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Type | Label | Description |
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Statement | ||
Theorem | onsucelsucexmid 4301* |
The converse of onsucelsucr 4280 implies excluded middle. On the other
hand, if ![]() |
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Theorem | ordsucunielexmid 4302* |
The converse of sucunielr 4282 (where ![]() |
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Theorem | regexmidlemm 4303* |
Lemma for regexmid 4306. ![]() |
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Theorem | regexmidlem1 4304* |
Lemma for regexmid 4306. If ![]() |
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Theorem | reg2exmidlema 4305* |
Lemma for reg2exmid 4307. If ![]() ![]() |
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Theorem | regexmid 4306* |
The axiom of foundation implies excluded middle.
By foundation (or regularity), we mean the principle that every
inhabited set has an element which is minimal (when arranged by
For this reason, IZF does not adopt foundation as an axiom and instead replaces it with ax-setind 4308. (Contributed by Jim Kingdon, 3-Sep-2019.) |
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Theorem | reg2exmid 4307* |
If any inhabited set has a minimal element (when expressed by ![]() |
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Axiom | ax-setind 4308* |
Axiom of ![]() For more on axioms which might be adopted which are incompatible with this axiom (that is, Non-wellfounded Set Theory but in the absence of excluded middle), see Chapter 20 of [AczelRathjen], p. 183. (Contributed by Jim Kingdon, 19-Oct-2018.) |
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Theorem | setindel 4309* |
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Theorem | setind 4310* | Set (epsilon) induction. Theorem 5.22 of [TakeutiZaring] p. 21. (Contributed by NM, 17-Sep-2003.) |
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Theorem | setind2 4311 | Set (epsilon) induction, stated compactly. Given as a homework problem in 1992 by George Boolos (1940-1996). (Contributed by NM, 17-Sep-2003.) |
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Theorem | elirr 4312 |
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 4308, we could redefine
(Contributed by NM, 7-Aug-1994.) (Proof rewritten by Mario Carneiro and Jim Kingdon, 26-Nov-2018.) (New usage is discouraged.) |
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Theorem | ordirr 4313 | 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 4308. If in the definition of ordinals df-iord 4149, we also required that membership be well-founded on any ordinal (see df-frind 4115), then we could prove ordirr 4313 without ax-setind 4308. (Contributed by NM, 2-Jan-1994.) |
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Theorem | onirri 4314 | An ordinal number is not a member of itself. Theorem 7M(c) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.) |
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Theorem | nordeq 4315 | A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998.) |
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Theorem | ordn2lp 4316 | 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.) |
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Theorem | orddisj 4317 | An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.) |
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Theorem | orddif 4318 | Ordinal derived from its successor. (Contributed by NM, 20-May-1998.) |
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Theorem | elirrv 4319 | The membership relation is irreflexive: no set is a member of itself. Theorem 105 of [Suppes] p. 54. (Contributed by NM, 19-Aug-1993.) |
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Theorem | sucprcreg 4320 | 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.) |
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Theorem | ruv 4321 |
The Russell class is equal to the universe ![]() |
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Theorem | ruALT 4322 | Alternate proof of Russell's Paradox ru 2823, simplified using (indirectly) the Axiom of Set Induction ax-setind 4308. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
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Theorem | onprc 4323 | 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 4258), 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.) |
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Theorem | sucon 4324 | The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.) |
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Theorem | en2lp 4325 | 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.) |
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Theorem | preleq 4326 | Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.) |
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Theorem | opthreg 4327 | Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4308 (via the preleq 4326 step). See df-op 3425 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.) |
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Theorem | suc11g 4328 | 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.) |
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Theorem | suc11 4329 | The successor operation behaves like a one-to-one function. Compare Exercise 16 of [Enderton] p. 194. (Contributed by NM, 3-Sep-2003.) |
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Theorem | dtruex 4330* |
At least two sets exist (or in terms of first-order logic, the universe
of discourse has two or more objects). Although dtruarb 3982 can also be
summarized as "at least two sets exist", the difference is
that
dtruarb 3982 shows the existence of two sets which are not
equal to each
other, but this theorem says that given a specific ![]() ![]() |
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Theorem | dtru 4331* | 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 4330. (Contributed by Jim Kingdon, 29-Dec-2018.) |
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Theorem | eunex 4332 | Existential uniqueness implies there is a value for which the wff argument is false. (Contributed by Jim Kingdon, 29-Dec-2018.) |
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Theorem | ordsoexmid 4333 | 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.) |
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Theorem | ordsuc 4334 | 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.) |
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Theorem | onsucuni2 4335 | A successor ordinal is the successor of its union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
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Theorem | 0elsucexmid 4336* | If the successor of any ordinal class contains the empty set, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2021.) |
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Theorem | nlimsucg 4337 | A successor is not a limit ordinal. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
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Theorem | ordpwsucss 4338 |
The collection of ordinals in the power class of an ordinal is a
superset of its successor.
We can think of
Constructively |
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Theorem | onnmin 4339 | 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.) |
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Theorem | ssnel 4340 | 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.) |
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Theorem | ordpwsucexmid 4341* | The subset in ordpwsucss 4338 cannot be equality. That is, strengthening it to equality implies excluded middle. (Contributed by Jim Kingdon, 30-Jul-2019.) |
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Theorem | ordtri2or2exmid 4342* | Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 29-Aug-2021.) |
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Theorem | onintexmid 4343* | 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.) |
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Theorem | zfregfr 4344 | The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.) |
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Theorem | ordfr 4345 | Epsilon is well-founded on an ordinal class. (Contributed by NM, 22-Apr-1994.) |
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Theorem | ordwe 4346 | Epsilon well-orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.) |
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Theorem | wetriext 4347* | A trichotomous well-order is extensional. (Contributed by Jim Kingdon, 26-Sep-2021.) |
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Theorem | wessep 4348 | A subset of a set well-ordered by set membership is well-ordered by set membership. (Contributed by Jim Kingdon, 30-Sep-2021.) |
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Theorem | reg3exmidlemwe 4349* |
Lemma for reg3exmid 4350. Our counterexample ![]() ![]() |
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Theorem | reg3exmid 4350* |
If any inhabited set satisfying df-wetr 4117 for ![]() |
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Theorem | tfi 4351* |
The Principle of Transfinite Induction. Theorem 7.17 of [TakeutiZaring]
p. 39. This principle states that if ![]() ![]() ![]() ![]() (Contributed by NM, 18-Feb-2004.) |
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Theorem | tfis 4352* |
Transfinite Induction Schema. If all ordinal numbers less than a given
number ![]() |
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Theorem | tfis2f 4353* | Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.) |
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Theorem | tfis2 4354* | Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.) |
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Theorem | tfis3 4355* | Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 4-Nov-2003.) |
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Theorem | tfisi 4356* | 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.) |
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Axiom | ax-iinf 4357* | Axiom of Infinity. Axiom 5 of [Crosilla] p. "Axioms of CZF and IZF". (Contributed by Jim Kingdon, 16-Nov-2018.) |
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Theorem | zfinf2 4358* | A standard version of the Axiom of Infinity, using definitions to abbreviate. Axiom Inf of [BellMachover] p. 472. (Contributed by NM, 30-Aug-1993.) |
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Syntax | com 4359 | Extend class notation to include the class of natural numbers. |
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Definition | df-iom 4360* |
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 |
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Theorem | dfom3 4361* | Alias for df-iom 4360. Use it instead of df-iom 4360 for naming consistency with set.mm. (Contributed by NM, 6-Aug-1994.) |
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Theorem | omex 4362 | The existence of omega (the class of natural numbers). Axiom 7 of [TakeutiZaring] p. 43. (Contributed by NM, 6-Aug-1994.) |
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Theorem | peano1 4363 | 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.) |
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Theorem | peano2 4364 | 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.) |
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Theorem | peano3 4365 | 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.) |
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Theorem | peano4 4366 | 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.) |
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Theorem | peano5 4367* | 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 4372. (Contributed by NM, 18-Feb-2004.) |
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Theorem | find 4368* |
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 ![]() ![]() ![]() ![]() ![]() |
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Theorem | finds 4369* | 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.) |
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Theorem | finds2 4370* | 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.) |
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Theorem | finds1 4371* | 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.) |
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Theorem | findes 4372 | 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.) |
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Theorem | nn0suc 4373* | 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.) |
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Theorem | elnn 4374 | A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.) |
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Theorem | ordom 4375 | Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.) |
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Theorem | omelon2 4376 | Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.) |
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Theorem | omelon 4377 | Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.) |
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Theorem | nnon 4378 | A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.) |
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Theorem | nnoni 4379 | A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.) |
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Theorem | nnord 4380 | A natural number is ordinal. (Contributed by NM, 17-Oct-1995.) |
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Theorem | omsson 4381 |
Omega is a subset of ![]() |
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Theorem | limom 4382 | 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.) |
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Theorem | peano2b 4383 | A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.) |
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Theorem | nnsuc 4384* | A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.) |
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Theorem | nndceq0 4385 | 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.) |
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Theorem | 0elnn 4386 | A natural number is either the empty set or has the empty set as an element. (Contributed by Jim Kingdon, 23-Aug-2019.) |
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Theorem | nn0eln0 4387 | 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.) |
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Theorem | nnregexmid 4388* | If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4306 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6164 or nntri3or 6158), sets of natural numbers are a different animal. (Contributed by Jim Kingdon, 6-Sep-2019.) |
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Syntax | cxp 4389 | Extend the definition of a class to include the cross product. |
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Syntax | ccnv 4390 | Extend the definition of a class to include the converse of a class. |
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Syntax | cdm 4391 | Extend the definition of a class to include the domain of a class. |
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Syntax | crn 4392 | Extend the definition of a class to include the range of a class. |
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Syntax | cres 4393 |
Extend the definition of a class to include the restriction of a class.
(Read: The restriction of ![]() ![]() |
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Syntax | cima 4394 |
Extend the definition of a class to include the image of a class. (Read:
The image of ![]() ![]() |
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Syntax | ccom 4395 |
Extend the definition of a class to include the composition of two
classes. (Read: The composition of ![]() ![]() |
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Syntax | wrel 4396 |
Extend the definition of a wff to include the relation predicate. (Read:
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Definition | df-xp 4397* |
Define the cross product of two classes. Definition 9.11 of [Quine]
p. 64. For example, ( { 1 , 5 } ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Definition | df-rel 4398 | Define the relation predicate. Definition 6.4(1) of [TakeutiZaring] p. 23. For alternate definitions, see dfrel2 4821 and dfrel3 4828. (Contributed by NM, 1-Aug-1994.) |
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Definition | df-cnv 4399* |
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 ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Definition | df-co 4400* |
Define the composition of two classes. Definition 6.6(3) of
[TakeutiZaring] p. 24. Note that
Definition 7 of [Suppes] p. 63
reverses ![]() ![]() ![]() |
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