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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ordsuc 4201 | 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 4202 | 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 4203 |
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 4033 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if A ∈ On then both ∪ suc A = A (onunisuci 4095) and ∪ {x ∈ On ∣ x ⊆ A} = A (onuniss2 4163). Constructively (𝒫 A ∩ On) and suc A cannot be shown to be equivalent (as proved at ordpwsucexmid 4206). (Contributed by Jim Kingdon, 21-Jul-2019.) |
⊢ (Ord A → suc A ⊆ (𝒫 A ∩ On)) | ||
Theorem | onnmin 4204 | 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 4205 | 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 4206* | The subset in ordpwsucss 4203 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 4207 | 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 4208* |
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 4209* | 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 4210* | 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 4211* | 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 4212* | 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 4213* | 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 4214* | 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 4215* | 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 4216 | Extend class notation to include the class of natural numbers. |
class 𝜔 | ||
Definition | df-iom 4217* |
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 4030. 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.) |
⊢ 𝜔 = ∩ {x ∣ (∅ ∈ x ∧ ∀y ∈ x suc y ∈ x)} | ||
Theorem | dfom3 4218* | Another name for df-iom 4217. (Contributed by NM, 6-Aug-1994.) |
⊢ 𝜔 = ∩ {x ∣ (∅ ∈ x ∧ ∀y ∈ x suc y ∈ x)} | ||
Theorem | omex 4219 | 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 4220 | Zero is a natural number. One of Peano's 5 postulates for arithmetic. Proposition 7.30(1) of [TakeutiZaring] p. 42. (Contributed by NM, 15-May-1994.) |
⊢ ∅ ∈ 𝜔 | ||
Theorem | peano2 4221 | The successor of any natural number is a natural number. One of Peano's 5 postulates for arithmetic. Proposition 7.30(2) of [TakeutiZaring] p. 42. (Contributed by NM, 3-Sep-2003.) |
⊢ (A ∈ 𝜔 → suc A ∈ 𝜔) | ||
Theorem | peano3 4222 | The successor of any natural number is not zero. One of Peano's 5 postulates for arithmetic. Proposition 7.30(3) of [TakeutiZaring] p. 42. (Contributed by NM, 3-Sep-2003.) |
⊢ (A ∈ 𝜔 → suc A ≠ ∅) | ||
Theorem | peano4 4223 | Two natural numbers are equal iff their successors are equal, i.e. the successor function is one-to-one. One of Peano's 5 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 4224* | The induction postulate: any class containing zero and closed under the successor operation contains all natural numbers. One of Peano's 5 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 hypothesis, is derived from this theorem as theorem findes 4229. (Contributed by NM, 18-Feb-2004.) |
⊢ ((∅ ∈ A ∧ ∀x ∈ 𝜔 (x ∈ A → suc x ∈ A)) → 𝜔 ⊆ A) | ||
Theorem | find 4225* | 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 4226* | 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 hypothesis. Theorem Schema 22 of [Suppes] p. 136. (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 4227* | 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 hypothesis. 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 4228* | 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 hypothesis. 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 4229 | Finite induction with explicit substitution. The first hypothesis is the basis and the second is the induction hypothesis. Theorem Schema 22 of [Suppes] p. 136. (Contributed by Raph Levien, 9-Jul-2003.) |
⊢ [∅ / x]φ & ⊢ (x ∈ 𝜔 → (φ → [suc x / x]φ)) ⇒ ⊢ (x ∈ 𝜔 → φ) | ||
Theorem | nn0suc 4230* | 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 4231 | A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.) |
⊢ ((A ∈ B ∧ B ∈ 𝜔) → A ∈ 𝜔) | ||
Theorem | ordom 4232 | Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.) |
⊢ Ord 𝜔 | ||
Theorem | omelon2 4233 | Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.) |
⊢ (𝜔 ∈ V → 𝜔 ∈ On) | ||
Theorem | omelon 4234 | Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.) |
⊢ 𝜔 ∈ On | ||
Theorem | nnon 4235 | A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.) |
⊢ (A ∈ 𝜔 → A ∈ On) | ||
Theorem | nnoni 4236 | A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.) |
⊢ A ∈ 𝜔 ⇒ ⊢ A ∈ On | ||
Theorem | nnord 4237 | A natural number is ordinal. (Contributed by NM, 17-Oct-1995.) |
⊢ (A ∈ 𝜔 → Ord A) | ||
Theorem | omsson 4238 | Omega is a subset of On. (Contributed by NM, 13-Jun-1994.) |
⊢ 𝜔 ⊆ On | ||
Theorem | limom 4239 | 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 4240 | A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.) |
⊢ (A ∈ 𝜔 ↔ suc A ∈ 𝜔) | ||
Theorem | nnsuc 4241* | A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.) |
⊢ ((A ∈ 𝜔 ∧ A ≠ ∅) → ∃x ∈ 𝜔 A = suc x) | ||
Theorem | nndceq0 4242 | 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 4243 | 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 4244 | 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 4245* | If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4179 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 5964 or nntri3or 5961), 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 4246 | Extend the definition of a class to include the cross product. |
class (A × B) | ||
Syntax | ccnv 4247 | Extend the definition of a class to include the converse of a class. |
class ^{◡}A | ||
Syntax | cdm 4248 | Extend the definition of a class to include the domain of a class. |
class dom A | ||
Syntax | crn 4249 | Extend the definition of a class to include the range of a class. |
class ran A | ||
Syntax | cres 4250 | 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 4251 | 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 4252 | 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 4253 | Extend the definition of a wff to include the relation predicate. (Read: A is a relation.) |
wff Rel A | ||
Definition | df-xp 4254* | 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 4255 | Define the relation predicate. Definition 6.4(1) of [TakeutiZaring] p. 23. For alternate definitions, see dfrel2 4675 and dfrel3 4682. (Contributed by NM, 1-Aug-1994.) |
⊢ (Rel A ↔ A ⊆ (V × V)) | ||
Definition | df-cnv 4256* | 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 4421 (see df-br 3718 and df-rel 4255 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 4257* | 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 4258* | 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 4259). For alternate definitions see dfdm2 4756, dfdm3 4425, and dfdm4 4430. 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 4259 | Define the range of a class. For example, F = { ⟨ 2 , 6 ⟩, ⟨ 3 , 9 ⟩ } -> ran F = { 6 , 9 } . Contrast with domain (defined in df-dm 4258). For alternate definitions, see dfrn2 4426, dfrn3 4427, and dfrn4 4685. 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 4260 | 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 4261 | 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 4260) and range (df-rn 4259). For an alternate definition, see dfima2 4574. (Contributed by NM, 2-Aug-1994.) |
⊢ (A “ B) = ran (A ↾ B) | ||
Theorem | xpeq1 4262 | Equality theorem for cross product. (Contributed by NM, 4-Jul-1994.) |
⊢ (A = B → (A × 𝐶) = (B × 𝐶)) | ||
Theorem | xpeq2 4263 | Equality theorem for cross product. (Contributed by NM, 5-Jul-1994.) |
⊢ (A = B → (𝐶 × A) = (𝐶 × B)) | ||
Theorem | elxpi 4264* | Membership in a cross product. Uses fewer axioms than elxp 4265. (Contributed by NM, 4-Jul-1994.) |
⊢ (A ∈ (B × 𝐶) → ∃x∃y(A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ 𝐶))) | ||
Theorem | elxp 4265* | 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 4266* | Membership in a cross product. (Contributed by NM, 23-Feb-2004.) |
⊢ (A ∈ (B × 𝐶) ↔ ∃x ∈ B ∃y ∈ 𝐶 A = ⟨x, y⟩) | ||
Theorem | xpeq12 4267 | Equality theorem for cross product. (Contributed by FL, 31-Aug-2009.) |
⊢ ((A = B ∧ 𝐶 = 𝐷) → (A × 𝐶) = (B × 𝐷)) | ||
Theorem | xpeq1i 4268 | Equality inference for cross product. (Contributed by NM, 21-Dec-2008.) |
⊢ A = B ⇒ ⊢ (A × 𝐶) = (B × 𝐶) | ||
Theorem | xpeq2i 4269 | Equality inference for cross product. (Contributed by NM, 21-Dec-2008.) |
⊢ A = B ⇒ ⊢ (𝐶 × A) = (𝐶 × B) | ||
Theorem | xpeq12i 4270 | Equality inference for cross product. (Contributed by FL, 31-Aug-2009.) |
⊢ A = B & ⊢ 𝐶 = 𝐷 ⇒ ⊢ (A × 𝐶) = (B × 𝐷) | ||
Theorem | xpeq1d 4271 | Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.) |
⊢ (φ → A = B) ⇒ ⊢ (φ → (A × 𝐶) = (B × 𝐶)) | ||
Theorem | xpeq2d 4272 | Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.) |
⊢ (φ → A = B) ⇒ ⊢ (φ → (𝐶 × A) = (𝐶 × B)) | ||
Theorem | xpeq12d 4273 | Equality deduction for cross product. (Contributed by NM, 8-Dec-2013.) |
⊢ (φ → A = B) & ⊢ (φ → 𝐶 = 𝐷) ⇒ ⊢ (φ → (A × 𝐶) = (B × 𝐷)) | ||
Theorem | nfxp 4274 | 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 4275 | 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 4276 | A member of a cross product (ordered pair) doesn't contain the empty set. (Contributed by NM, 15-Dec-2008.) |
⊢ (𝐶 ∈ (A × B) → ¬ ∅ ∈ 𝐶) | ||
Theorem | opelxp 4277 | Ordered pair membership in a cross product. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ (⟨A, B⟩ ∈ (𝐶 × 𝐷) ↔ (A ∈ 𝐶 ∧ B ∈ 𝐷)) | ||
Theorem | brxp 4278 | Binary relation on a cross product. (Contributed by NM, 22-Apr-2004.) |
⊢ (A(𝐶 × 𝐷)B ↔ (A ∈ 𝐶 ∧ B ∈ 𝐷)) | ||
Theorem | opelxpi 4279 | Ordered pair membership in a cross product (implication). (Contributed by NM, 28-May-1995.) |
⊢ ((A ∈ 𝐶 ∧ B ∈ 𝐷) → ⟨A, B⟩ ∈ (𝐶 × 𝐷)) | ||
Theorem | opelxp1 4280 | The first member of an ordered pair of classes in a cross product belongs to first cross product argument. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ (⟨A, B⟩ ∈ (𝐶 × 𝐷) → A ∈ 𝐶) | ||
Theorem | opelxp2 4281 | The second member of an ordered pair of classes in a cross product belongs to second cross product argument. (Contributed by Mario Carneiro, 26-Apr-2015.) |
⊢ (⟨A, B⟩ ∈ (𝐶 × 𝐷) → B ∈ 𝐷) | ||
Theorem | otelxp1 4282 | The first member of an ordered triple of classes in a cross product belongs to first cross product argument. (Contributed by NM, 28-May-2008.) |
⊢ (⟨⟨A, B⟩, 𝐶⟩ ∈ ((𝑅 × 𝑆) × 𝑇) → A ∈ 𝑅) | ||
Theorem | rabxp 4283* | Membership in a class builder restricted to a cross product. (Contributed by NM, 20-Feb-2014.) |
⊢ (x = ⟨y, z⟩ → (φ ↔ ψ)) ⇒ ⊢ {x ∈ (A × B) ∣ φ} = {⟨y, z⟩ ∣ (y ∈ A ∧ z ∈ B ∧ ψ)} | ||
Theorem | brrelex12 4284 | A true binary relation on a relation implies the arguments are sets. (This is a property of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.) |
⊢ ((Rel 𝑅 ∧ A𝑅B) → (A ∈ V ∧ B ∈ V)) | ||
Theorem | brrelex 4285 | A true binary relation on a relation implies the first argument is a set. (This is a property of our ordered pair definition.) (Contributed by NM, 18-May-2004.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ ((Rel 𝑅 ∧ A𝑅B) → A ∈ V) | ||
Theorem | brrelex2 4286 | A true binary relation on a relation implies the second argument is a set. (This is a property of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.) |
⊢ ((Rel 𝑅 ∧ A𝑅B) → B ∈ V) | ||
Theorem | brrelexi 4287 | The first argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by NM, 4-Jun-1998.) |
⊢ Rel 𝑅 ⇒ ⊢ (A𝑅B → A ∈ V) | ||
Theorem | brrelex2i 4288 | The second argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.) |
⊢ Rel 𝑅 ⇒ ⊢ (A𝑅B → B ∈ V) | ||
Theorem | nprrel 4289 | No proper class is related to anything via any relation. (Contributed by Roy F. Longton, 30-Jul-2005.) |
⊢ Rel 𝑅 & ⊢ ¬ A ∈ V ⇒ ⊢ ¬ A𝑅B | ||
Theorem | fconstmpt 4290* | Representation of a constant function using the mapping operation. (Note that x cannot appear free in B.) (Contributed by NM, 12-Oct-1999.) (Revised by Mario Carneiro, 16-Nov-2013.) |
⊢ (A × {B}) = (x ∈ A ↦ B) | ||
Theorem | vtoclr 4291* | Variable to class conversion of transitive relation. (Contributed by NM, 9-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ Rel 𝑅 & ⊢ ((x𝑅y ∧ y𝑅z) → x𝑅z) ⇒ ⊢ ((A𝑅B ∧ B𝑅𝐶) → A𝑅𝐶) | ||
Theorem | opelvvg 4292 | Ordered pair membership in the universal class of ordered pairs. (Contributed by Mario Carneiro, 3-May-2015.) |
⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → ⟨A, B⟩ ∈ (V × V)) | ||
Theorem | opelvv 4293 | Ordered pair membership in the universal class of ordered pairs. (Contributed by NM, 22-Aug-2013.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ A ∈ V & ⊢ B ∈ V ⇒ ⊢ ⟨A, B⟩ ∈ (V × V) | ||
Theorem | opthprc 4294 | Justification theorem for an ordered pair definition that works for any classes, including proper classes. This is a possible definition implied by the footnote in [Jech] p. 78, which says, "The sophisticated reader will not object to our use of a pair of classes." (Contributed by NM, 28-Sep-2003.) |
⊢ (((A × {∅}) ∪ (B × {{∅}})) = ((𝐶 × {∅}) ∪ (𝐷 × {{∅}})) ↔ (A = 𝐶 ∧ B = 𝐷)) | ||
Theorem | brel 4295 | Two things in a binary relation belong to the relation's domain. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ 𝑅 ⊆ (𝐶 × 𝐷) ⇒ ⊢ (A𝑅B → (A ∈ 𝐶 ∧ B ∈ 𝐷)) | ||
Theorem | brab2a 4296* | Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 9-Nov-2015.) |
⊢ ((x = A ∧ y = B) → (φ ↔ ψ)) & ⊢ 𝑅 = {⟨x, y⟩ ∣ ((x ∈ 𝐶 ∧ y ∈ 𝐷) ∧ φ)} ⇒ ⊢ (A𝑅B ↔ ((A ∈ 𝐶 ∧ B ∈ 𝐷) ∧ ψ)) | ||
Theorem | elxp3 4297* | Membership in a cross product. (Contributed by NM, 5-Mar-1995.) |
⊢ (A ∈ (B × 𝐶) ↔ ∃x∃y(⟨x, y⟩ = A ∧ ⟨x, y⟩ ∈ (B × 𝐶))) | ||
Theorem | opeliunxp 4298 | Membership in a union of cross products. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 1-Jan-2017.) |
⊢ (⟨x, 𝐶⟩ ∈ ∪ x ∈ A ({x} × B) ↔ (x ∈ A ∧ 𝐶 ∈ B)) | ||
Theorem | xpundi 4299 | Distributive law for cross product over union. Theorem 103 of [Suppes] p. 52. (Contributed by NM, 12-Aug-2004.) |
⊢ (A × (B ∪ 𝐶)) = ((A × B) ∪ (A × 𝐶)) | ||
Theorem | xpundir 4300 | Distributive law for cross product over union. Similar to Theorem 103 of [Suppes] p. 52. (Contributed by NM, 30-Sep-2002.) |
⊢ ((A ∪ B) × 𝐶) = ((A × 𝐶) ∪ (B × 𝐶)) |
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