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Type | Label | Description |
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Statement | ||
Theorem | elirr 4501 |
No class is a member of itself. Exercise 6 of [TakeutiZaring] p. 22.
The reason that this theorem is marked as discouraged is a bit subtle. If we wanted to reduce usage of ax-setind 4497, we could redefine (df-iord 4327) to also require (df-frind 4293) and in that case any theorem related to irreflexivity of ordinals could use ordirr 4502 (which under that definition would presumably not need ax-setind 4497 to prove it). But since ordinals have not yet been defined that way, we cannot rely on the "don't add additional axiom use" feature of the minimizer to get theorems to use ordirr 4502. To encourage ordirr 4502 when possible, we mark this theorem as discouraged. (Contributed by NM, 7-Aug-1994.) (Proof rewritten by Mario Carneiro and Jim Kingdon, 26-Nov-2018.) (New usage is discouraged.) |
Theorem | ordirr 4502 | Epsilon irreflexivity of ordinals: no ordinal class is a member of itself. Theorem 2.2(i) of [BellMachover] p. 469, generalized to classes. The present proof requires ax-setind 4497. If in the definition of ordinals df-iord 4327, we also required that membership be well-founded on any ordinal (see df-frind 4293), then we could prove ordirr 4502 without ax-setind 4497. (Contributed by NM, 2-Jan-1994.) |
Theorem | onirri 4503 | An ordinal number is not a member of itself. Theorem 7M(c) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.) |
Theorem | nordeq 4504 | A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998.) |
Theorem | ordn2lp 4505 | An ordinal class cannot be an element of one of its members. Variant of first part of Theorem 2.2(vii) of [BellMachover] p. 469. (Contributed by NM, 3-Apr-1994.) |
Theorem | orddisj 4506 | An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.) |
Theorem | orddif 4507 | Ordinal derived from its successor. (Contributed by NM, 20-May-1998.) |
Theorem | elirrv 4508 | The membership relation is irreflexive: no set is a member of itself. Theorem 105 of [Suppes] p. 54. (Contributed by NM, 19-Aug-1993.) |
Theorem | sucprcreg 4509 | A class is equal to its successor iff it is a proper class (assuming the Axiom of Set Induction). (Contributed by NM, 9-Jul-2004.) |
Theorem | ruv 4510 | The Russell class is equal to the universe . Exercise 5 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 4-Oct-2008.) |
Theorem | ruALT 4511 | Alternate proof of Russell's Paradox ru 2936, simplified using (indirectly) the Axiom of Set Induction ax-setind 4497. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | onprc 4512 | 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 4446), 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.) |
Theorem | sucon 4513 | The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.) |
Theorem | en2lp 4514 | 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.) |
Theorem | preleq 4515 | Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.) |
Theorem | opthreg 4516 | Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4497 (via the preleq 4515 step). See df-op 3569 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.) |
Theorem | suc11g 4517 | 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.) |
Theorem | suc11 4518 | The successor operation behaves like a one-to-one function. Compare Exercise 16 of [Enderton] p. 194. (Contributed by NM, 3-Sep-2003.) |
Theorem | dtruex 4519* | At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 4153 can also be summarized as "at least two sets exist", the difference is that dtruarb 4153 shows the existence of two sets which are not equal to each other, but this theorem says that given a specific , we can construct a set which does not equal it. (Contributed by Jim Kingdon, 29-Dec-2018.) |
Theorem | dtru 4520* | 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 4519. (Contributed by Jim Kingdon, 29-Dec-2018.) |
Theorem | eunex 4521 | Existential uniqueness implies there is a value for which the wff argument is false. (Contributed by Jim Kingdon, 29-Dec-2018.) |
Theorem | ordsoexmid 4522 | 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.) |
Theorem | ordsuc 4523 | 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.) |
Theorem | onsucuni2 4524 | A successor ordinal is the successor of its union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | 0elsucexmid 4525* | If the successor of any ordinal class contains the empty set, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2021.) |
Theorem | nlimsucg 4526 | A successor is not a limit ordinal. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | ordpwsucss 4527 |
The collection of ordinals in the power class of an ordinal is a
superset of its successor.
We can think of as another possible definition of successor, which would be equivalent to df-suc 4332 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if then both (onunisuci 4393) and (onuniss2 4472). Constructively and cannot be shown to be equivalent (as proved at ordpwsucexmid 4530). (Contributed by Jim Kingdon, 21-Jul-2019.) |
Theorem | onnmin 4528 | 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.) |
Theorem | ssnel 4529 | 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.) |
Theorem | ordpwsucexmid 4530* | The subset in ordpwsucss 4527 cannot be equality. That is, strengthening it to equality implies excluded middle. (Contributed by Jim Kingdon, 30-Jul-2019.) |
Theorem | ordtri2or2exmid 4531* | Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 29-Aug-2021.) |
Theorem | ontri2orexmidim 4532* | Ordinal trichotomy implies excluded middle. Closed form of ordtri2or2exmid 4531. (Contributed by Jim Kingdon, 26-Aug-2024.) |
DECID | ||
Theorem | onintexmid 4533* | 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.) |
Theorem | zfregfr 4534 | The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.) |
Theorem | ordfr 4535 | Epsilon is well-founded on an ordinal class. (Contributed by NM, 22-Apr-1994.) |
Theorem | ordwe 4536 | Epsilon well-orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.) |
Theorem | wetriext 4537* | A trichotomous well-order is extensional. (Contributed by Jim Kingdon, 26-Sep-2021.) |
Theorem | wessep 4538 | A subset of a set well-ordered by set membership is well-ordered by set membership. (Contributed by Jim Kingdon, 30-Sep-2021.) |
Theorem | reg3exmidlemwe 4539* | Lemma for reg3exmid 4540. Our counterexample satisfies . (Contributed by Jim Kingdon, 3-Oct-2021.) |
Theorem | reg3exmid 4540* | If any inhabited set satisfying df-wetr 4295 for has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Oct-2021.) |
Theorem | dcextest 4541* | If it is decidable whether is a set, then is decidable (where does not occur in ). From this fact, we can deduce (outside the formal system, since we cannot quantify over classes) that if it is decidable whether any class is a set, then "weak excluded middle" (that is, any negated proposition is decidable) holds. (Contributed by Jim Kingdon, 3-Jul-2022.) |
DECID DECID | ||
Theorem | tfi 4542* |
The Principle of Transfinite Induction. Theorem 7.17 of [TakeutiZaring]
p. 39. This principle states that if is a class of ordinal
numbers with the property that every ordinal number included in
also belongs to , then every ordinal number is in .
(Contributed by NM, 18-Feb-2004.) |
Theorem | tfis 4543* | Transfinite Induction Schema. If all ordinal numbers less than a given number 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.) |
Theorem | tfis2f 4544* | Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.) |
Theorem | tfis2 4545* | Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.) |
Theorem | tfis3 4546* | Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 4-Nov-2003.) |
Theorem | tfisi 4547* | 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.) |
Axiom | ax-iinf 4548* | Axiom of Infinity. Axiom 5 of [Crosilla] p. "Axioms of CZF and IZF". (Contributed by Jim Kingdon, 16-Nov-2018.) |
Theorem | zfinf2 4549* | A standard version of the Axiom of Infinity, using definitions to abbreviate. Axiom Inf of [BellMachover] p. 472. (Contributed by NM, 30-Aug-1993.) |
Syntax | com 4550 | Extend class notation to include the class of natural numbers. |
Definition | df-iom 4551* |
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 are a subset of the ordinal numbers df-on 4329. Later, when we define complex numbers, we will be able to also define a subset of the complex numbers (df-inn 8835) 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 7130. (Contributed by NM, 6-Aug-1994.) Use its alias dfom3 4552 instead for naming consistency with set.mm. (New usage is discouraged.) |
Theorem | dfom3 4552* | Alias for df-iom 4551. Use it instead of df-iom 4551 for naming consistency with set.mm. (Contributed by NM, 6-Aug-1994.) |
Theorem | omex 4553 | The existence of omega (the class of natural numbers). Axiom 7 of [TakeutiZaring] p. 43. (Contributed by NM, 6-Aug-1994.) |
Theorem | peano1 4554 | 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.) |
Theorem | peano2 4555 | 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.) |
Theorem | peano3 4556 | 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.) |
Theorem | peano4 4557 | 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.) |
Theorem | peano5 4558* | 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 4563. (Contributed by NM, 18-Feb-2004.) |
Theorem | find 4559* | 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 is a set of natural numbers, zero belongs to , and given any member of the member's successor also belongs to . The conclusion is that every natural number is in . (Contributed by NM, 22-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Theorem | finds 4560* | 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.) |
Theorem | finds2 4561* | 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.) |
Theorem | finds1 4562* | 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.) |
Theorem | findes 4563 | 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.) |
Theorem | nn0suc 4564* | 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.) |
Theorem | elomssom 4565 | 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 4566. (Revised by BJ, 7-Aug-2024.) |
Theorem | elnn 4566 | A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.) |
Theorem | ordom 4567 | Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.) |
Theorem | omelon2 4568 | Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.) |
Theorem | omelon 4569 | Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.) |
Theorem | nnon 4570 | A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.) |
Theorem | nnoni 4571 | A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.) |
Theorem | nnord 4572 | A natural number is ordinal. (Contributed by NM, 17-Oct-1995.) |
Theorem | omsson 4573 | Omega is a subset of . (Contributed by NM, 13-Jun-1994.) |
Theorem | limom 4574 | 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.) |
Theorem | peano2b 4575 | A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.) |
Theorem | nnsuc 4576* | A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.) |
Theorem | nnsucpred 4577 | The successor of the precedessor of a nonzero natural number. (Contributed by Jim Kingdon, 31-Jul-2022.) |
Theorem | nndceq0 4578 | 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.) |
DECID | ||
Theorem | 0elnn 4579 | A natural number is either the empty set or has the empty set as an element. (Contributed by Jim Kingdon, 23-Aug-2019.) |
Theorem | nn0eln0 4580 | 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.) |
Theorem | nnregexmid 4581* | If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4495 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6447 or nntri3or 6441), sets of natural numbers are a different animal. (Contributed by Jim Kingdon, 6-Sep-2019.) |
Theorem | omsinds 4582* | Strong (or "total") induction principle over . (Contributed by Scott Fenton, 17-Jul-2015.) |
Theorem | nnpredcl 4583 | 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 4524) but also holds when it is by uni0 3800. (Contributed by Jim Kingdon, 31-Jul-2022.) |
Theorem | nnpredlt 4584 | The predecessor (see nnpredcl 4583) of a nonzero natural number is less than (see df-iord 4327) that number. (Contributed by Jim Kingdon, 14-Sep-2024.) |
Syntax | cxp 4585 | Extend the definition of a class to include the cross product. |
Syntax | ccnv 4586 | Extend the definition of a class to include the converse of a class. |
Syntax | cdm 4587 | Extend the definition of a class to include the domain of a class. |
Syntax | crn 4588 | Extend the definition of a class to include the range of a class. |
Syntax | cres 4589 | Extend the definition of a class to include the restriction of a class. (Read: The restriction of to .) |
Syntax | cima 4590 | Extend the definition of a class to include the image of a class. (Read: The image of under .) |
Syntax | ccom 4591 | Extend the definition of a class to include the composition of two classes. (Read: The composition of and .) |
Syntax | wrel 4592 | Extend the definition of a wff to include the relation predicate. (Read: is a relation.) |
Definition | df-xp 4593* | 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, . Another example is that the set of rational numbers is defined using the Cartesian product as ; 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.) |
Definition | df-rel 4594 | Define the relation predicate. Definition 6.4(1) of [TakeutiZaring] p. 23. For alternate definitions, see dfrel2 5037 and dfrel3 5044. (Contributed by NM, 1-Aug-1994.) |
Definition | df-cnv 4595* |
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
and then , as proven in brcnv 4770
(see df-br 3967 and df-rel 4594 for more on relations). For example,
.
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.) |
Definition | df-co 4596* | Define the composition of two classes. Definition 6.6(3) of [TakeutiZaring] p. 24. Note that Definition 7 of [Suppes] p. 63 reverses and , uses a slash instead of , and calls the operation "relative product". (Contributed by NM, 4-Jul-1994.) |
Definition | df-dm 4597* | 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 4598). For alternate definitions see dfdm2 5121, dfdm3 4774, and dfdm4 4779. The notation " " is used by Enderton; other authors sometimes use script D. (Contributed by NM, 1-Aug-1994.) |
Definition | df-rn 4598 | Define the range of a class. For example, F = { 2 , 6 , 3 , 9 } -> ran F = { 6 , 9 } . Contrast with domain (defined in df-dm 4597). For alternate definitions, see dfrn2 4775, dfrn3 4776, and dfrn4 5047. The notation " " is used by Enderton; other authors sometimes use script R or script W. (Contributed by NM, 1-Aug-1994.) |
Definition | df-res 4599 | Define the restriction of a class. Definition 6.6(1) of [TakeutiZaring] p. 24. For example, . We do not introduce a special syntax for the corestriction of a class: it will be expressed either as the intersection or as the converse of the restricted converse. (Contributed by NM, 2-Aug-1994.) |
Definition | df-ima 4600 | 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 4599) and range (df-rn 4598). For an alternate definition, see dfima2 4931. (Contributed by NM, 2-Aug-1994.) |
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