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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | wetriext 4301* | A trichotomous well-order is extensional. (Contributed by Jim Kingdon, 26-Sep-2021.) |
⊢ (𝜑 → 𝑅 We 𝐴) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎)) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 (𝑧𝑅𝐵 ↔ 𝑧𝑅𝐶)) ⇒ ⊢ (𝜑 → 𝐵 = 𝐶) | ||
Theorem | wessep 4302 | A subset of a set well-ordered by set membership is well-ordered by set membership. (Contributed by Jim Kingdon, 30-Sep-2021.) |
⊢ (( E We 𝐴 ∧ 𝐵 ⊆ 𝐴) → E We 𝐵) | ||
Theorem | reg3exmidlemwe 4303* | Lemma for reg3exmid 4304. Our counterexample 𝐴 satisfies We. (Contributed by Jim Kingdon, 3-Oct-2021.) |
⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))} ⇒ ⊢ E We 𝐴 | ||
Theorem | reg3exmid 4304* | If any inhabited set satisfying df-wetr 4071 for E has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Oct-2021.) |
⊢ (( E We 𝑧 ∧ ∃𝑤 𝑤 ∈ 𝑧) → ∃𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 𝑥 ⊆ 𝑦) ⇒ ⊢ (𝜑 ∨ ¬ 𝜑) | ||
Theorem | tfi 4305* |
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 4306* | 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 4307* | 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 4308* | 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 4309* | 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 4310* | 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 4311* | 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 4312* | 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 4313 | Extend class notation to include the class of natural numbers. |
class ω | ||
Definition | df-iom 4314* |
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 4105. 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 4315 instead for naming consistency with set.mm. (New usage is discouraged.) |
⊢ ω = ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} | ||
Theorem | dfom3 4315* | Alias for df-iom 4314. Use it instead of df-iom 4314 for naming consistency with set.mm. (Contributed by NM, 6-Aug-1994.) |
⊢ ω = ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} | ||
Theorem | omex 4316 | 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 4317 | 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 4318 | 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 4319 | 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 4320 | 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 4321* | 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 4326. (Contributed by NM, 18-Feb-2004.) |
⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴) | ||
Theorem | find 4322* | 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 4323* | 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 4324* | 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 4325* | 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 4326 | 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 4327* | 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 4328 | A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.) |
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω) | ||
Theorem | ordom 4329 | Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.) |
⊢ Ord ω | ||
Theorem | omelon2 4330 | Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.) |
⊢ (ω ∈ V → ω ∈ On) | ||
Theorem | omelon 4331 | Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.) |
⊢ ω ∈ On | ||
Theorem | nnon 4332 | A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.) |
⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | ||
Theorem | nnoni 4333 | A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.) |
⊢ 𝐴 ∈ ω ⇒ ⊢ 𝐴 ∈ On | ||
Theorem | nnord 4334 | A natural number is ordinal. (Contributed by NM, 17-Oct-1995.) |
⊢ (𝐴 ∈ ω → Ord 𝐴) | ||
Theorem | omsson 4335 | Omega is a subset of On. (Contributed by NM, 13-Jun-1994.) |
⊢ ω ⊆ On | ||
Theorem | limom 4336 | 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 4337 | A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.) |
⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω) | ||
Theorem | nnsuc 4338* | A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.) |
⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥) | ||
Theorem | nndceq0 4339 | 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 4340 | 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 4341 | 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 4342* | If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4260 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6077 or nntri3or 6072), 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 4343 | Extend the definition of a class to include the cross product. |
class (𝐴 × 𝐵) | ||
Syntax | ccnv 4344 | Extend the definition of a class to include the converse of a class. |
class ^{◡}𝐴 | ||
Syntax | cdm 4345 | Extend the definition of a class to include the domain of a class. |
class dom 𝐴 | ||
Syntax | crn 4346 | Extend the definition of a class to include the range of a class. |
class ran 𝐴 | ||
Syntax | cres 4347 | 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 4348 | 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 4349 | 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 4350 | Extend the definition of a wff to include the relation predicate. (Read: 𝐴 is a relation.) |
wff Rel 𝐴 | ||
Definition | df-xp 4351* | 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 4352 | Define the relation predicate. Definition 6.4(1) of [TakeutiZaring] p. 23. For alternate definitions, see dfrel2 4771 and dfrel3 4778. (Contributed by NM, 1-Aug-1994.) |
⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) | ||
Definition | df-cnv 4353* | 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 4518 (see df-br 3765 and df-rel 4352 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 4354* | 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 4355* | 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 4356). For alternate definitions see dfdm2 4852, dfdm3 4522, and dfdm4 4527. 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 4356 | Define the range of a class. For example, F = { ⟨ 2 , 6 ⟩, ⟨ 3 , 9 ⟩ } -> ran F = { 6 , 9 } . Contrast with domain (defined in df-dm 4355). For alternate definitions, see dfrn2 4523, dfrn3 4524, and dfrn4 4781. 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 4357 | 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 4358 | 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 4357) and range (df-rn 4356). For an alternate definition, see dfima2 4670. (Contributed by NM, 2-Aug-1994.) |
⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | ||
Theorem | xpeq1 4359 | Equality theorem for cross product. (Contributed by NM, 4-Jul-1994.) |
⊢ (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶)) | ||
Theorem | xpeq2 4360 | Equality theorem for cross product. (Contributed by NM, 5-Jul-1994.) |
⊢ (𝐴 = 𝐵 → (𝐶 × 𝐴) = (𝐶 × 𝐵)) | ||
Theorem | elxpi 4361* | Membership in a cross product. Uses fewer axioms than elxp 4362. (Contributed by NM, 4-Jul-1994.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) | ||
Theorem | elxp 4362* | 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 4363* | Membership in a cross product. (Contributed by NM, 23-Feb-2004.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐴 = ⟨𝑥, 𝑦⟩) | ||
Theorem | xpeq12 4364 | Equality theorem for cross product. (Contributed by FL, 31-Aug-2009.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷)) | ||
Theorem | xpeq1i 4365 | Equality inference for cross product. (Contributed by NM, 21-Dec-2008.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐶) | ||
Theorem | xpeq2i 4366 | Equality inference for cross product. (Contributed by NM, 21-Dec-2008.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 × 𝐴) = (𝐶 × 𝐵) | ||
Theorem | xpeq12i 4367 | Equality inference for cross product. (Contributed by FL, 31-Aug-2009.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐷) | ||
Theorem | xpeq1d 4368 | Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐶)) | ||
Theorem | xpeq2d 4369 | Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 × 𝐴) = (𝐶 × 𝐵)) | ||
Theorem | xpeq12d 4370 | Equality deduction for cross product. (Contributed by NM, 8-Dec-2013.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐷)) | ||
Theorem | nfxp 4371 | 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 4372 | 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 4373 | 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 4374 | 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 4375 | Binary relation on a cross product. (Contributed by NM, 22-Apr-2004.) |
⊢ (𝐴(𝐶 × 𝐷)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) | ||
Theorem | opelxpi 4376 | Ordered pair membership in a cross product (implication). (Contributed by NM, 28-May-1995.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷)) | ||
Theorem | opelxp1 4377 | 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 4378 | 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 4379 | 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 4380* | 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 4381 | 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 4382 | 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 4383 | 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 4384 | 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 4385 | 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 4386 | 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 4387* | 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 4388* | 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 4389 | 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 4390 | 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 4391 | 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 4392 | 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 4393* | 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 4394* | 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 4395 | 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 4396 | 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 4397 | 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 4398* | 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 4399* | 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 4400* | 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) |
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