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Theorem List for Intuitionistic Logic Explorer - 4201-4300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremuniex2 4201* The Axiom of Union using the standard abbreviation for union. Given any set  x, its union  y exists. (Contributed by NM, 4-Jun-2006.)
 |- 
 E. y  y  = 
 U. x
 
Theoremuniex 4202 The Axiom of Union in class notation. This says that if  A is a set i.e.  A  e.  _V (see isset 2578), then the union of  A is also a set. Same as Axiom 3 of [TakeutiZaring] p. 16. (Contributed by NM, 11-Aug-1993.)
 |-  A  e.  _V   =>    |-  U. A  e.  _V
 
Theoremuniexg 4203 The ZF Axiom of Union in class notation, in the form of a theorem instead of an inference. We use the antecedent  A  e.  V instead of  A  e.  _V to make the theorem more general and thus shorten some proofs; obviously the universal class constant  _V is one possible substitution for class variable  V. (Contributed by NM, 25-Nov-1994.)
 |-  ( A  e.  V  ->  U. A  e.  _V )
 
Theoremunex 4204 The union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 1-Jul-1994.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  u.  B )  e.  _V
 
Theoremunexb 4205 Existence of union is equivalent to existence of its components. (Contributed by NM, 11-Jun-1998.)
 |-  ( ( A  e.  _V 
 /\  B  e.  _V ) 
 <->  ( A  u.  B )  e.  _V )
 
Theoremunexg 4206 A union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 18-Sep-2006.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  u.  B )  e.  _V )
 
Theoremtpexg 4207 An unordered triple of classes exists. (Contributed by NM, 10-Apr-1994.)
 |-  ( ( A  e.  U  /\  B  e.  V  /\  C  e.  W ) 
 ->  { A ,  B ,  C }  e.  _V )
 
Theoremunisn3 4208* Union of a singleton in the form of a restricted class abstraction. (Contributed by NM, 3-Jul-2008.)
 |-  ( A  e.  B  ->  U. { x  e.  B  |  x  =  A }  =  A )
 
Theoremsnnex 4209* The class of all singletons is a proper class. (Contributed by NM, 10-Oct-2008.) (Proof shortened by Eric Schmidt, 7-Dec-2008.)
 |- 
 { x  |  E. y  x  =  {
 y } }  e/  _V
 
Theoremopeluu 4210 Each member of an ordered pair belongs to the union of the union of a class to which the ordered pair belongs. Lemma 3D of [Enderton] p. 41. (Contributed by NM, 31-Mar-1995.) (Revised by Mario Carneiro, 27-Feb-2016.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( <. A ,  B >.  e.  C  ->  ( A  e.  U. U. C  /\  B  e.  U. U. C ) )
 
Theoremuniuni 4211* Expression for double union that moves union into a class builder. (Contributed by FL, 28-May-2007.)
 |- 
 U. U. A  =  U. { x  |  E. y
 ( x  =  U. y  /\  y  e.  A ) }
 
Theoremeusv1 4212* Two ways to express single-valuedness of a class expression  A ( x ). (Contributed by NM, 14-Oct-2010.)
 |-  ( E! y A. x  y  =  A  <->  E. y A. x  y  =  A )
 
Theoremeusvnf 4213* Even if  x is free in  A, it is effectively bound when  A ( x ) is single-valued. (Contributed by NM, 14-Oct-2010.) (Revised by Mario Carneiro, 14-Oct-2016.)
 |-  ( E! y A. x  y  =  A  -> 
 F/_ x A )
 
Theoremeusvnfb 4214* Two ways to say that  A ( x ) is a set expression that does not depend on  x. (Contributed by Mario Carneiro, 18-Nov-2016.)
 |-  ( E! y A. x  y  =  A  <->  (
 F/_ x A  /\  A  e.  _V )
 )
 
Theoremeusv2i 4215* Two ways to express single-valuedness of a class expression  A ( x ). (Contributed by NM, 14-Oct-2010.) (Revised by Mario Carneiro, 18-Nov-2016.)
 |-  ( E! y A. x  y  =  A  ->  E! y E. x  y  =  A )
 
Theoremeusv2nf 4216* Two ways to express single-valuedness of a class expression  A ( x ). (Contributed by Mario Carneiro, 18-Nov-2016.)
 |-  A  e.  _V   =>    |-  ( E! y E. x  y  =  A 
 <-> 
 F/_ x A )
 
Theoremeusv2 4217* Two ways to express single-valuedness of a class expression  A ( x ). (Contributed by NM, 15-Oct-2010.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
 |-  A  e.  _V   =>    |-  ( E! y E. x  y  =  A 
 <->  E! y A. x  y  =  A )
 
Theoremreusv1 4218* Two ways to express single-valuedness of a class expression  C ( y ). (Contributed by NM, 16-Dec-2012.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
 |-  ( E. y  e.  B  ph  ->  ( E! x  e.  A  A. y  e.  B  ( ph  ->  x  =  C ) 
 <-> 
 E. x  e.  A  A. y  e.  B  (
 ph  ->  x  =  C ) ) )
 
Theoremreusv3i 4219* Two ways of expressing existential uniqueness via an indirect equality. (Contributed by NM, 23-Dec-2012.)
 |-  ( y  =  z 
 ->  ( ph  <->  ps ) )   &    |-  (
 y  =  z  ->  C  =  D )   =>    |-  ( E. x  e.  A  A. y  e.  B  (
 ph  ->  x  =  C )  ->  A. y  e.  B  A. z  e.  B  ( ( ph  /\  ps )  ->  C  =  D ) )
 
Theoremreusv3 4220* Two ways to express single-valuedness of a class expression  C ( y ). See reusv1 4218 for the connection to uniqueness. (Contributed by NM, 27-Dec-2012.)
 |-  ( y  =  z 
 ->  ( ph  <->  ps ) )   &    |-  (
 y  =  z  ->  C  =  D )   =>    |-  ( E. y  e.  B  ( ph  /\  C  e.  A )  ->  ( A. y  e.  B  A. z  e.  B  ( ( ph  /\ 
 ps )  ->  C  =  D )  <->  E. x  e.  A  A. y  e.  B  (
 ph  ->  x  =  C ) ) )
 
Theoremalxfr 4221* Transfer universal quantification from a variable  x to another variable  y contained in expression  A. (Contributed by NM, 18-Feb-2007.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( ( A. y  A  e.  B  /\  A. x E. y  x  =  A )  ->  ( A. x ph  <->  A. y ps ) )
 
Theoremralxfrd 4222* Transfer universal quantification from a variable  x to another variable  y contained in expression  A. (Contributed by NM, 15-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
 |-  ( ( ph  /\  y  e.  C )  ->  A  e.  B )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  E. y  e.  C  x  =  A )   &    |-  (
 ( ph  /\  x  =  A )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  B  ps 
 <-> 
 A. y  e.  C  ch ) )
 
Theoremrexxfrd 4223* Transfer universal quantification from a variable  x to another variable  y contained in expression  A. (Contributed by FL, 10-Apr-2007.) (Revised by Mario Carneiro, 15-Aug-2014.)
 |-  ( ( ph  /\  y  e.  C )  ->  A  e.  B )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  E. y  e.  C  x  =  A )   &    |-  (
 ( ph  /\  x  =  A )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E. x  e.  B  ps 
 <-> 
 E. y  e.  C  ch ) )
 
Theoremralxfr2d 4224* Transfer universal quantification from a variable  x to another variable  y contained in expression  A. (Contributed by Mario Carneiro, 20-Aug-2014.)
 |-  ( ( ph  /\  y  e.  C )  ->  A  e.  V )   &    |-  ( ph  ->  ( x  e.  B  <->  E. y  e.  C  x  =  A )
 )   &    |-  ( ( ph  /\  x  =  A )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  B  ps 
 <-> 
 A. y  e.  C  ch ) )
 
Theoremrexxfr2d 4225* Transfer universal quantification from a variable  x to another variable  y contained in expression  A. (Contributed by Mario Carneiro, 20-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
 |-  ( ( ph  /\  y  e.  C )  ->  A  e.  V )   &    |-  ( ph  ->  ( x  e.  B  <->  E. y  e.  C  x  =  A )
 )   &    |-  ( ( ph  /\  x  =  A )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E. x  e.  B  ps 
 <-> 
 E. y  e.  C  ch ) )
 
Theoremralxfr 4226* Transfer universal quantification from a variable  x to another variable  y contained in expression  A. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.)
 |-  ( y  e.  C  ->  A  e.  B )   &    |-  ( x  e.  B  ->  E. y  e.  C  x  =  A )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x  e.  B  ph  <->  A. y  e.  C  ps )
 
TheoremralxfrALT 4227* Transfer universal quantification from a variable  x to another variable  y contained in expression  A. This proof does not use ralxfrd 4222. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( y  e.  C  ->  A  e.  B )   &    |-  ( x  e.  B  ->  E. y  e.  C  x  =  A )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x  e.  B  ph  <->  A. y  e.  C  ps )
 
Theoremrexxfr 4228* Transfer existence from a variable 
x to another variable  y contained in expression  A. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.)
 |-  ( y  e.  C  ->  A  e.  B )   &    |-  ( x  e.  B  ->  E. y  e.  C  x  =  A )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E. x  e.  B  ph  <->  E. y  e.  C  ps )
 
Theoremrabxfrd 4229* Class builder membership after substituting an expression  A (containing  y) for  x in the class expression  ch. (Contributed by NM, 16-Jan-2012.)
 |-  F/_ y B   &    |-  F/_ y C   &    |-  (
 ( ph  /\  y  e.  D )  ->  A  e.  D )   &    |-  ( x  =  A  ->  ( ps  <->  ch ) )   &    |-  ( y  =  B  ->  A  =  C )   =>    |-  ( ( ph  /\  B  e.  D )  ->  ( C  e.  { x  e.  D  |  ps }  <->  B  e.  { y  e.  D  |  ch }
 ) )
 
Theoremrabxfr 4230* Class builder membership after substituting an expression  A (containing  y) for  x in the class expression  ph. (Contributed by NM, 10-Jun-2005.)
 |-  F/_ y B   &    |-  F/_ y C   &    |-  (
 y  e.  D  ->  A  e.  D )   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   &    |-  (
 y  =  B  ->  A  =  C )   =>    |-  ( B  e.  D  ->  ( C  e.  { x  e.  D  |  ph
 } 
 <->  B  e.  { y  e.  D  |  ps }
 ) )
 
Theoremreuhypd 4231* A theorem useful for eliminating restricted existential uniqueness hypotheses. (Contributed by NM, 16-Jan-2012.)
 |-  ( ( ph  /\  x  e.  C )  ->  B  e.  C )   &    |-  ( ( ph  /\  x  e.  C  /\  y  e.  C )  ->  ( x  =  A  <->  y  =  B ) )   =>    |-  ( ( ph  /\  x  e.  C )  ->  E! y  e.  C  x  =  A )
 
Theoremreuhyp 4232* A theorem useful for eliminating restricted existential uniqueness hypotheses. (Contributed by NM, 15-Nov-2004.)
 |-  ( x  e.  C  ->  B  e.  C )   &    |-  ( ( x  e.  C  /\  y  e.  C )  ->  ( x  =  A  <->  y  =  B ) )   =>    |-  ( x  e.  C  ->  E! y  e.  C  x  =  A )
 
Theoremuniexb 4233 The Axiom of Union and its converse. A class is a set iff its union is a set. (Contributed by NM, 11-Nov-2003.)
 |-  ( A  e.  _V  <->  U. A  e.  _V )
 
Theorempwexb 4234 The Axiom of Power Sets and its converse. A class is a set iff its power class is a set. (Contributed by NM, 11-Nov-2003.)
 |-  ( A  e.  _V  <->  ~P A  e.  _V )
 
Theoremuniv 4235 The union of the universe is the universe. Exercise 4.12(c) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.)
 |- 
 U. _V  =  _V
 
Theoremeldifpw 4236 Membership in a power class difference. (Contributed by NM, 25-Mar-2007.)
 |-  C  e.  _V   =>    |-  ( ( A  e.  ~P B  /\  -.  C  C_  B )  ->  ( A  u.  C )  e.  ( ~P ( B  u.  C )  \  ~P B ) )
 
Theoremop1stb 4237 Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (Contributed by NM, 25-Nov-2003.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 |^| |^| <. A ,  B >.  =  A
 
Theoremop1stbg 4238 Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (Contributed by Jim Kingdon, 17-Dec-2018.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| |^| <. A ,  B >.  =  A )
 
Theoremiunpw 4239* An indexed union of a power class in terms of the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33. (Contributed by NM, 29-Nov-2003.)
 |-  A  e.  _V   =>    |-  ( E. x  e.  A  x  =  U. A 
 <->  ~P U. A  =  U_ x  e.  A  ~P x )
 
2.4.2  Ordinals (continued)
 
Theoremordon 4240 The class of all ordinal numbers is ordinal. Proposition 7.12 of [TakeutiZaring] p. 38, but without using the Axiom of Regularity. (Contributed by NM, 17-May-1994.)
 |- 
 Ord  On
 
Theoremssorduni 4241 The union of a class of ordinal numbers is ordinal. Proposition 7.19 of [TakeutiZaring] p. 40. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
 |-  ( A  C_  On  ->  Ord  U. A )
 
Theoremssonuni 4242 The union of a set of ordinal numbers is an ordinal number. Theorem 9 of [Suppes] p. 132. (Contributed by NM, 1-Nov-2003.)
 |-  ( A  e.  V  ->  ( A  C_  On  ->  U. A  e.  On ) )
 
Theoremssonunii 4243 The union of a set of ordinal numbers is an ordinal number. Corollary 7N(d) of [Enderton] p. 193. (Contributed by NM, 20-Sep-2003.)
 |-  A  e.  _V   =>    |-  ( A  C_  On  ->  U. A  e.  On )
 
Theoremonun2 4244 The union of two ordinal numbers is an ordinal number. (Contributed by Jim Kingdon, 25-Jul-2019.)
 |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  u.  B )  e.  On )
 
Theoremonun2i 4245 The union of two ordinal numbers is an ordinal number. (Contributed by NM, 13-Jun-1994.) (Constructive proof by Jim Kingdon, 25-Jul-2019.)
 |-  A  e.  On   &    |-  B  e.  On   =>    |-  ( A  u.  B )  e.  On
 
Theoremordsson 4246 Any ordinal class is a subclass of the class of ordinal numbers. Corollary 7.15 of [TakeutiZaring] p. 38. (Contributed by NM, 18-May-1994.)
 |-  ( Ord  A  ->  A 
 C_  On )
 
Theoremonss 4247 An ordinal number is a subset of the class of ordinal numbers. (Contributed by NM, 5-Jun-1994.)
 |-  ( A  e.  On  ->  A  C_  On )
 
Theoremonuni 4248 The union of an ordinal number is an ordinal number. (Contributed by NM, 29-Sep-2006.)
 |-  ( A  e.  On  ->  U. A  e.  On )
 
Theoremorduni 4249 The union of an ordinal class is ordinal. (Contributed by NM, 12-Sep-2003.)
 |-  ( Ord  A  ->  Ord  U. A )
 
Theorembm2.5ii 4250* 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 4251 A successor exists iff its class argument exists. (Contributed by NM, 22-Jun-1998.)
 |-  ( A  e.  _V  <->  suc  A  e.  _V )
 
Theoremsucexg 4252 The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.)
 |-  ( A  e.  V  ->  suc  A  e.  _V )
 
Theoremsucex 4253 The successor of a set is a set. (Contributed by NM, 30-Aug-1993.)
 |-  A  e.  _V   =>    |-  suc  A  e.  _V
 
Theoremordsucim 4254 The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 8-Nov-2018.)
 |-  ( Ord  A  ->  Ord 
 suc  A )
 
Theoremsuceloni 4255 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 4256 The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 20-Nov-2018.)
 |-  ( A  e.  _V  ->  ( Ord  A  <->  Ord  suc  A )
 )
 
Theoremsucelon 4257 The successor of an ordinal number is an ordinal number. (Contributed by NM, 9-Sep-2003.)
 |-  ( A  e.  On  <->  suc  A  e.  On )
 
Theoremordsucss 4258 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 4259 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 4260 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 4261* 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 4262 Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4283. However, the converse does hold where  B is a natural number, as seen at nnsucelsuc 6101. (Contributed by Jim Kingdon, 17-Jul-2019.)
 |-  ( B  e.  On  ->  ( suc  A  e.  suc 
 B  ->  A  e.  B ) )
 
Theoremonsucsssucr 4263 The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4280. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.)
 |-  ( ( A  e.  On  /\  Ord  B )  ->  ( suc  A  C_  suc 
 B  ->  A  C_  B ) )
 
Theoremsucunielr 4264 Successor and union. The converse (where  B is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4284. (Contributed by Jim Kingdon, 2-Aug-2019.)
 |-  ( suc  A  e.  B  ->  A  e.  U. B )
 
Theoremunon 4265 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 4266* 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 4267 The class of ordinal numbers is a limit ordinal. (Contributed by NM, 24-Mar-1995.)
 |- 
 Lim  On
 
Theoremordunisuc2r 4268* 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 4269 An ordinal number is a subset of 
On. (Contributed by NM, 11-Aug-1994.)
 |-  A  e.  On   =>    |-  A  C_  On
 
Theoremonsuci 4270 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 4271* 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 4272 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 4273 Lemma for decidability and ordinals. The set  { x  e.  { (/) }  |  ph } is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4275 or weak linearity in ordsoexmid 4314) 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 4274* Lemma for decidability and ordinals. The set  { x  e.  { (/) }  |  ph } is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4275 or weak linearity in ordsoexmid 4314) 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 4275* 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 4276* 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 4277 Version of 2on 6040 with the definition of  2o expanded and expressed in terms of  Ord. (Contributed by Jim Kingdon, 29-Aug-2021.)
 |- 
 Ord  { (/) ,  { (/) } }
 
Theoremontr2exmid 4278* 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 4279* 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 4280* The converse of onsucsssucr 4263 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 4281* Lemma for onsucelsucexmid 4283. (Contributed by Jim Kingdon, 2-Aug-2019.)
 |-  (/)  e.  { x  e. 
 { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }
 
Theoremonsucelsucexmidlem 4282* Lemma for onsucelsucexmid 4283. The set  { x  e. 
{ (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) } appears as  A in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5531), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4273. (Contributed by Jim Kingdon, 2-Aug-2019.)
 |- 
 { x  e.  { (/)
 ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }  e.  On
 
Theoremonsucelsucexmid 4283* The converse of onsucelsucr 4262 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 4262 does hold, as seen at nnsucelsuc 6101. (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 4284* The converse of sucunielr 4264 (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 4285* Lemma for regexmid 4288. 
A is inhabited. (Contributed by Jim Kingdon, 3-Sep-2019.)
 |-  A  =  { x  e.  { (/) ,  { (/) } }  |  ( x  =  { (/)
 }  \/  ( x  =  (/)  /\  ph ) ) }   =>    |- 
 E. y  y  e.  A
 
Theoremregexmidlem1 4286* Lemma for regexmid 4288. 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 4287* Lemma for reg2exmid 4289. 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 4288* 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 4290. (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 4289* 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 4290* 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 4291*  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 4292* 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 4293 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 4294 No class is a member of itself. Exercise 6 of [TakeutiZaring] p. 22. (Contributed by NM, 7-Aug-1994.) (Proof rewritten by Mario Carneiro and Jim Kingdon, 26-Nov-2018.)
 |- 
 -.  A  e.  A
 
Theoremordirr 4295 Epsilon irreflexivity of ordinals: no ordinal class is a member of itself. Theorem 2.2(i) of [BellMachover] p. 469, generalized to classes. (Contributed by NM, 2-Jan-1994.)
 |-  ( Ord  A  ->  -.  A  e.  A )
 
Theoremnordeq 4296 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 4297 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 4298 An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.)
 |-  ( Ord  A  ->  ( A  i^i  { A } )  =  (/) )
 
Theoremorddif 4299 Ordinal derived from its successor. (Contributed by NM, 20-May-1998.)
 |-  ( Ord  A  ->  A  =  ( suc  A  \  { A } )
 )
 
Theoremelirrv 4300 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
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