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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | tfis2f 4601* | 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 4602* | 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 4603* | 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 4604* | 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 4605* | 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 4606* | 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 4607 | Extend class notation to include the class of natural numbers. |
class ω | ||
Definition | df-iom 4608* |
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 4386. Later, when we define complex numbers, we will be able to also define a subset of the complex numbers (df-inn 8951) 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 7224. (Contributed by NM, 6-Aug-1994.) Use its alias dfom3 4609 instead for naming consistency with set.mm. (New usage is discouraged.) |
⊢ ω = ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} | ||
Theorem | dfom3 4609* | Alias for df-iom 4608. Use it instead of df-iom 4608 for naming consistency with set.mm. (Contributed by NM, 6-Aug-1994.) |
⊢ ω = ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} | ||
Theorem | omex 4610 | 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 4611 | 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 4612 | 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 4613 | 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 4614 | 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 4615* | 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 4620. (Contributed by NM, 18-Feb-2004.) |
⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴) | ||
Theorem | find 4616* | 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 4617* | 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 4618* | 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 4619* | 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 4620 | 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 4621* | 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 4622 | 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 4623. (Revised by BJ, 7-Aug-2024.) |
⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω) | ||
Theorem | elnn 4623 | A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.) |
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω) | ||
Theorem | ordom 4624 | Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.) |
⊢ Ord ω | ||
Theorem | omelon2 4625 | Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.) |
⊢ (ω ∈ V → ω ∈ On) | ||
Theorem | omelon 4626 | Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.) |
⊢ ω ∈ On | ||
Theorem | nnon 4627 | A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.) |
⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | ||
Theorem | nnoni 4628 | A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.) |
⊢ 𝐴 ∈ ω ⇒ ⊢ 𝐴 ∈ On | ||
Theorem | nnord 4629 | A natural number is ordinal. (Contributed by NM, 17-Oct-1995.) |
⊢ (𝐴 ∈ ω → Ord 𝐴) | ||
Theorem | omsson 4630 | Omega is a subset of On. (Contributed by NM, 13-Jun-1994.) |
⊢ ω ⊆ On | ||
Theorem | limom 4631 | 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 4632 | A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.) |
⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω) | ||
Theorem | nnsuc 4633* | A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.) |
⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥) | ||
Theorem | nnsucpred 4634 | 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 4635 | 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 4636 | 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 4637 | 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 4638* | If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4552 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6525 or nntri3or 6519), 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 4639* | 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 4640 | 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 4581) but also holds when it is ∅ by uni0 3851. (Contributed by Jim Kingdon, 31-Jul-2022.) |
⊢ (𝐴 ∈ ω → ∪ 𝐴 ∈ ω) | ||
Theorem | nnpredlt 4641 | The predecessor (see nnpredcl 4640) of a nonzero natural number is less than (see df-iord 4384) that number. (Contributed by Jim Kingdon, 14-Sep-2024.) |
⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∪ 𝐴 ∈ 𝐴) | ||
Syntax | cxp 4642 | Extend the definition of a class to include the cross product. |
class (𝐴 × 𝐵) | ||
Syntax | ccnv 4643 | Extend the definition of a class to include the converse of a class. |
class ◡𝐴 | ||
Syntax | cdm 4644 | Extend the definition of a class to include the domain of a class. |
class dom 𝐴 | ||
Syntax | crn 4645 | Extend the definition of a class to include the range of a class. |
class ran 𝐴 | ||
Syntax | cres 4646 | 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 4647 | 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 4648 | 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 4649 | Extend the definition of a wff to include the relation predicate. (Read: 𝐴 is a relation.) |
wff Rel 𝐴 | ||
Definition | df-xp 4650* | Define the Cartesian product of two classes. This is also sometimes called the "cross product" but that term also has other meanings; we intentionally choose a less ambiguous term. 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 is defined using the Cartesian product as (ℤ × ℕ); the left- and right-hand sides of the Cartesian 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 4651 | Define the relation predicate. Definition 6.4(1) of [TakeutiZaring] p. 23. For alternate definitions, see dfrel2 5097 and dfrel3 5104. (Contributed by NM, 1-Aug-1994.) |
⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) | ||
Definition | df-cnv 4652* |
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 4828
(see df-br 4019 and df-rel 4651 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. "Converse" is Quine's terminology. Some authors use a "minus one" exponent and call it "inverse", especially when the argument is a function, although this is not in general a genuine inverse. (Contributed by NM, 4-Jul-1994.) |
⊢ ◡𝐴 = {〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} | ||
Definition | df-co 4653* | 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 4654* | 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 4655). For alternate definitions see dfdm2 5181, dfdm3 4832, and dfdm4 4837. 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 4655 | Define the range of a class. For example, F = { 〈 2 , 6 〉, 〈 3 , 9 〉 } -> ran F = { 6 , 9 } . Contrast with domain (defined in df-dm 4654). For alternate definitions, see dfrn2 4833, dfrn3 4834, and dfrn4 5107. 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 4656 | 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〉}. We do not introduce a special syntax for the corestriction of a class: it will be expressed either as the intersection (𝐴 ∩ (V × 𝐵)) or as the converse of the restricted converse. (Contributed by NM, 2-Aug-1994.) |
⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V)) | ||
Definition | df-ima 4657 | 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 4656) and range (df-rn 4655). For an alternate definition, see dfima2 4990. (Contributed by NM, 2-Aug-1994.) |
⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | ||
Theorem | xpeq1 4658 | Equality theorem for cross product. (Contributed by NM, 4-Jul-1994.) |
⊢ (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶)) | ||
Theorem | xpeq2 4659 | Equality theorem for cross product. (Contributed by NM, 5-Jul-1994.) |
⊢ (𝐴 = 𝐵 → (𝐶 × 𝐴) = (𝐶 × 𝐵)) | ||
Theorem | elxpi 4660* | Membership in a cross product. Uses fewer axioms than elxp 4661. (Contributed by NM, 4-Jul-1994.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) | ||
Theorem | elxp 4661* | 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 4662* | Membership in a cross product. (Contributed by NM, 23-Feb-2004.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐴 = 〈𝑥, 𝑦〉) | ||
Theorem | xpeq12 4663 | Equality theorem for cross product. (Contributed by FL, 31-Aug-2009.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷)) | ||
Theorem | xpeq1i 4664 | Equality inference for cross product. (Contributed by NM, 21-Dec-2008.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐶) | ||
Theorem | xpeq2i 4665 | Equality inference for cross product. (Contributed by NM, 21-Dec-2008.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 × 𝐴) = (𝐶 × 𝐵) | ||
Theorem | xpeq12i 4666 | Equality inference for cross product. (Contributed by FL, 31-Aug-2009.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐷) | ||
Theorem | xpeq1d 4667 | Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐶)) | ||
Theorem | xpeq2d 4668 | Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 × 𝐴) = (𝐶 × 𝐵)) | ||
Theorem | xpeq12d 4669 | Equality deduction for Cartesian product. (Contributed by NM, 8-Dec-2013.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐷)) | ||
Theorem | sqxpeqd 4670 | Equality deduction for a Cartesian square, see Wikipedia "Cartesian product", https://en.wikipedia.org/wiki/Cartesian_product#n-ary_Cartesian_power. (Contributed by AV, 13-Jan-2020.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 × 𝐴) = (𝐵 × 𝐵)) | ||
Theorem | nfxp 4671 | 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 4672 | 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 4673 | 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 4674 | 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 4675 | Binary relation on a cross product. (Contributed by NM, 22-Apr-2004.) |
⊢ (𝐴(𝐶 × 𝐷)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) | ||
Theorem | opelxpi 4676 | Ordered pair membership in a cross product (implication). (Contributed by NM, 28-May-1995.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷)) | ||
Theorem | opelxpd 4677 | Ordered pair membership in a Cartesian product, deduction form. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
⊢ (𝜑 → 𝐴 ∈ 𝐶) & ⊢ (𝜑 → 𝐵 ∈ 𝐷) ⇒ ⊢ (𝜑 → 〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷)) | ||
Theorem | opelxp1 4678 | 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 4679 | 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 4680 | 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 4681* | 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 4682 | 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 | brrelex1 4683 | 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 | brrelex 4684 | 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 4685 | 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 | brrelex12i 4686 | Two classes that are related by a binary relation are sets. (An artifact of our ordered pair definition.) (Contributed by BJ, 3-Oct-2022.) |
⊢ Rel 𝑅 ⇒ ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | ||
Theorem | brrelex1i 4687 | 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 4688 | 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 4689 | No proper class is related to anything via any relation. (Contributed by Roy F. Longton, 30-Jul-2005.) |
⊢ Rel 𝑅 & ⊢ ¬ 𝐴 ∈ V ⇒ ⊢ ¬ 𝐴𝑅𝐵 | ||
Theorem | 0nelrel 4690 | A binary relation does not contain the empty set. (Contributed by AV, 15-Nov-2021.) |
⊢ (Rel 𝑅 → ∅ ∉ 𝑅) | ||
Theorem | fconstmpt 4691* | 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 4692* | 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 4693 | 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 4694 | 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 4695 | 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 4696 | 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 4697* | 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 4698* | 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 4699 | 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 4700 | 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)) |
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