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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | tfisi 4501* | 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 4502* | 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 4503* | 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 4504 | Extend class notation to include the class of natural numbers. |
class ω | ||
Definition | df-iom 4505* |
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 4290. 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 4506 instead for naming consistency with set.mm. (New usage is discouraged.) |
⊢ ω = ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} | ||
Theorem | dfom3 4506* | Alias for df-iom 4505. Use it instead of df-iom 4505 for naming consistency with set.mm. (Contributed by NM, 6-Aug-1994.) |
⊢ ω = ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} | ||
Theorem | omex 4507 | 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 4508 | 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 4509 | 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 4510 | 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 4511 | 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 4512* | 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 4517. (Contributed by NM, 18-Feb-2004.) |
⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴) | ||
Theorem | find 4513* | 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 4514* | 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 4515* | 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 4516* | 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 4517 | 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 4518* | 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 4519 | A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.) |
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω) | ||
Theorem | ordom 4520 | Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.) |
⊢ Ord ω | ||
Theorem | omelon2 4521 | Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.) |
⊢ (ω ∈ V → ω ∈ On) | ||
Theorem | omelon 4522 | Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.) |
⊢ ω ∈ On | ||
Theorem | nnon 4523 | A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.) |
⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | ||
Theorem | nnoni 4524 | A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.) |
⊢ 𝐴 ∈ ω ⇒ ⊢ 𝐴 ∈ On | ||
Theorem | nnord 4525 | A natural number is ordinal. (Contributed by NM, 17-Oct-1995.) |
⊢ (𝐴 ∈ ω → Ord 𝐴) | ||
Theorem | omsson 4526 | Omega is a subset of On. (Contributed by NM, 13-Jun-1994.) |
⊢ ω ⊆ On | ||
Theorem | limom 4527 | 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 4528 | A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.) |
⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω) | ||
Theorem | nnsuc 4529* | A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.) |
⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥) | ||
Theorem | nnsucpred 4530 | 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 4531 | 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 4532 | 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 4533 | 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 4534* | If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4450 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6395 or nntri3or 6389), 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 4535* | 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 4536 | 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 4479) but also holds when it is ∅ by uni0 3763. (Contributed by Jim Kingdon, 31-Jul-2022.) |
⊢ (𝐴 ∈ ω → ∪ 𝐴 ∈ ω) | ||
Syntax | cxp 4537 | Extend the definition of a class to include the cross product. |
class (𝐴 × 𝐵) | ||
Syntax | ccnv 4538 | Extend the definition of a class to include the converse of a class. |
class ^{◡}𝐴 | ||
Syntax | cdm 4539 | Extend the definition of a class to include the domain of a class. |
class dom 𝐴 | ||
Syntax | crn 4540 | Extend the definition of a class to include the range of a class. |
class ran 𝐴 | ||
Syntax | cres 4541 | 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 4542 | 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 4543 | 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 4544 | Extend the definition of a wff to include the relation predicate. (Read: 𝐴 is a relation.) |
wff Rel 𝐴 | ||
Definition | df-xp 4545* | 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 4546 | Define the relation predicate. Definition 6.4(1) of [TakeutiZaring] p. 23. For alternate definitions, see dfrel2 4989 and dfrel3 4996. (Contributed by NM, 1-Aug-1994.) |
⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) | ||
Definition | df-cnv 4547* | 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 4722 (see df-br 3930 and df-rel 4546 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 4548* | 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 4549* | 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 4550). For alternate definitions see dfdm2 5073, dfdm3 4726, and dfdm4 4731. 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 4550 | Define the range of a class. For example, F = { ⟨ 2 , 6 ⟩, ⟨ 3 , 9 ⟩ } -> ran F = { 6 , 9 } . Contrast with domain (defined in df-dm 4549). For alternate definitions, see dfrn2 4727, dfrn3 4728, and dfrn4 4999. 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 4551 | 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 4552 | 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 4551) and range (df-rn 4550). For an alternate definition, see dfima2 4883. (Contributed by NM, 2-Aug-1994.) |
⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | ||
Theorem | xpeq1 4553 | Equality theorem for cross product. (Contributed by NM, 4-Jul-1994.) |
⊢ (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶)) | ||
Theorem | xpeq2 4554 | Equality theorem for cross product. (Contributed by NM, 5-Jul-1994.) |
⊢ (𝐴 = 𝐵 → (𝐶 × 𝐴) = (𝐶 × 𝐵)) | ||
Theorem | elxpi 4555* | Membership in a cross product. Uses fewer axioms than elxp 4556. (Contributed by NM, 4-Jul-1994.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) | ||
Theorem | elxp 4556* | 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 4557* | Membership in a cross product. (Contributed by NM, 23-Feb-2004.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐴 = ⟨𝑥, 𝑦⟩) | ||
Theorem | xpeq12 4558 | Equality theorem for cross product. (Contributed by FL, 31-Aug-2009.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷)) | ||
Theorem | xpeq1i 4559 | Equality inference for cross product. (Contributed by NM, 21-Dec-2008.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐶) | ||
Theorem | xpeq2i 4560 | Equality inference for cross product. (Contributed by NM, 21-Dec-2008.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 × 𝐴) = (𝐶 × 𝐵) | ||
Theorem | xpeq12i 4561 | Equality inference for cross product. (Contributed by FL, 31-Aug-2009.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐷) | ||
Theorem | xpeq1d 4562 | Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐶)) | ||
Theorem | xpeq2d 4563 | Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 × 𝐴) = (𝐶 × 𝐵)) | ||
Theorem | xpeq12d 4564 | Equality deduction for Cartesian product. (Contributed by NM, 8-Dec-2013.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐷)) | ||
Theorem | sqxpeqd 4565 | 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 4566 | 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 4567 | 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 4568 | 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 4569 | 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 4570 | Binary relation on a cross product. (Contributed by NM, 22-Apr-2004.) |
⊢ (𝐴(𝐶 × 𝐷)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) | ||
Theorem | opelxpi 4571 | Ordered pair membership in a cross product (implication). (Contributed by NM, 28-May-1995.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷)) | ||
Theorem | opelxpd 4572 | 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 4573 | 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 4574 | 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 4575 | 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 4576* | 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 4577 | 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 4578 | 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 4579 | 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 4580 | 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 4581 | 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 4582 | 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 4583 | 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 4584 | 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 4585 | A binary relation does not contain the empty set. (Contributed by AV, 15-Nov-2021.) |
⊢ (Rel 𝑅 → ∅ ∉ 𝑅) | ||
Theorem | fconstmpt 4586* | 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 4587* | 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 4588 | 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 4589 | 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 4590 | 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 4591 | 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 4592* | 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 4593* | 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 4594 | 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 4595 | 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 4596 | 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)) | ||
Theorem | xpiundi 4597* | Distributive law for cross product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.) |
⊢ (𝐶 × ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 (𝐶 × 𝐵) | ||
Theorem | xpiundir 4598* | Distributive law for cross product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.) |
⊢ (∪ 𝑥 ∈ 𝐴 𝐵 × 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 × 𝐶) | ||
Theorem | iunxpconst 4599* | Membership in a union of cross products when the second factor is constant. (Contributed by Mario Carneiro, 29-Dec-2014.) |
⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵) | ||
Theorem | xpun 4600 | The cross product of two unions. (Contributed by NM, 12-Aug-2004.) |
⊢ ((𝐴 ∪ 𝐵) × (𝐶 ∪ 𝐷)) = (((𝐴 × 𝐶) ∪ (𝐴 × 𝐷)) ∪ ((𝐵 × 𝐶) ∪ (𝐵 × 𝐷))) |
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