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Theorem List for Intuitionistic Logic Explorer - 4301-4400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremonsucelsucexmid 4301* The converse of onsucelsucr 4280 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 4280 does hold, as seen at nnsucelsuc 6156. (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 4302* The converse of sucunielr 4282 (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 4303* Lemma for regexmid 4306. 
A is inhabited. (Contributed by Jim Kingdon, 3-Sep-2019.)
 |-  A  =  { x  e.  { (/) ,  { (/) } }  |  ( x  =  { (/)
 }  \/  ( x  =  (/)  /\  ph ) ) }   =>    |- 
 E. y  y  e.  A
 
Theoremregexmidlem1 4304* Lemma for regexmid 4306. 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 4305* Lemma for reg2exmid 4307. 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 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  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 4308. (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 4307* 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 4308* 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 4309*  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 4310* 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 4311 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 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  Ord  A (df-iord 4149) to also require  _E 
Fr  A (df-frind 4115) and in that case any theorem related to irreflexivity of ordinals could use ordirr 4313 (which under that definition would presumably not need ax-setind 4308 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 4313. To encourage ordirr 4313 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 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.)
 |-  ( Ord  A  ->  -.  A  e.  A )
 
Theoremonirri 4314 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 4315 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 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.)
 |-  ( Ord  A  ->  -.  ( A  e.  B  /\  B  e.  A ) )
 
Theoremorddisj 4317 An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.)
 |-  ( Ord  A  ->  ( A  i^i  { A } )  =  (/) )
 
Theoremorddif 4318 Ordinal derived from its successor. (Contributed by NM, 20-May-1998.)
 |-  ( Ord  A  ->  A  =  ( suc  A  \  { A } )
 )
 
Theoremelirrv 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.)
 |- 
 -.  x  e.  x
 
Theoremsucprcreg 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.)
 |-  ( -.  A  e.  _V  <->  suc 
 A  =  A )
 
Theoremruv 4321 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 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.)
 |- 
 { x  |  x  e/  x }  e/  _V
 
Theoremonprc 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.)
 |- 
 -.  On  e.  _V
 
Theoremsucon 4324 The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.)
 |- 
 suc  On  =  On
 
Theoremen2lp 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.)
 |- 
 -.  ( A  e.  B  /\  B  e.  A )
 
Theorempreleq 4326 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 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.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( { A ,  { A ,  B } }  =  { C ,  { C ,  D } }  <->  ( A  =  C  /\  B  =  D ) )
 
Theoremsuc11g 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.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( suc  A  =  suc  B  <->  A  =  B ) )
 
Theoremsuc11 4329 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 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  y, we can construct a set  x which does not equal it. (Contributed by Jim Kingdon, 29-Dec-2018.)
 |- 
 E. x  -.  x  =  y
 
Theoremdtru 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.)
 |- 
 -.  A. x  x  =  y
 
Theoremeunex 4332 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 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.)
 |- 
 _E  Or  On   =>    |-  ( ph  \/  -.  ph )
 
Theoremordsuc 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.)
 |-  ( Ord  A  <->  Ord  suc  A )
 
Theoremonsucuni2 4335 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 4336* 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 4337 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 4338 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 4154 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 4215) and  U. { x  e.  On  |  x  C_  A }  =  A (onuniss2 4284).

Constructively  ( ~P A  i^i  On ) and  suc  A cannot be shown to be equivalent (as proved at ordpwsucexmid 4341). (Contributed by Jim Kingdon, 21-Jul-2019.)

 |-  ( Ord  A  ->  suc 
 A  C_  ( ~P A  i^i  On ) )
 
Theoremonnmin 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.)
 |-  ( ( A  C_  On  /\  B  e.  A )  ->  -.  B  e.  |^|
 A )
 
Theoremssnel 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.)
 |-  ( A  C_  B  ->  -.  B  e.  A )
 
Theoremordpwsucexmid 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.)
 |- 
 A. x  e.  On  suc 
 x  =  ( ~P x  i^i  On )   =>    |-  ( ph  \/  -.  ph )
 
Theoremordtri2or2exmid 4342* 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 )
 
Theoremonintexmid 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.)
 |-  ( ( y  C_  On  /\  E. x  x  e.  y )  ->  |^| y  e.  y
 )   =>    |-  ( ph  \/  -.  ph )
 
Theoremzfregfr 4344 The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.)
 |- 
 _E  Fr  A
 
Theoremordfr 4345 Epsilon is well-founded on an ordinal class. (Contributed by NM, 22-Apr-1994.)
 |-  ( Ord  A  ->  _E 
 Fr  A )
 
Theoremordwe 4346 Epsilon well-orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.)
 |-  ( Ord  A  ->  _E 
 We  A )
 
Theoremwetriext 4347* 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 4348 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 4349* Lemma for reg3exmid 4350. Our counterexample  A satisfies  We. (Contributed by Jim Kingdon, 3-Oct-2021.)
 |-  A  =  { x  e.  { (/) ,  { (/) } }  |  ( x  =  { (/)
 }  \/  ( x  =  (/)  /\  ph ) ) }   =>    |- 
 _E  We  A
 
Theoremreg3exmid 4350* If any inhabited set satisfying df-wetr 4117 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 )
 
2.5.3  Transfinite induction
 
Theoremtfi 4351* 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 4352* 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 4353* 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 4354* 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 4355* 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 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.)
 |-  ( 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 4357* 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 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.)
 |- 
 E. x ( (/)  e.  x  /\  A. y  e.  x  suc  y  e.  x )
 
2.6.2  The natural numbers (i.e. finite ordinals)
 
Syntaxcom 4359 Extend class notation to include the class of natural numbers.
 class  om
 
Definitiondf-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  om are a subset of the ordinal numbers df-on 4151. 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 4361 instead for naming consistency with set.mm. (New usage is discouraged.)

 |- 
 om  =  |^| { x  |  ( (/)  e.  x  /\  A. y  e.  x  suc  y  e.  x ) }
 
Theoremdfom3 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.)
 |- 
 om  =  |^| { x  |  ( (/)  e.  x  /\  A. y  e.  x  suc  y  e.  x ) }
 
Theoremomex 4362 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 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.)
 |-  (/)  e.  om
 
Theorempeano2 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.)
 |-  ( A  e.  om  ->  suc  A  e.  om )
 
Theorempeano3 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.)
 |-  ( A  e.  om  ->  suc  A  =/=  (/) )
 
Theorempeano4 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.)
 |-  ( ( A  e.  om 
 /\  B  e.  om )  ->  ( suc  A  =  suc  B  <->  A  =  B ) )
 
Theorempeano5 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.)
 |-  ( ( (/)  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 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  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 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.)
 |-  ( 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 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.)
 |-  ( 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 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.)
 |-  ( x  =  (/)  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  (
 ph 
 <->  ch ) )   &    |-  ( x  =  suc  y  ->  ( ph  <->  th ) )   &    |-  ps   &    |-  (
 y  e.  om  ->  ( ch  ->  th )
 )   =>    |-  ( x  e.  om  -> 
 ph )
 
Theoremfindes 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.)
 |-  [. (/)  /  x ]. ph   &    |-  ( x  e.  om  ->  (
 ph  ->  [. suc  x  /  x ]. ph ) )   =>    |-  ( x  e.  om  ->  ph )
 
2.6.5  The Natural Numbers (continued)
 
Theoremnn0suc 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.)
 |-  ( A  e.  om  ->  ( A  =  (/)  \/ 
 E. x  e.  om  A  =  suc  x ) )
 
Theoremelnn 4374 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 4375 Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.)
 |- 
 Ord  om
 
Theoremomelon2 4376 Omega is an ordinal number. (Contributed by Mario Carneiro, 30-Jan-2013.)
 |-  ( om  e.  _V  ->  om  e.  On )
 
Theoremomelon 4377 Omega is an ordinal number. (Contributed by NM, 10-May-1998.) (Revised by Mario Carneiro, 30-Jan-2013.)
 |- 
 om  e.  On
 
Theoremnnon 4378 A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
 |-  ( A  e.  om  ->  A  e.  On )
 
Theoremnnoni 4379 A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.)
 |-  A  e.  om   =>    |-  A  e.  On
 
Theoremnnord 4380 A natural number is ordinal. (Contributed by NM, 17-Oct-1995.)
 |-  ( A  e.  om  ->  Ord  A )
 
Theoremomsson 4381 Omega is a subset of  On. (Contributed by NM, 13-Jun-1994.)
 |- 
 om  C_  On
 
Theoremlimom 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.)
 |- 
 Lim  om
 
Theorempeano2b 4383 A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.)
 |-  ( A  e.  om  <->  suc  A  e.  om )
 
Theoremnnsuc 4384* A nonzero natural number is a successor. (Contributed by NM, 18-Feb-2004.)
 |-  ( ( A  e.  om 
 /\  A  =/=  (/) )  ->  E. x  e.  om  A  =  suc  x )
 
Theoremnndceq0 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.)
 |-  ( A  e.  om  -> DECID  A  =  (/) )
 
Theorem0elnn 4386 A natural number is either the empty set or has the empty set as an element. (Contributed by Jim Kingdon, 23-Aug-2019.)
 |-  ( A  e.  om  ->  ( A  =  (/)  \/  (/)  e.  A ) )
 
Theoremnn0eln0 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.)
 |-  ( A  e.  om  ->  ( (/)  e.  A  <->  A  =/=  (/) ) )
 
Theoremnnregexmid 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.)
 |-  ( ( x  C_  om 
 /\  E. y  y  e.  x )  ->  E. y
 ( y  e.  x  /\  A. z ( z  e.  y  ->  -.  z  e.  x ) ) )   =>    |-  ( ph  \/  -.  ph )
 
2.6.6  Relations
 
Syntaxcxp 4389 Extend the definition of a class to include the cross product.
 class  ( A  X.  B )
 
Syntaxccnv 4390 Extend the definition of a class to include the converse of a class.
 class  `' A
 
Syntaxcdm 4391 Extend the definition of a class to include the domain of a class.
 class  dom  A
 
Syntaxcrn 4392 Extend the definition of a class to include the range of a class.
 class  ran  A
 
Syntaxcres 4393 Extend the definition of a class to include the restriction of a class. (Read: The restriction of  A to  B.)
 class  ( A  |`  B )
 
Syntaxcima 4394 Extend the definition of a class to include the image of a class. (Read: The image of  B under  A.)
 class  ( A " B )
 
Syntaxccom 4395 Extend the definition of a class to include the composition of two classes. (Read: The composition of  A and  B.)
 class  ( A  o.  B )
 
Syntaxwrel 4396 Extend the definition of a wff to include the relation predicate. (Read:  A is a relation.)
 wff  Rel  A
 
Definitiondf-xp 4397* Define the cross product of two classes. Definition 9.11 of [Quine] p. 64. For example, ( { 1 , 5 }  X. { 2 , 7 } ) = ( {  <. 1 , 2  >.,  <. 1 , 7  >. }  u. {  <. 5 , 2  >.,  <. 5 , 7  >. } ) . Another example is that the set of rational numbers are defined in using the cross-product ( Z  X. N ) ; the left- and right-hand sides of the cross-product represent the top (integer) and bottom (natural) numbers of a fraction. (Contributed by NM, 4-Jul-1994.)
 |-  ( A  X.  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  B ) }
 
Definitiondf-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.)
 |-  ( Rel  A  <->  A  C_  ( _V 
 X.  _V ) )
 
Definitiondf-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  A  e. 
_V and  B  e.  _V then  ( A `' R B  <-> 
B R A ), as proven in brcnv 4566 (see df-br 3806 and df-rel 4398 for more on relations). For example,  `' {  <. 2 , 6  >.,  <. 3 , 9  >. } = {  <. 6 , 2  >.,  <. 9 , 3  >. } . We use Quine's breve accent (smile) notation. Like Quine, we use it as a prefix, which eliminates the need for parentheses. Many authors use the postfix superscript "to the minus one." "Converse" is Quine's terminology; some authors call it "inverse," especially when the argument is a function. (Contributed by NM, 4-Jul-1994.)
 |-  `' A  =  { <. x ,  y >.  |  y A x }
 
Definitiondf-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  A and  B, uses a slash instead of  o., and calls the operation "relative product." (Contributed by NM, 4-Jul-1994.)
 |-  ( A  o.  B )  =  { <. x ,  y >.  |  E. z
 ( x B z 
 /\  z A y ) }
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