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Type | Label | Description |
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Statement | ||
Theorem | ruv 4301 | The Russell class is equal to the universe V. Exercise 5 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 4-Oct-2008.) |
⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V | ||
Theorem | ruALT 4302 | Alternate proof of Russell's Paradox ru 2785, simplified using (indirectly) the Axiom of Set Induction ax-setind 4289. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V | ||
Theorem | onprc 4303 | No set contains all ordinal numbers. Proposition 7.13 of [TakeutiZaring] p. 38. This is also known as the Burali-Forti paradox (remark in [Enderton] p. 194). In 1897, Cesare Burali-Forti noticed that since the "set" of all ordinal numbers is an ordinal class (ordon 4239), it must be both an element of the set of all ordinal numbers yet greater than every such element. ZF set theory resolves this paradox by not allowing the class of all ordinal numbers to be a set (so instead it is a proper class). Here we prove the denial of its existence. (Contributed by NM, 18-May-1994.) |
⊢ ¬ On ∈ V | ||
Theorem | sucon 4304 | The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.) |
⊢ suc On = On | ||
Theorem | en2lp 4305 | No class has 2-cycle membership loops. Theorem 7X(b) of [Enderton] p. 206. (Contributed by NM, 16-Oct-1996.) (Proof rewritten by Mario Carneiro and Jim Kingdon, 27-Nov-2018.) |
⊢ ¬ (𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) | ||
Theorem | preleq 4306 | Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ (((𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐷) ∧ {𝐴, 𝐵} = {𝐶, 𝐷}) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
Theorem | opthreg 4307 | Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4289 (via the preleq 4306 step). See df-op 3411 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ ({𝐴, {𝐴, 𝐵}} = {𝐶, {𝐶, 𝐷}} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
Theorem | suc11g 4308 | The successor operation behaves like a one-to-one function (assuming the Axiom of Set Induction). Similar to Exercise 35 of [Enderton] p. 208 and its converse. (Contributed by NM, 25-Oct-2003.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵)) | ||
Theorem | suc11 4309 | The successor operation behaves like a one-to-one function. Compare Exercise 16 of [Enderton] p. 194. (Contributed by NM, 3-Sep-2003.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc 𝐴 = suc 𝐵 ↔ 𝐴 = 𝐵)) | ||
Theorem | dtruex 4310* | At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 3969 can also be summarized as "at least two sets exist", the difference is that dtruarb 3969 shows the existence of two sets which are not equal to each other, but this theorem says that given a specific 𝑦, we can construct a set 𝑥 which does not equal it. (Contributed by Jim Kingdon, 29-Dec-2018.) |
⊢ ∃𝑥 ¬ 𝑥 = 𝑦 | ||
Theorem | dtru 4311* | At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). If we assumed the law of the excluded middle this would be equivalent to dtruex 4310. (Contributed by Jim Kingdon, 29-Dec-2018.) |
⊢ ¬ ∀𝑥 𝑥 = 𝑦 | ||
Theorem | eunex 4312 | Existential uniqueness implies there is a value for which the wff argument is false. (Contributed by Jim Kingdon, 29-Dec-2018.) |
⊢ (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑) | ||
Theorem | ordsoexmid 4313 | Weak linearity of ordinals implies the law of the excluded middle (that is, decidability of an arbitrary proposition). (Contributed by Mario Carneiro and Jim Kingdon, 29-Jan-2019.) |
⊢ E Or On ⇒ ⊢ (𝜑 ∨ ¬ 𝜑) | ||
Theorem | ordsuc 4314 | 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 | onsucuni2 4315 | A successor ordinal is the successor of its union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc ∪ 𝐴 = 𝐴) | ||
Theorem | 0elsucexmid 4316* | If the successor of any ordinal class contains the empty set, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2021.) |
⊢ ∀𝑥 ∈ On ∅ ∈ suc 𝑥 ⇒ ⊢ (𝜑 ∨ ¬ 𝜑) | ||
Theorem | nlimsucg 4317 | 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 4318 |
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 4135 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if 𝐴 ∈ On then both ∪ suc 𝐴 = 𝐴 (onunisuci 4196) and ∪ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = 𝐴 (onuniss2 4265). Constructively (𝒫 𝐴 ∩ On) and suc 𝐴 cannot be shown to be equivalent (as proved at ordpwsucexmid 4321). (Contributed by Jim Kingdon, 21-Jul-2019.) |
⊢ (Ord 𝐴 → suc 𝐴 ⊆ (𝒫 𝐴 ∩ On)) | ||
Theorem | onnmin 4319 | 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 4320 | 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 4321* | The subset in ordpwsucss 4318 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 4322 | An ordinal number is a proper subset of its successor. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
⊢ (𝐴 ∈ On → 𝐴 ⊊ suc 𝐴) | ||
Theorem | ordtri2or2exmid 4323* | Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 29-Aug-2021.) |
⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥) ⇒ ⊢ (𝜑 ∨ ¬ 𝜑) | ||
Theorem | onintexmid 4324* | If the intersection (infimum) of an inhabited class of ordinal numbers belongs to the class, excluded middle follows. The hypothesis would be provable given excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Aug-2021.) |
⊢ ((𝑦 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝑦) → ∩ 𝑦 ∈ 𝑦) ⇒ ⊢ (𝜑 ∨ ¬ 𝜑) | ||
Theorem | zfregfr 4325 | The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.) |
⊢ E Fr 𝐴 | ||
Theorem | ordfr 4326 | Epsilon is well-founded on an ordinal class. (Contributed by NM, 22-Apr-1994.) |
⊢ (Ord 𝐴 → E Fr 𝐴) | ||
Theorem | ordwe 4327 | Epsilon well-orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.) |
⊢ (Ord 𝐴 → E We 𝐴) | ||
Theorem | wetriext 4328* | 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 4329 | 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 4330* | Lemma for reg3exmid 4331. Our counterexample 𝐴 satisfies We. (Contributed by Jim Kingdon, 3-Oct-2021.) |
⊢ 𝐴 = {𝑥 ∈ {∅, {∅}} ∣ (𝑥 = {∅} ∨ (𝑥 = ∅ ∧ 𝜑))} ⇒ ⊢ E We 𝐴 | ||
Theorem | reg3exmid 4331* | If any inhabited set satisfying df-wetr 4098 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 4332* |
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 4333* | 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 4334* | 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 4335* | 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 4336* | 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 4337* | 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 4338* | 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 4339* | 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 4340 | Extend class notation to include the class of natural numbers. |
class ω | ||
Definition | df-iom 4341* |
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 4132. 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 4342 instead for naming consistency with set.mm. (New usage is discouraged.) |
⊢ ω = ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} | ||
Theorem | dfom3 4342* | Alias for df-iom 4341. Use it instead of df-iom 4341 for naming consistency with set.mm. (Contributed by NM, 6-Aug-1994.) |
⊢ ω = ∩ {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥)} | ||
Theorem | omex 4343 | 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 4344 | 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 4345 | 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 4346 | 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 4347 | 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 4348* | 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 4353. (Contributed by NM, 18-Feb-2004.) |
⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴) | ||
Theorem | find 4349* | 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 4350* | 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 4351* | 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 4352* | 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 4353 | 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 4354* | 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 4355 | A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.) |
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω) | ||
Theorem | ordom 4356 | Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.) |
⊢ Ord ω | ||
Theorem | omelon2 4357 | Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.) |
⊢ (ω ∈ V → ω ∈ On) | ||
Theorem | omelon 4358 | Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.) |
⊢ ω ∈ On | ||
Theorem | nnon 4359 | A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.) |
⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | ||
Theorem | nnoni 4360 | A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.) |
⊢ 𝐴 ∈ ω ⇒ ⊢ 𝐴 ∈ On | ||
Theorem | nnord 4361 | A natural number is ordinal. (Contributed by NM, 17-Oct-1995.) |
⊢ (𝐴 ∈ ω → Ord 𝐴) | ||
Theorem | omsson 4362 | Omega is a subset of On. (Contributed by NM, 13-Jun-1994.) |
⊢ ω ⊆ On | ||
Theorem | limom 4363 | 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 4364 | A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.) |
⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω) | ||
Theorem | nnsuc 4365* | A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.) |
⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ ω 𝐴 = suc 𝑥) | ||
Theorem | nndceq0 4366 | 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 4367 | 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 4368 | 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 4369* | If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4287 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6107 or nntri3or 6102), 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 4370 | Extend the definition of a class to include the cross product. |
class (𝐴 × 𝐵) | ||
Syntax | ccnv 4371 | Extend the definition of a class to include the converse of a class. |
class ^{◡}𝐴 | ||
Syntax | cdm 4372 | Extend the definition of a class to include the domain of a class. |
class dom 𝐴 | ||
Syntax | crn 4373 | Extend the definition of a class to include the range of a class. |
class ran 𝐴 | ||
Syntax | cres 4374 | 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 4375 | 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 4376 | 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 4377 | Extend the definition of a wff to include the relation predicate. (Read: 𝐴 is a relation.) |
wff Rel 𝐴 | ||
Definition | df-xp 4378* | 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 4379 | Define the relation predicate. Definition 6.4(1) of [TakeutiZaring] p. 23. For alternate definitions, see dfrel2 4798 and dfrel3 4805. (Contributed by NM, 1-Aug-1994.) |
⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) | ||
Definition | df-cnv 4380* | 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 4545 (see df-br 3792 and df-rel 4379 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 4381* | 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 4382* | 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 4383). For alternate definitions see dfdm2 4879, dfdm3 4549, and dfdm4 4554. 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 4383 | Define the range of a class. For example, F = { ⟨ 2 , 6 ⟩, ⟨ 3 , 9 ⟩ } -> ran F = { 6 , 9 } . Contrast with domain (defined in df-dm 4382). For alternate definitions, see dfrn2 4550, dfrn3 4551, and dfrn4 4808. 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 4384 | 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 4385 | 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 4384) and range (df-rn 4383). For an alternate definition, see dfima2 4697. (Contributed by NM, 2-Aug-1994.) |
⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | ||
Theorem | xpeq1 4386 | Equality theorem for cross product. (Contributed by NM, 4-Jul-1994.) |
⊢ (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶)) | ||
Theorem | xpeq2 4387 | Equality theorem for cross product. (Contributed by NM, 5-Jul-1994.) |
⊢ (𝐴 = 𝐵 → (𝐶 × 𝐴) = (𝐶 × 𝐵)) | ||
Theorem | elxpi 4388* | Membership in a cross product. Uses fewer axioms than elxp 4389. (Contributed by NM, 4-Jul-1994.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) | ||
Theorem | elxp 4389* | 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 4390* | Membership in a cross product. (Contributed by NM, 23-Feb-2004.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐴 = ⟨𝑥, 𝑦⟩) | ||
Theorem | xpeq12 4391 | Equality theorem for cross product. (Contributed by FL, 31-Aug-2009.) |
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 × 𝐶) = (𝐵 × 𝐷)) | ||
Theorem | xpeq1i 4392 | Equality inference for cross product. (Contributed by NM, 21-Dec-2008.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐶) | ||
Theorem | xpeq2i 4393 | Equality inference for cross product. (Contributed by NM, 21-Dec-2008.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 × 𝐴) = (𝐶 × 𝐵) | ||
Theorem | xpeq12i 4394 | Equality inference for cross product. (Contributed by FL, 31-Aug-2009.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ (𝐴 × 𝐶) = (𝐵 × 𝐷) | ||
Theorem | xpeq1d 4395 | Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐶)) | ||
Theorem | xpeq2d 4396 | Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 × 𝐴) = (𝐶 × 𝐵)) | ||
Theorem | xpeq12d 4397 | Equality deduction for cross product. (Contributed by NM, 8-Dec-2013.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 × 𝐶) = (𝐵 × 𝐷)) | ||
Theorem | nfxp 4398 | 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 4399 | 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 4400 | A member of a cross product (ordered pair) doesn't contain the empty set. (Contributed by NM, 15-Dec-2008.) |
⊢ (𝐶 ∈ (𝐴 × 𝐵) → ¬ ∅ ∈ 𝐶) |
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