HomeHome Intuitionistic Logic Explorer
Theorem List (p. 46 of 142)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 4501-4600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremonintonm 4501* The intersection of an inhabited collection of ordinal numbers is an ordinal number. Compare Exercise 6 of [TakeutiZaring] p. 44. (Contributed by Mario Carneiro and Jim Kingdon, 30-Aug-2021.)
 |-  ( ( A  C_  On  /\  E. x  x  e.  A )  ->  |^| A  e.  On )
 
Theoremonintrab2im 4502 An existence condition which implies an intersection is an ordinal number. (Contributed by Jim Kingdon, 30-Aug-2021.)
 |-  ( E. x  e. 
 On  ph  ->  |^| { x  e.  On  |  ph }  e.  On )
 
Theoremordtriexmidlem 4503 Lemma for decidability and ordinals. The set  { x  e.  { (/) }  |  ph } is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4505 or weak linearity in ordsoexmid 4546) with a proposition  ph. Our lemma states that it is an ordinal number. (Contributed by Jim Kingdon, 28-Jan-2019.)
 |- 
 { x  e.  { (/)
 }  |  ph }  e.  On
 
Theoremordtriexmidlem2 4504* Lemma for decidability and ordinals. The set  { x  e.  { (/) }  |  ph } is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4505 or weak linearity in ordsoexmid 4546) with a proposition  ph. Our lemma helps connect that set to excluded middle. (Contributed by Jim Kingdon, 28-Jan-2019.)
 |-  ( { x  e. 
 { (/) }  |  ph }  =  (/)  ->  -.  ph )
 
Theoremordtriexmid 4505* Ordinal trichotomy implies the law of the excluded middle (that is, decidability of an arbitrary proposition).

This theorem is stated in "Constructive ordinals", [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic".

Also see exmidontri 7216 which is much the same theorem but biconditionalized and using the EXMID notation. (Contributed by Mario Carneiro and Jim Kingdon, 14-Nov-2018.)

 |- 
 A. x  e.  On  A. y  e.  On  ( x  e.  y  \/  x  =  y  \/  y  e.  x )   =>    |-  ( ph  \/  -.  ph )
 
Theoremontriexmidim 4506* Ordinal trichotomy implies excluded middle. Closed form of ordtriexmid 4505. (Contributed by Jim Kingdon, 26-Aug-2024.)
 |-  ( A. x  e. 
 On  A. y  e.  On  ( x  e.  y  \/  x  =  y  \/  y  e.  x )  -> DECID  ph )
 
Theoremordtri2orexmid 4507* Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 31-Jul-2019.)
 |- 
 A. x  e.  On  A. y  e.  On  ( x  e.  y  \/  y  C_  x )   =>    |-  ( ph  \/  -.  ph )
 
Theorem2ordpr 4508 Version of 2on 6404 with the definition of  2o expanded and expressed in terms of  Ord. (Contributed by Jim Kingdon, 29-Aug-2021.)
 |- 
 Ord  { (/) ,  { (/) } }
 
Theoremontr2exmid 4509* An ordinal transitivity law which implies excluded middle. (Contributed by Jim Kingdon, 17-Sep-2021.)
 |- 
 A. x  e.  On  A. y A. z  e. 
 On  ( ( x 
 C_  y  /\  y  e.  z )  ->  x  e.  z )   =>    |-  ( ph  \/  -.  ph )
 
Theoremordtri2or2exmidlem 4510* A set which is  2o if  ph or  (/) if  -.  ph is an ordinal. (Contributed by Jim Kingdon, 29-Aug-2021.)
 |- 
 { x  e.  { (/)
 ,  { (/) } }  |  ph }  e.  On
 
Theoremonsucsssucexmid 4511* The converse of onsucsssucr 4493 implies excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.)
 |- 
 A. x  e.  On  A. y  e.  On  ( x  C_  y  ->  suc  x  C_ 
 suc  y )   =>    |-  ( ph  \/  -.  ph )
 
Theoremonsucelsucexmidlem1 4512* Lemma for onsucelsucexmid 4514. (Contributed by Jim Kingdon, 2-Aug-2019.)
 |-  (/)  e.  { x  e. 
 { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }
 
Theoremonsucelsucexmidlem 4513* Lemma for onsucelsucexmid 4514. The set  { x  e. 
{ (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) } appears as  A in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5844), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4503. (Contributed by Jim Kingdon, 2-Aug-2019.)
 |- 
 { x  e.  { (/)
 ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }  e.  On
 
Theoremonsucelsucexmid 4514* The converse of onsucelsucr 4492 implies excluded middle. On the other hand, if  y is constrained to be a natural number, instead of an arbitrary ordinal, then the converse of onsucelsucr 4492 does hold, as seen at nnsucelsuc 6470. (Contributed by Jim Kingdon, 2-Aug-2019.)
 |- 
 A. x  e.  On  A. y  e.  On  ( x  e.  y  ->  suc 
 x  e.  suc  y
 )   =>    |-  ( ph  \/  -.  ph )
 
Theoremordsucunielexmid 4515* The converse of sucunielr 4494 (where  B is an ordinal) implies excluded middle. (Contributed by Jim Kingdon, 2-Aug-2019.)
 |- 
 A. x  e.  On  A. y  e.  On  ( x  e.  U. y  ->  suc  x  e.  y )   =>    |-  ( ph  \/  -.  ph )
 
2.5  IZF Set Theory - add the Axiom of Set Induction
 
2.5.1  The ZF Axiom of Foundation would imply Excluded Middle
 
Theoremregexmidlemm 4516* Lemma for regexmid 4519. 
A is inhabited. (Contributed by Jim Kingdon, 3-Sep-2019.)
 |-  A  =  { x  e.  { (/) ,  { (/) } }  |  ( x  =  { (/)
 }  \/  ( x  =  (/)  /\  ph ) ) }   =>    |- 
 E. y  y  e.  A
 
Theoremregexmidlem1 4517* Lemma for regexmid 4519. If  A has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2019.)
 |-  A  =  { x  e.  { (/) ,  { (/) } }  |  ( x  =  { (/)
 }  \/  ( x  =  (/)  /\  ph ) ) }   =>    |-  ( E. y ( y  e.  A  /\  A. z ( z  e.  y  ->  -.  z  e.  A ) )  ->  ( ph  \/  -.  ph ) )
 
Theoremreg2exmidlema 4518* Lemma for reg2exmid 4520. If  A has a minimal element (expressed by  C_), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.)
 |-  A  =  { x  e.  { (/) ,  { (/) } }  |  ( x  =  { (/)
 }  \/  ( x  =  (/)  /\  ph ) ) }   =>    |-  ( E. u  e.  A  A. v  e.  A  u  C_  v  ->  ( ph  \/  -.  ph ) )
 
Theoremregexmid 4519* 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  e.). The statement of foundation here is taken from Metamath Proof Explorer's ax-reg, and is identical (modulo one unnecessary quantifier) to the statement of foundation in Theorem "Foundation implies instances of EM" of [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic".

For this reason, IZF does not adopt foundation as an axiom and instead replaces it with ax-setind 4521. (Contributed by Jim Kingdon, 3-Sep-2019.)

 |-  ( E. y  y  e.  x  ->  E. y
 ( y  e.  x  /\  A. z ( z  e.  y  ->  -.  z  e.  x ) ) )   =>    |-  ( ph  \/  -.  ph )
 
Theoremreg2exmid 4520* If any inhabited set has a minimal element (when expressed by  C_), excluded middle follows. (Contributed by Jim Kingdon, 2-Oct-2021.)
 |- 
 A. z ( E. w  w  e.  z  ->  E. x  e.  z  A. y  e.  z  x  C_  y )   =>    |-  ( ph  \/  -.  ph )
 
2.5.2  Introduce the Axiom of Set Induction
 
Axiomax-setind 4521* Axiom of  e.-Induction (also known as set induction). An axiom of Intuitionistic Zermelo-Fraenkel set theory. Axiom 9 of [Crosilla] p. "Axioms of CZF and IZF". This replaces the Axiom of Foundation (also called Regularity) from Zermelo-Fraenkel set theory.

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.)

 |-  ( A. a (
 A. y  e.  a  [ y  /  a ] ph  ->  ph )  ->  A. a ph )
 
Theoremsetindel 4522*  e.-Induction in terms of membership in a class. (Contributed by Mario Carneiro and Jim Kingdon, 22-Oct-2018.)
 |-  ( A. x (
 A. y ( y  e.  x  ->  y  e.  S )  ->  x  e.  S )  ->  S  =  _V )
 
Theoremsetind 4523* Set (epsilon) induction. Theorem 5.22 of [TakeutiZaring] p. 21. (Contributed by NM, 17-Sep-2003.)
 |-  ( A. x ( x  C_  A  ->  x  e.  A )  ->  A  =  _V )
 
Theoremsetind2 4524 Set (epsilon) induction, stated compactly. Given as a homework problem in 1992 by George Boolos (1940-1996). (Contributed by NM, 17-Sep-2003.)
 |-  ( ~P A  C_  A  ->  A  =  _V )
 
Theoremelirr 4525 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 4521, we could redefine  Ord  A (df-iord 4351) to also require  _E 
Fr  A (df-frind 4317) and in that case any theorem related to irreflexivity of ordinals could use ordirr 4526 (which under that definition would presumably not need ax-setind 4521 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 4526. To encourage ordirr 4526 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.)

 |- 
 -.  A  e.  A
 
Theoremordirr 4526 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 4521. If in the definition of ordinals df-iord 4351, we also required that membership be well-founded on any ordinal (see df-frind 4317), then we could prove ordirr 4526 without ax-setind 4521. (Contributed by NM, 2-Jan-1994.)
 |-  ( Ord  A  ->  -.  A  e.  A )
 
Theoremonirri 4527 An ordinal number is not a member of itself. Theorem 7M(c) of [Enderton] p. 192. (Contributed by NM, 11-Jun-1994.)
 |-  A  e.  On   =>    |-  -.  A  e.  A
 
Theoremnordeq 4528 A member of an ordinal class is not equal to it. (Contributed by NM, 25-May-1998.)
 |-  ( ( Ord  A  /\  B  e.  A ) 
 ->  A  =/=  B )
 
Theoremordn2lp 4529 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.)
 |-  ( Ord  A  ->  -.  ( A  e.  B  /\  B  e.  A ) )
 
Theoremorddisj 4530 An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.)
 |-  ( Ord  A  ->  ( A  i^i  { A } )  =  (/) )
 
Theoremorddif 4531 Ordinal derived from its successor. (Contributed by NM, 20-May-1998.)
 |-  ( Ord  A  ->  A  =  ( suc  A  \  { A } )
 )
 
Theoremelirrv 4532 The membership relation is irreflexive: no set is a member of itself. Theorem 105 of [Suppes] p. 54. (Contributed by NM, 19-Aug-1993.)
 |- 
 -.  x  e.  x
 
Theoremsucprcreg 4533 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.)
 |-  ( -.  A  e.  _V  <->  suc 
 A  =  A )
 
Theoremruv 4534 The Russell class is equal to the universe  _V. Exercise 5 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 4-Oct-2008.)
 |- 
 { x  |  x  e/  x }  =  _V
 
TheoremruALT 4535 Alternate proof of Russell's Paradox ru 2954, simplified using (indirectly) the Axiom of Set Induction ax-setind 4521. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 { x  |  x  e/  x }  e/  _V
 
Theoremonprc 4536 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 4470), 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  e.  _V
 
Theoremsucon 4537 The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.)
 |- 
 suc  On  =  On
 
Theoremen2lp 4538 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  e.  B  /\  B  e.  A )
 
Theorempreleq 4539 Equality of two unordered pairs when one member of each pair contains the other member. (Contributed by NM, 16-Oct-1996.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( ( ( A  e.  B  /\  C  e.  D )  /\  { A ,  B }  =  { C ,  D } )  ->  ( A  =  C  /\  B  =  D ) )
 
Theoremopthreg 4540 Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4521 (via the preleq 4539 step). See df-op 3592 for a description of other ordered pair representations. Exercise 34 of [Enderton] p. 207. (Contributed by NM, 16-Oct-1996.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( { A ,  { A ,  B } }  =  { C ,  { C ,  D } }  <->  ( A  =  C  /\  B  =  D ) )
 
Theoremsuc11g 4541 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  e.  V  /\  B  e.  W )  ->  ( suc  A  =  suc  B  <->  A  =  B ) )
 
Theoremsuc11 4542 The successor operation behaves like a one-to-one function. Compare Exercise 16 of [Enderton] p. 194. (Contributed by NM, 3-Sep-2003.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( suc  A  =  suc  B  <->  A  =  B ) )
 
Theoremdtruex 4543* At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 4177 can also be summarized as "at least two sets exist", the difference is that dtruarb 4177 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.)
 |- 
 E. x  -.  x  =  y
 
Theoremdtru 4544* 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 4543. (Contributed by Jim Kingdon, 29-Dec-2018.)
 |- 
 -.  A. x  x  =  y
 
Theoremeunex 4545 Existential uniqueness implies there is a value for which the wff argument is false. (Contributed by Jim Kingdon, 29-Dec-2018.)
 |-  ( E! x ph  ->  E. x  -.  ph )
 
Theoremordsoexmid 4546 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   =>    |-  ( ph  \/  -.  ph )
 
Theoremordsuc 4547 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 )
 
Theoremonsucuni2 4548 A successor ordinal is the successor of its union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( ( A  e.  On  /\  A  =  suc  B )  ->  suc  U. A  =  A )
 
Theorem0elsucexmid 4549* If the successor of any ordinal class contains the empty set, excluded middle follows. (Contributed by Jim Kingdon, 3-Sep-2021.)
 |- 
 A. x  e.  On  (/) 
 e.  suc  x   =>    |-  ( ph  \/  -.  ph )
 
Theoremnlimsucg 4550 A successor is not a limit ordinal. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
 |-  ( A  e.  V  ->  -.  Lim  suc  A )
 
Theoremordpwsucss 4551 The collection of ordinals in the power class of an ordinal is a superset of its successor.

We can think of  ( ~P A  i^i  On ) as another possible definition of successor, which would be equivalent to df-suc 4356 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if  A  e.  On then both  U. suc  A  =  A (onunisuci 4417) and  U. { x  e.  On  |  x  C_  A }  =  A (onuniss2 4496).

Constructively  ( ~P A  i^i  On ) and  suc  A cannot be shown to be equivalent (as proved at ordpwsucexmid 4554). (Contributed by Jim Kingdon, 21-Jul-2019.)

 |-  ( Ord  A  ->  suc 
 A  C_  ( ~P A  i^i  On ) )
 
Theoremonnmin 4552 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  C_  On  /\  B  e.  A )  ->  -.  B  e.  |^|
 A )
 
Theoremssnel 4553 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  C_  B  ->  -.  B  e.  A )
 
Theoremordpwsucexmid 4554* The subset in ordpwsucss 4551 cannot be equality. That is, strengthening it to equality implies excluded middle. (Contributed by Jim Kingdon, 30-Jul-2019.)
 |- 
 A. x  e.  On  suc 
 x  =  ( ~P x  i^i  On )   =>    |-  ( ph  \/  -.  ph )
 
Theoremordtri2or2exmid 4555* Ordinal trichotomy implies excluded middle. (Contributed by Jim Kingdon, 29-Aug-2021.)
 |- 
 A. x  e.  On  A. y  e.  On  ( x  C_  y  \/  y  C_  x )   =>    |-  ( ph  \/  -.  ph )
 
Theoremontri2orexmidim 4556* Ordinal trichotomy implies excluded middle. Closed form of ordtri2or2exmid 4555. (Contributed by Jim Kingdon, 26-Aug-2024.)
 |-  ( A. x  e. 
 On  A. y  e.  On  ( x  C_  y  \/  y  C_  x )  -> DECID  ph )
 
Theoremonintexmid 4557* 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  C_  On  /\  E. x  x  e.  y )  ->  |^| y  e.  y
 )   =>    |-  ( ph  \/  -.  ph )
 
Theoremzfregfr 4558 The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.)
 |- 
 _E  Fr  A
 
Theoremordfr 4559 Epsilon is well-founded on an ordinal class. (Contributed by NM, 22-Apr-1994.)
 |-  ( Ord  A  ->  _E 
 Fr  A )
 
Theoremordwe 4560 Epsilon well-orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.)
 |-  ( Ord  A  ->  _E 
 We  A )
 
Theoremwetriext 4561* A trichotomous well-order is extensional. (Contributed by Jim Kingdon, 26-Sep-2021.)
 |-  ( ph  ->  R  We  A )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  A. a  e.  A  A. b  e.  A  ( a R b  \/  a  =  b  \/  b R a ) )   &    |-  ( ph  ->  B  e.  A )   &    |-  ( ph  ->  C  e.  A )   &    |-  ( ph  ->  A. z  e.  A  ( z R B  <->  z R C ) )   =>    |-  ( ph  ->  B  =  C )
 
Theoremwessep 4562 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  C_  A )  ->  _E  We  B )
 
Theoremreg3exmidlemwe 4563* Lemma for reg3exmid 4564. Our counterexample  A satisfies  We. (Contributed by Jim Kingdon, 3-Oct-2021.)
 |-  A  =  { x  e.  { (/) ,  { (/) } }  |  ( x  =  { (/)
 }  \/  ( x  =  (/)  /\  ph ) ) }   =>    |- 
 _E  We  A
 
Theoremreg3exmid 4564* If any inhabited set satisfying df-wetr 4319 for  _E has a minimal element, excluded middle follows. (Contributed by Jim Kingdon, 3-Oct-2021.)
 |-  ( (  _E  We  z  /\  E. w  w  e.  z )  ->  E. x  e.  z  A. y  e.  z  x  C_  y )   =>    |-  ( ph  \/  -.  ph )
 
Theoremdcextest 4565* If it is decidable whether  { x  |  ph } is a set, then 
-.  ph is decidable (where  x does not occur in 
ph). 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  -.  ph is decidable) holds. (Contributed by Jim Kingdon, 3-Jul-2022.)
 |- DECID  { x  |  ph }  e.  _V   =>    |- DECID  -.  ph
 
2.5.3  Transfinite induction
 
Theoremtfi 4566* 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  C_  On  /\  A. x  e. 
 On  ( x  C_  A  ->  x  e.  A ) )  ->  A  =  On )
 
Theoremtfis 4567* 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  e.  On  ->  ( A. y  e.  x  [ y  /  x ] ph  ->  ph )
 )   =>    |-  ( x  e.  On  -> 
 ph )
 
Theoremtfis2f 4568* Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
 |- 
 F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   &    |-  ( x  e.  On  ->  (
 A. y  e.  x  ps  ->  ph ) )   =>    |-  ( x  e. 
 On  ->  ph )
 
Theoremtfis2 4569* Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 18-Aug-1994.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  ( x  e.  On  ->  (
 A. y  e.  x  ps  ->  ph ) )   =>    |-  ( x  e. 
 On  ->  ph )
 
Theoremtfis3 4570* Transfinite Induction Schema, using implicit substitution. (Contributed by NM, 4-Nov-2003.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  e.  On  ->  (
 A. y  e.  x  ps  ->  ph ) )   =>    |-  ( A  e.  On  ->  ch )
 
Theoremtfisi 4571* 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.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  T  e.  On )   &    |-  (
 ( ph  /\  ( R  e.  On  /\  R  C_  T )  /\  A. y ( S  e.  R  ->  ch ) )  ->  ps )   &    |-  ( x  =  y  ->  ( ps  <->  ch ) )   &    |-  ( x  =  A  ->  ( ps  <->  th ) )   &    |-  ( x  =  y  ->  R  =  S )   &    |-  ( x  =  A  ->  R  =  T )   =>    |-  ( ph  ->  th )
 
2.6  IZF Set Theory - add the Axiom of Infinity
 
2.6.1  Introduce the Axiom of Infinity
 
Axiomax-iinf 4572* Axiom of Infinity. Axiom 5 of [Crosilla] p. "Axioms of CZF and IZF". (Contributed by Jim Kingdon, 16-Nov-2018.)
 |- 
 E. x ( (/)  e.  x  /\  A. y
 ( y  e.  x  ->  suc  y  e.  x ) )
 
Theoremzfinf2 4573* A standard version of the Axiom of Infinity, using definitions to abbreviate. Axiom Inf of [BellMachover] p. 472. (Contributed by NM, 30-Aug-1993.)
 |- 
 E. x ( (/)  e.  x  /\  A. y  e.  x  suc  y  e.  x )
 
2.6.2  The natural numbers
 
Syntaxcom 4574 Extend class notation to include the class of natural numbers.
 class  om
 
Definitiondf-iom 4575* 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  om are a subset of the ordinal numbers df-on 4353. Later, when we define complex numbers, we will be able to also define a subset of the complex numbers (df-inn 8879) 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 7171. (Contributed by NM, 6-Aug-1994.) Use its alias dfom3 4576 instead for naming consistency with set.mm. (New usage is discouraged.)

 |- 
 om  =  |^| { x  |  ( (/)  e.  x  /\  A. y  e.  x  suc  y  e.  x ) }
 
Theoremdfom3 4576* Alias for df-iom 4575. Use it instead of df-iom 4575 for naming consistency with set.mm. (Contributed by NM, 6-Aug-1994.)
 |- 
 om  =  |^| { x  |  ( (/)  e.  x  /\  A. y  e.  x  suc  y  e.  x ) }
 
Theoremomex 4577 The existence of omega (the class of natural numbers). Axiom 7 of [TakeutiZaring] p. 43. (Contributed by NM, 6-Aug-1994.)
 |- 
 om  e.  _V
 
2.6.3  Peano's postulates
 
Theorempeano1 4578 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.)
 |-  (/)  e.  om
 
Theorempeano2 4579 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  e.  om  ->  suc  A  e.  om )
 
Theorempeano3 4580 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  e.  om  ->  suc  A  =/=  (/) )
 
Theorempeano4 4581 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  e.  om 
 /\  B  e.  om )  ->  ( suc  A  =  suc  B  <->  A  =  B ) )
 
Theorempeano5 4582* 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 4587. (Contributed by NM, 18-Feb-2004.)
 |-  ( ( (/)  e.  A  /\  A. x  e.  om  ( x  e.  A  ->  suc  x  e.  A ) )  ->  om  C_  A )
 
2.6.4  Finite induction (for finite ordinals)
 
Theoremfind 4583* 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  C_  om  /\  (/) 
 e.  A  /\  A. x  e.  A  suc  x  e.  A )   =>    |-  A  =  om
 
Theoremfinds 4584* 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  =  (/)  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  suc  y  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ta ) )   &    |-  ps   &    |-  (
 y  e.  om  ->  ( ch  ->  th )
 )   =>    |-  ( A  e.  om  ->  ta )
 
Theoremfinds2 4585* 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  =  (/)  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  suc  y  ->  ( ph  <->  th ) )   &    |-  ( ta  ->  ps )   &    |-  ( y  e. 
 om  ->  ( ta  ->  ( ch  ->  th )
 ) )   =>    |-  ( x  e.  om  ->  ( ta  ->  ph )
 )
 
Theoremfinds1 4586* 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  =  (/)  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  suc  y  ->  ( ph  <->  th ) )   &    |-  ps   &    |-  (
 y  e.  om  ->  ( ch  ->  th )
 )   =>    |-  ( x  e.  om  -> 
 ph )
 
Theoremfindes 4587 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 ]. ph   &    |-  ( x  e.  om  ->  (
 ph  ->  [. suc  x  /  x ]. ph ) )   =>    |-  ( x  e.  om  ->  ph )
 
2.6.5  The Natural Numbers (continued)
 
Theoremnn0suc 4588* 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  e.  om  ->  ( A  =  (/)  \/ 
 E. x  e.  om  A  =  suc  x ) )
 
Theoremelomssom 4589 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 4590. (Revised by BJ, 7-Aug-2024.)
 |-  ( A  e.  om  ->  A  C_  om )
 
Theoremelnn 4590 A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.)
 |-  ( ( A  e.  B  /\  B  e.  om )  ->  A  e.  om )
 
Theoremordom 4591 Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.)
 |- 
 Ord  om
 
Theoremomelon2 4592 Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.)
 |-  ( om  e.  _V  ->  om  e.  On )
 
Theoremomelon 4593 Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.)
 |- 
 om  e.  On
 
Theoremnnon 4594 A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
 |-  ( A  e.  om  ->  A  e.  On )
 
Theoremnnoni 4595 A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
 |-  A  e.  om   =>    |-  A  e.  On
 
Theoremnnord 4596 A natural number is ordinal. (Contributed by NM, 17-Oct-1995.)
 |-  ( A  e.  om  ->  Ord  A )
 
Theoremomsson 4597 Omega is a subset of  On. (Contributed by NM, 13-Jun-1994.)
 |- 
 om  C_  On
 
Theoremlimom 4598 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  om
 
Theorempeano2b 4599 A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.)
 |-  ( A  e.  om  <->  suc  A  e.  om )
 
Theoremnnsuc 4600* A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.)
 |-  ( ( A  e.  om 
 /\  A  =/=  (/) )  ->  E. x  e.  om  A  =  suc  x )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14113
  Copyright terms: Public domain < Previous  Next >