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Theorem List for Intuitionistic Logic Explorer - 4301-4400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremonuni 4301 The union of an ordinal number is an ordinal number. (Contributed by NM, 29-Sep-2006.)
 |-  ( A  e.  On  ->  U. A  e.  On )
 
Theoremorduni 4302 The union of an ordinal class is ordinal. (Contributed by NM, 12-Sep-2003.)
 |-  ( Ord  A  ->  Ord  U. A )
 
Theorembm2.5ii 4303* Problem 2.5(ii) of [BellMachover] p. 471. (Contributed by NM, 20-Sep-2003.)
 |-  A  e.  _V   =>    |-  ( A  C_  On  ->  U. A  =  |^| { x  e.  On  |  A. y  e.  A  y  C_  x } )
 
Theoremsucexb 4304 A successor exists iff its class argument exists. (Contributed by NM, 22-Jun-1998.)
 |-  ( A  e.  _V  <->  suc  A  e.  _V )
 
Theoremsucexg 4305 The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.)
 |-  ( A  e.  V  ->  suc  A  e.  _V )
 
Theoremsucex 4306 The successor of a set is a set. (Contributed by NM, 30-Aug-1993.)
 |-  A  e.  _V   =>    |-  suc  A  e.  _V
 
Theoremordsucim 4307 The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 8-Nov-2018.)
 |-  ( Ord  A  ->  Ord 
 suc  A )
 
Theoremsuceloni 4308 The successor of an ordinal number is an ordinal number. Proposition 7.24 of [TakeutiZaring] p. 41. (Contributed by NM, 6-Jun-1994.)
 |-  ( A  e.  On  ->  suc  A  e.  On )
 
Theoremordsucg 4309 The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 20-Nov-2018.)
 |-  ( A  e.  _V  ->  ( Ord  A  <->  Ord  suc  A )
 )
 
Theoremsucelon 4310 The successor of an ordinal number is an ordinal number. (Contributed by NM, 9-Sep-2003.)
 |-  ( A  e.  On  <->  suc  A  e.  On )
 
Theoremordsucss 4311 The successor of an element of an ordinal class is a subset of it. (Contributed by NM, 21-Jun-1998.)
 |-  ( Ord  B  ->  ( A  e.  B  ->  suc 
 A  C_  B )
 )
 
Theoremordelsuc 4312 A set belongs to an ordinal iff its successor is a subset of the ordinal. Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM, 29-Nov-2003.)
 |-  ( ( A  e.  C  /\  Ord  B )  ->  ( A  e.  B  <->  suc 
 A  C_  B )
 )
 
Theoremonsucssi 4313 A set belongs to an ordinal number iff its successor is a subset of the ordinal number. Exercise 8 of [TakeutiZaring] p. 42 and its converse. (Contributed by NM, 16-Sep-1995.)
 |-  A  e.  On   &    |-  B  e.  On   =>    |-  ( A  e.  B  <->  suc 
 A  C_  B )
 
Theoremonsucmin 4314* The successor of an ordinal number is the smallest larger ordinal number. (Contributed by NM, 28-Nov-2003.)
 |-  ( A  e.  On  ->  suc  A  =  |^| { x  e.  On  |  A  e.  x }
 )
 
Theoremonsucelsucr 4315 Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4336. However, the converse does hold where  B is a natural number, as seen at nnsucelsuc 6234. (Contributed by Jim Kingdon, 17-Jul-2019.)
 |-  ( B  e.  On  ->  ( suc  A  e.  suc 
 B  ->  A  e.  B ) )
 
Theoremonsucsssucr 4316 The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4333. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.)
 |-  ( ( A  e.  On  /\  Ord  B )  ->  ( suc  A  C_  suc 
 B  ->  A  C_  B ) )
 
Theoremsucunielr 4317 Successor and union. The converse (where  B is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4337. (Contributed by Jim Kingdon, 2-Aug-2019.)
 |-  ( suc  A  e.  B  ->  A  e.  U. B )
 
Theoremunon 4318 The class of all ordinal numbers is its own union. Exercise 11 of [TakeutiZaring] p. 40. (Contributed by NM, 12-Nov-2003.)
 |- 
 U. On  =  On
 
Theoremonuniss2 4319* The union of the ordinal subsets of an ordinal number is that number. (Contributed by Jim Kingdon, 2-Aug-2019.)
 |-  ( A  e.  On  ->  U. { x  e. 
 On  |  x  C_  A }  =  A )
 
Theoremlimon 4320 The class of ordinal numbers is a limit ordinal. (Contributed by NM, 24-Mar-1995.)
 |- 
 Lim  On
 
Theoremordunisuc2r 4321* An ordinal which contains the successor of each of its members is equal to its union. (Contributed by Jim Kingdon, 14-Nov-2018.)
 |-  ( Ord  A  ->  (
 A. x  e.  A  suc  x  e.  A  ->  A  =  U. A ) )
 
Theoremonssi 4322 An ordinal number is a subset of 
On. (Contributed by NM, 11-Aug-1994.)
 |-  A  e.  On   =>    |-  A  C_  On
 
Theoremonsuci 4323 The successor of an ordinal number is an ordinal number. Corollary 7N(c) of [Enderton] p. 193. (Contributed by NM, 12-Jun-1994.)
 |-  A  e.  On   =>    |-  suc  A  e.  On
 
Theoremonintonm 4324* 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 4325 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 4326 Lemma for decidability and ordinals. The set  { x  e.  { (/) }  |  ph } is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4328 or weak linearity in ordsoexmid 4368) 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 4327* Lemma for decidability and ordinals. The set  { x  e.  { (/) }  |  ph } is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4328 or weak linearity in ordsoexmid 4368) 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 4328* 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".

(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 )
 
Theoremordtri2orexmid 4329* 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 4330 Version of 2on 6172 with the definition of  2o expanded and expressed in terms of  Ord. (Contributed by Jim Kingdon, 29-Aug-2021.)
 |- 
 Ord  { (/) ,  { (/) } }
 
Theoremontr2exmid 4331* 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 4332* 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 4333* The converse of onsucsssucr 4316 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 4334* Lemma for onsucelsucexmid 4336. (Contributed by Jim Kingdon, 2-Aug-2019.)
 |-  (/)  e.  { x  e. 
 { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }
 
Theoremonsucelsucexmidlem 4335* Lemma for onsucelsucexmid 4336. The set  { x  e. 
{ (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) } appears as  A in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5625), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4326. (Contributed by Jim Kingdon, 2-Aug-2019.)
 |- 
 { x  e.  { (/)
 ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }  e.  On
 
Theoremonsucelsucexmid 4336* The converse of onsucelsucr 4315 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 4315 does hold, as seen at nnsucelsuc 6234. (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 4337* The converse of sucunielr 4317 (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 4338* Lemma for regexmid 4341. 
A is inhabited. (Contributed by Jim Kingdon, 3-Sep-2019.)
 |-  A  =  { x  e.  { (/) ,  { (/) } }  |  ( x  =  { (/)
 }  \/  ( x  =  (/)  /\  ph ) ) }   =>    |- 
 E. y  y  e.  A
 
Theoremregexmidlem1 4339* Lemma for regexmid 4341. 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 4340* Lemma for reg2exmid 4342. 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 4341* 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 4343. (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 4342* 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 4343* 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 4344*  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 4345* 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 4346 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 4347 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 4343, we could redefine  Ord  A (df-iord 4184) to also require  _E 
Fr  A (df-frind 4150) and in that case any theorem related to irreflexivity of ordinals could use ordirr 4348 (which under that definition would presumably not need ax-setind 4343 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 4348. To encourage ordirr 4348 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 4348 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 4343. If in the definition of ordinals df-iord 4184, we also required that membership be well-founded on any ordinal (see df-frind 4150), then we could prove ordirr 4348 without ax-setind 4343. (Contributed by NM, 2-Jan-1994.)
 |-  ( Ord  A  ->  -.  A  e.  A )
 
Theoremonirri 4349 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 4350 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 4351 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 4352 An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.)
 |-  ( Ord  A  ->  ( A  i^i  { A } )  =  (/) )
 
Theoremorddif 4353 Ordinal derived from its successor. (Contributed by NM, 20-May-1998.)
 |-  ( Ord  A  ->  A  =  ( suc  A  \  { A } )
 )
 
Theoremelirrv 4354 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 4355 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 4356 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 4357 Alternate proof of Russell's Paradox ru 2837, simplified using (indirectly) the Axiom of Set Induction ax-setind 4343. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 { x  |  x  e/  x }  e/  _V
 
Theoremonprc 4358 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 4293), 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 4359 The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.)
 |- 
 suc  On  =  On
 
Theoremen2lp 4360 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 4361 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 4362 Theorem for alternate representation of ordered pairs, requiring the Axiom of Set Induction ax-setind 4343 (via the preleq 4361 step). See df-op 3450 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 4363 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 4364 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 4365* At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 4017 can also be summarized as "at least two sets exist", the difference is that dtruarb 4017 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 4366* 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 4365. (Contributed by Jim Kingdon, 29-Dec-2018.)
 |- 
 -.  A. x  x  =  y
 
Theoremeunex 4367 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 4368 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 4369 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 4370 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 4371* 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 4372 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 4373 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 4189 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 4250) and  U. { x  e.  On  |  x  C_  A }  =  A (onuniss2 4319).

Constructively  ( ~P A  i^i  On ) and  suc  A cannot be shown to be equivalent (as proved at ordpwsucexmid 4376). (Contributed by Jim Kingdon, 21-Jul-2019.)

 |-  ( Ord  A  ->  suc 
 A  C_  ( ~P A  i^i  On ) )
 
Theoremonnmin 4374 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 4375 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 4376* The subset in ordpwsucss 4373 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 4377* 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 4378* 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 4379 The epsilon relation is well-founded on any class. (Contributed by NM, 26-Nov-1995.)
 |- 
 _E  Fr  A
 
Theoremordfr 4380 Epsilon is well-founded on an ordinal class. (Contributed by NM, 22-Apr-1994.)
 |-  ( Ord  A  ->  _E 
 Fr  A )
 
Theoremordwe 4381 Epsilon well-orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.)
 |-  ( Ord  A  ->  _E 
 We  A )
 
Theoremwetriext 4382* 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 4383 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 4384* Lemma for reg3exmid 4385. Our counterexample  A satisfies  We. (Contributed by Jim Kingdon, 3-Oct-2021.)
 |-  A  =  { x  e.  { (/) ,  { (/) } }  |  ( x  =  { (/)
 }  \/  ( x  =  (/)  /\  ph ) ) }   =>    |- 
 _E  We  A
 
Theoremreg3exmid 4385* If any inhabited set satisfying df-wetr 4152 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 4386* 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 4387* 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 4388* 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 4389* 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 4390* 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 4391* 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 4392* 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 4393* 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 4394* 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 4395 Extend class notation to include the class of natural numbers.
 class  om
 
Definitiondf-iom 4396* 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 4186. 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 4397 instead for naming consistency with set.mm. (New usage is discouraged.)

 |- 
 om  =  |^| { x  |  ( (/)  e.  x  /\  A. y  e.  x  suc  y  e.  x ) }
 
Theoremdfom3 4397* Alias for df-iom 4396. Use it instead of df-iom 4396 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 4398 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 4399 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 4400 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 )
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