i 4293 | Equality inference for cross product. (Contributed by NM, 21-Dec-2008.) |
⊢ A = B ⇒ ⊢ (𝐶 × A) = (𝐶 × B) | ||
Theorem | xpeq12i 4294 | Equality inference for cross product. (Contributed by FL, 31-Aug-2009.) |
⊢ A = B & ⊢ 𝐶 = 𝐷 ⇒ ⊢ (A × 𝐶) = (B × 𝐷) | ||
Theorem | xpeq1d 4295 | Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.) |
⊢ (φ → A = B) ⇒ ⊢ (φ → (A × 𝐶) = (B × 𝐶)) | ||
Theorem | xpeq2d 4296 | Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.) |
⊢ (φ → A = B) ⇒ ⊢ (φ → (𝐶 × A) = (𝐶 × B)) | ||
Theorem | xpeq12d 4297 | Equality deduction for cross product. (Contributed by NM, 8-Dec-2013.) |
⊢ (φ → A = B) & ⊢ (φ → 𝐶 = 𝐷) ⇒ ⊢ (φ → (A × 𝐶) = (B × 𝐷)) | ||
Theorem | nfxp 4298 | Bound-variable hypothesis builder for cross product. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 15-Oct-2016.) |
⊢ ℲxA & ⊢ ℲxB ⇒ ⊢ Ⅎx(A × B) | ||
Theorem | 0nelxp 4299 | The empty set is not a member of a cross product. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ ¬ ∅ ∈ (A × B) | ||
Theorem | 0nelelxp 4300 | A member of a cross product (ordered pair) doesn't contain the empty set. (Contributed by NM, 15-Dec-2008.) |
⊢ (𝐶 ∈ (A × B) → ¬ ∅ ∈ 𝐶) |
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