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Type | Label | Description |
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Statement | ||
Theorem | onintexmid 4401* | 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 4402 | The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.) |
Theorem | ordfr 4403 | Epsilon is well-founded on an ordinal class. (Contributed by NM, 22-Apr-1994.) |
Theorem | ordwe 4404 | Epsilon well-orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.) |
Theorem | wetriext 4405* | A trichotomous well-order is extensional. (Contributed by Jim Kingdon, 26-Sep-2021.) |
Theorem | wessep 4406 | 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 4407* | Lemma for reg3exmid 4408. Our counterexample satisfies . (Contributed by Jim Kingdon, 3-Oct-2021.) |
Theorem | reg3exmid 4408* | If any inhabited set satisfying df-wetr 4170 for has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Oct-2021.) |
Theorem | dcextest 4409* | 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 4410* |
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 4411* | 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 4412* | Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.) |
Theorem | tfis2 4413* | Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.) |
Theorem | tfis3 4414* | Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 4-Nov-2003.) |
Theorem | tfisi 4415* | 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 4416* | Axiom of Infinity. Axiom 5 of [Crosilla] p. "Axioms of CZF and IZF". (Contributed by Jim Kingdon, 16-Nov-2018.) |
Theorem | zfinf2 4417* | 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 4418 | Extend class notation to include the class of natural numbers. |
Definition | df-iom 4419* |
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 4204. 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 4420 instead for naming consistency with set.mm. (New usage is discouraged.) |
Theorem | dfom3 4420* | Alias for df-iom 4419. Use it instead of df-iom 4419 for naming consistency with set.mm. (Contributed by NM, 6-Aug-1994.) |
Theorem | omex 4421 | The existence of omega (the class of natural numbers). Axiom 7 of [TakeutiZaring] p. 43. (Contributed by NM, 6-Aug-1994.) |
Theorem | peano1 4422 | 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 4423 | 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 4424 | 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 4425 | 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 4426* | 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 4431. (Contributed by NM, 18-Feb-2004.) |
Theorem | find 4427* | 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 4428* | 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 4429* | 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 4430* | 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 4431 | 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 4432* | 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 | elnn 4433 | A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.) |
Theorem | ordom 4434 | Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.) |
Theorem | omelon2 4435 | Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.) |
Theorem | omelon 4436 | Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.) |
Theorem | nnon 4437 | A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.) |
Theorem | nnoni 4438 | A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.) |
Theorem | nnord 4439 | A natural number is ordinal. (Contributed by NM, 17-Oct-1995.) |
Theorem | omsson 4440 | Omega is a subset of . (Contributed by NM, 13-Jun-1994.) |
Theorem | limom 4441 | 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 4442 | A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.) |
Theorem | nnsuc 4443* | A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.) |
Theorem | nndceq0 4444 | 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 4445 | 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 4446 | 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 4447* | If inhabited sets of natural numbers always have minimal elements, excluded middle follows. The argument is essentially the same as regexmid 4364 and the larger lesson is that although natural numbers may behave "non-constructively" even in a constructive set theory (for example see nndceq 6274 or nntri3or 6268), sets of natural numbers are a different animal. (Contributed by Jim Kingdon, 6-Sep-2019.) |
Theorem | omsinds 4448* | Strong (or "total") induction principle over . (Contributed by Scott Fenton, 17-Jul-2015.) |
Theorem | nnpredcl 4449 | 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 4393) but also holds when it is by uni0 3686. (Contributed by Jim Kingdon, 31-Jul-2022.) |
Syntax | cxp 4450 | Extend the definition of a class to include the cross product. |
Syntax | ccnv 4451 | Extend the definition of a class to include the converse of a class. |
Syntax | cdm 4452 | Extend the definition of a class to include the domain of a class. |
Syntax | crn 4453 | Extend the definition of a class to include the range of a class. |
Syntax | cres 4454 | Extend the definition of a class to include the restriction of a class. (Read: The restriction of to .) |
Syntax | cima 4455 | Extend the definition of a class to include the image of a class. (Read: The image of under .) |
Syntax | ccom 4456 | Extend the definition of a class to include the composition of two classes. (Read: The composition of and .) |
Syntax | wrel 4457 | Extend the definition of a wff to include the relation predicate. (Read: is a relation.) |
Definition | df-xp 4458* | 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.) |
Definition | df-rel 4459 | Define the relation predicate. Definition 6.4(1) of [TakeutiZaring] p. 23. For alternate definitions, see dfrel2 4894 and dfrel3 4901. (Contributed by NM, 1-Aug-1994.) |
Definition | df-cnv 4460* | 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 4632 (see df-br 3852 and df-rel 4459 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.) |
Definition | df-co 4461* | 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 4462* | 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 4463). For alternate definitions see dfdm2 4978, dfdm3 4636, and dfdm4 4641. The notation " " is used by Enderton; other authors sometimes use script D. (Contributed by NM, 1-Aug-1994.) |
Definition | df-rn 4463 | Define the range of a class. For example, F = { 2 , 6 , 3 , 9 } -> ran F = { 6 , 9 } . Contrast with domain (defined in df-dm 4462). For alternate definitions, see dfrn2 4637, dfrn3 4638, and dfrn4 4904. The notation " " is used by Enderton; other authors sometimes use script R or script W. (Contributed by NM, 1-Aug-1994.) |
Definition | df-res 4464 | 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.) |
Definition | df-ima 4465 | 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 4464) and range (df-rn 4463). For an alternate definition, see dfima2 4789. (Contributed by NM, 2-Aug-1994.) |
Theorem | xpeq1 4466 | Equality theorem for cross product. (Contributed by NM, 4-Jul-1994.) |
Theorem | xpeq2 4467 | Equality theorem for cross product. (Contributed by NM, 5-Jul-1994.) |
Theorem | elxpi 4468* | Membership in a cross product. Uses fewer axioms than elxp 4469. (Contributed by NM, 4-Jul-1994.) |
Theorem | elxp 4469* | Membership in a cross product. (Contributed by NM, 4-Jul-1994.) |
Theorem | elxp2 4470* | Membership in a cross product. (Contributed by NM, 23-Feb-2004.) |
Theorem | xpeq12 4471 | Equality theorem for cross product. (Contributed by FL, 31-Aug-2009.) |
Theorem | xpeq1i 4472 | Equality inference for cross product. (Contributed by NM, 21-Dec-2008.) |
Theorem | xpeq2i 4473 | Equality inference for cross product. (Contributed by NM, 21-Dec-2008.) |
Theorem | xpeq12i 4474 | Equality inference for cross product. (Contributed by FL, 31-Aug-2009.) |
Theorem | xpeq1d 4475 | Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.) |
Theorem | xpeq2d 4476 | Equality deduction for cross product. (Contributed by Jeff Madsen, 17-Jun-2010.) |
Theorem | xpeq12d 4477 | Equality deduction for cross product. (Contributed by NM, 8-Dec-2013.) |
Theorem | nfxp 4478 | Bound-variable hypothesis builder for cross product. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Theorem | 0nelxp 4479 | The empty set is not a member of a cross product. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Theorem | 0nelelxp 4480 | A member of a cross product (ordered pair) doesn't contain the empty set. (Contributed by NM, 15-Dec-2008.) |
Theorem | opelxp 4481 | 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.) |
Theorem | brxp 4482 | Binary relation on a cross product. (Contributed by NM, 22-Apr-2004.) |
Theorem | opelxpi 4483 | Ordered pair membership in a cross product (implication). (Contributed by NM, 28-May-1995.) |
Theorem | opelxpd 4484 | Ordered pair membership in a Cartesian product, deduction form. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Theorem | opelxp1 4485 | 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.) |
Theorem | opelxp2 4486 | 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.) |
Theorem | otelxp1 4487 | 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.) |
Theorem | rabxp 4488* | Membership in a class builder restricted to a cross product. (Contributed by NM, 20-Feb-2014.) |
Theorem | brrelex12 4489 | 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.) |
Theorem | brrelex1 4490 | 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.) |
Theorem | brrelex 4491 | 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.) |
Theorem | brrelex2 4492 | 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.) |
Theorem | brrelex12i 4493 | Two classes that are related by a binary relation are sets. (An artifact of our ordered pair definition.) (Contributed by BJ, 3-Oct-2022.) |
Theorem | brrelex1i 4494 | The first argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by NM, 4-Jun-1998.) |
Theorem | brrelex2i 4495 | The second argument of a binary relation exists. (An artifact of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.) |
Theorem | nprrel 4496 | No proper class is related to anything via any relation. (Contributed by Roy F. Longton, 30-Jul-2005.) |
Theorem | 0nelrel 4497 | A binary relation does not contain the empty set. (Contributed by AV, 15-Nov-2021.) |
Theorem | fconstmpt 4498* | Representation of a constant function using the mapping operation. (Note that cannot appear free in .) (Contributed by NM, 12-Oct-1999.) (Revised by Mario Carneiro, 16-Nov-2013.) |
Theorem | vtoclr 4499* | Variable to class conversion of transitive relation. (Contributed by NM, 9-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Theorem | opelvvg 4500 | Ordered pair membership in the universal class of ordered pairs. (Contributed by Mario Carneiro, 3-May-2015.) |
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