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Theorem List for Intuitionistic Logic Explorer - 4001-4100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdftr4 4001 An alternate way of defining a transitive class. Definition of [Enderton] p. 71. (Contributed by NM, 29-Aug-1993.)
 |-  ( Tr  A  <->  A  C_  ~P A )
 
Theoremtreq 4002 Equality theorem for the transitive class predicate. (Contributed by NM, 17-Sep-1993.)
 |-  ( A  =  B  ->  ( Tr  A  <->  Tr  B ) )
 
Theoremtrel 4003 In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( Tr  A  ->  ( ( B  e.  C  /\  C  e.  A ) 
 ->  B  e.  A ) )
 
Theoremtrel3 4004 In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.)
 |-  ( Tr  A  ->  ( ( B  e.  C  /\  C  e.  D  /\  D  e.  A )  ->  B  e.  A ) )
 
Theoremtrss 4005 An element of a transitive class is a subset of the class. (Contributed by NM, 7-Aug-1994.)
 |-  ( Tr  A  ->  ( B  e.  A  ->  B 
 C_  A ) )
 
Theoremtrin 4006 The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.)
 |-  ( ( Tr  A  /\  Tr  B )  ->  Tr  ( A  i^i  B ) )
 
Theoremtr0 4007 The empty set is transitive. (Contributed by NM, 16-Sep-1993.)
 |- 
 Tr  (/)
 
Theoremtrv 4008 The universe is transitive. (Contributed by NM, 14-Sep-2003.)
 |- 
 Tr  _V
 
Theoremtriun 4009* The indexed union of a class of transitive sets is transitive. (Contributed by Mario Carneiro, 16-Nov-2014.)
 |-  ( A. x  e.  A  Tr  B  ->  Tr  U_ x  e.  A  B )
 
Theoremtruni 4010* The union of a class of transitive sets is transitive. Exercise 5(a) of [Enderton] p. 73. (Contributed by Scott Fenton, 21-Feb-2011.) (Proof shortened by Mario Carneiro, 26-Apr-2014.)
 |-  ( A. x  e.  A  Tr  x  ->  Tr  U. A )
 
Theoremtrint 4011* The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. (Contributed by Scott Fenton, 25-Feb-2011.)
 |-  ( A. x  e.  A  Tr  x  ->  Tr  |^| A )
 
Theoremtrintssm 4012* Any inhabited transitive class includes its intersection. Similar to Exercise 3 in [TakeutiZaring] p. 44 (which mistakenly does not include the inhabitedness hypothesis). (Contributed by Jim Kingdon, 22-Aug-2018.)
 |-  ( ( Tr  A  /\  E. x  x  e.  A )  ->  |^| A  C_  A )
 
2.2  IZF Set Theory - add the Axioms of Collection and Separation
 
2.2.1  Introduce the Axiom of Collection
 
Axiomax-coll 4013* Axiom of Collection. Axiom 7 of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed). It is similar to bnd 4066 but uses a freeness hypothesis in place of one of the distinct variable constraints. (Contributed by Jim Kingdon, 23-Aug-2018.)
 |- 
 F/ b ph   =>    |-  ( A. x  e.  a  E. y ph  ->  E. b A. x  e.  a  E. y  e.  b  ph )
 
Theoremrepizf 4014* Axiom of Replacement. Axiom 7' of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed). In our context this is not an axiom, but a theorem proved from ax-coll 4013. It is identical to zfrep6 4015 except for the choice of a freeness hypothesis rather than a distinct variable constraint between  b and  ph. (Contributed by Jim Kingdon, 23-Aug-2018.)
 |- 
 F/ b ph   =>    |-  ( A. x  e.  a  E! y ph  ->  E. b A. x  e.  a  E. y  e.  b  ph )
 
Theoremzfrep6 4015* A version of the Axiom of Replacement. Normally  ph would have free variables  x and  y. Axiom 6 of [Kunen] p. 12. The Separation Scheme ax-sep 4016 cannot be derived from this version and must be stated as a separate axiom in an axiom system (such as Kunen's) that uses this version. (Contributed by NM, 10-Oct-2003.)
 |-  ( A. x  e.  z  E! y ph  ->  E. w A. x  e.  z  E. y  e.  w  ph )
 
2.2.2  Introduce the Axiom of Separation
 
Axiomax-sep 4016* The Axiom of Separation of IZF set theory. Axiom 6 of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed, and with a  F/ y ph condition replaced by a distinct variable constraint between  y and  ph).

The Separation Scheme is a weak form of Frege's Axiom of Comprehension, conditioning it (with  x  e.  z) so that it asserts the existence of a collection only if it is smaller than some other collection  z that already exists. This prevents Russell's paradox ru 2881. In some texts, this scheme is called "Aussonderung" or the Subset Axiom.

(Contributed by NM, 11-Sep-2006.)

 |- 
 E. y A. x ( x  e.  y  <->  ( x  e.  z  /\  ph ) )
 
Theoremaxsep2 4017* A less restrictive version of the Separation Scheme ax-sep 4016, where variables  x and  z can both appear free in the wff  ph, which can therefore be thought of as  ph ( x ,  z ). This version was derived from the more restrictive ax-sep 4016 with no additional set theory axioms. (Contributed by NM, 10-Dec-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
 |- 
 E. y A. x ( x  e.  y  <->  ( x  e.  z  /\  ph ) )
 
Theoremzfauscl 4018* Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 4016, we invoke the Axiom of Extensionality (indirectly via vtocl 2714), which is needed for the justification of class variable notation. (Contributed by NM, 5-Aug-1993.)
 |-  A  e.  _V   =>    |-  E. y A. x ( x  e.  y  <->  ( x  e.  A  /\  ph )
 )
 
Theorembm1.3ii 4019* Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 4016. Similar to Theorem 1.3ii of [BellMachover] p. 463. (Contributed by NM, 5-Aug-1993.)
 |- 
 E. x A. y
 ( ph  ->  y  e.  x )   =>    |- 
 E. x A. y
 ( y  e.  x  <->  ph )
 
Theorema9evsep 4020* Derive a weakened version of ax-i9 1495, where  x and  y must be distinct, from Separation ax-sep 4016 and Extensionality ax-ext 2099. The theorem  -.  A. x -.  x  =  y also holds (ax9vsep 4021), but in intuitionistic logic  E. x x  =  y is stronger. (Contributed by Jim Kingdon, 25-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 E. x  x  =  y
 
Theoremax9vsep 4021* Derive a weakened version of ax-9 1496, where  x and  y must be distinct, from Separation ax-sep 4016 and Extensionality ax-ext 2099. In intuitionistic logic a9evsep 4020 is stronger and also holds. (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 -.  A. x  -.  x  =  y
 
2.2.3  Derive the Null Set Axiom
 
Theoremzfnuleu 4022* Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 2102 to strengthen the hypothesis in the form of axnul 4023). (Contributed by NM, 22-Dec-2007.)
 |- 
 E. x A. y  -.  y  e.  x   =>    |-  E! x A. y  -.  y  e.  x
 
Theoremaxnul 4023* The Null Set Axiom of ZF set theory: there exists a set with no elements. Axiom of Empty Set of [Enderton] p. 18. In some textbooks, this is presented as a separate axiom; here we show it can be derived from Separation ax-sep 4016. This version of the Null Set Axiom tells us that at least one empty set exists, but does not tell us that it is unique - we need the Axiom of Extensionality to do that (see zfnuleu 4022).

This theorem should not be referenced by any proof. Instead, use ax-nul 4024 below so that the uses of the Null Set Axiom can be more easily identified. (Contributed by Jeff Hoffman, 3-Feb-2008.) (Revised by NM, 4-Feb-2008.) (New usage is discouraged.) (Proof modification is discouraged.)

 |- 
 E. x A. y  -.  y  e.  x
 
Axiomax-nul 4024* The Null Set Axiom of IZF set theory. It was derived as axnul 4023 above and is therefore redundant, but we state it as a separate axiom here so that its uses can be identified more easily. Axiom 4 of [Crosilla] p. "Axioms of CZF and IZF". (Contributed by NM, 7-Aug-2003.)
 |- 
 E. x A. y  -.  y  e.  x
 
Theorem0ex 4025 The Null Set Axiom of ZF set theory: the empty set exists. Corollary 5.16 of [TakeutiZaring] p. 20. For the unabbreviated version, see ax-nul 4024. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  (/)  e.  _V
 
Theoremcsbexga 4026 The existence of proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
 |-  ( ( A  e.  V  /\  A. x  B  e.  W )  ->  [_ A  /  x ]_ B  e.  _V )
 
Theoremcsbexa 4027 The existence of proper substitution into a class. (Contributed by NM, 7-Aug-2007.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  [_ A  /  x ]_ B  e.  _V
 
2.2.4  Theorems requiring subset and intersection existence
 
Theoremnalset 4028* No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.)
 |- 
 -.  E. x A. y  y  e.  x
 
Theoremvnex 4029 The universal class does not exist as a set. (Contributed by NM, 4-Jul-2005.)
 |- 
 -.  E. x  x  =  _V
 
Theoremvprc 4030 The universal class is not a member of itself (and thus is not a set). Proposition 5.21 of [TakeutiZaring] p. 21; our proof, however, does not depend on the Axiom of Regularity. (Contributed by NM, 23-Aug-1993.)
 |- 
 -.  _V  e.  _V
 
Theoremnvel 4031 The universal class does not belong to any class. (Contributed by FL, 31-Dec-2006.)
 |- 
 -.  _V  e.  A
 
Theoreminex1 4032 Separation Scheme (Aussonderung) using class notation. Compare Exercise 4 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.)
 |-  A  e.  _V   =>    |-  ( A  i^i  B )  e.  _V
 
Theoreminex2 4033 Separation Scheme (Aussonderung) using class notation. (Contributed by NM, 27-Apr-1994.)
 |-  A  e.  _V   =>    |-  ( B  i^i  A )  e.  _V
 
Theoreminex1g 4034 Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.)
 |-  ( A  e.  V  ->  ( A  i^i  B )  e.  _V )
 
Theoremssex 4035 The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22. This is one way to express the Axiom of Separation ax-sep 4016 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.)
 |-  B  e.  _V   =>    |-  ( A  C_  B  ->  A  e.  _V )
 
Theoremssexi 4036 The subset of a set is also a set. (Contributed by NM, 9-Sep-1993.)
 |-  B  e.  _V   &    |-  A  C_  B   =>    |-  A  e.  _V
 
Theoremssexg 4037 The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22 (generalized). (Contributed by NM, 14-Aug-1994.)
 |-  ( ( A  C_  B  /\  B  e.  C )  ->  A  e.  _V )
 
Theoremssexd 4038 A subclass of a set is a set. Deduction form of ssexg 4037. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  B  e.  C )   &    |-  ( ph  ->  A 
 C_  B )   =>    |-  ( ph  ->  A  e.  _V )
 
Theoremdifexg 4039 Existence of a difference. (Contributed by NM, 26-May-1998.)
 |-  ( A  e.  V  ->  ( A  \  B )  e.  _V )
 
Theoremzfausab 4040* Separation Scheme (Aussonderung) in terms of a class abstraction. (Contributed by NM, 8-Jun-1994.)
 |-  A  e.  _V   =>    |-  { x  |  ( x  e.  A  /\  ph ) }  e.  _V
 
Theoremrabexg 4041* Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 23-Oct-1999.)
 |-  ( A  e.  V  ->  { x  e.  A  |  ph }  e.  _V )
 
Theoremrabex 4042* Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 19-Jul-1996.)
 |-  A  e.  _V   =>    |-  { x  e.  A  |  ph }  e.  _V
 
Theoremelssabg 4043* Membership in a class abstraction involving a subset. Unlike elabg 2803,  A does not have to be a set. (Contributed by NM, 29-Aug-2006.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( B  e.  V  ->  ( A  e.  { x  |  ( x 
 C_  B  /\  ph ) } 
 <->  ( A  C_  B  /\  ps ) ) )
 
Theoreminteximm 4044* The intersection of an inhabited class exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
 |-  ( E. x  x  e.  A  ->  |^| A  e.  _V )
 
Theoremintexr 4045 If the intersection of a class exists, the class is nonempty. (Contributed by Jim Kingdon, 27-Aug-2018.)
 |-  ( |^| A  e.  _V 
 ->  A  =/=  (/) )
 
Theoremintnexr 4046 If a class intersection is the universe, it is not a set. In classical logic this would be an equivalence. (Contributed by Jim Kingdon, 27-Aug-2018.)
 |-  ( |^| A  =  _V  ->  -.  |^| A  e.  _V )
 
Theoremintexabim 4047 The intersection of an inhabited class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
 |-  ( E. x ph  -> 
 |^| { x  |  ph }  e.  _V )
 
Theoremintexrabim 4048 The intersection of an inhabited restricted class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
 |-  ( E. x  e.  A  ph  ->  |^| { x  e.  A  |  ph }  e.  _V )
 
Theoremiinexgm 4049* The existence of an indexed union. 
x is normally a free-variable parameter in  B, which should be read  B ( x ). (Contributed by Jim Kingdon, 28-Aug-2018.)
 |-  ( ( E. x  x  e.  A  /\  A. x  e.  A  B  e.  C )  ->  |^|_ x  e.  A  B  e.  _V )
 
Theoreminuni 4050* The intersection of a union  U. A with a class  B is equal to the union of the intersections of each element of  A with  B. (Contributed by FL, 24-Mar-2007.)
 |-  ( U. A  i^i  B )  =  U. { x  |  E. y  e.  A  x  =  ( y  i^i  B ) }
 
Theoremelpw2g 4051 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 7-Aug-2000.)
 |-  ( B  e.  V  ->  ( A  e.  ~P B 
 <->  A  C_  B )
 )
 
Theoremelpw2 4052 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 11-Oct-2007.)
 |-  B  e.  _V   =>    |-  ( A  e.  ~P B  <->  A  C_  B )
 
Theorempwnss 4053 The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.)
 |-  ( A  e.  V  ->  -.  ~P A  C_  A )
 
Theorempwne 4054 No set equals its power set. The sethood antecedent is necessary; compare pwv 3705. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
 |-  ( A  e.  V  ->  ~P A  =/=  A )
 
Theoremrepizf2lem 4055 Lemma for repizf2 4056. If we have a function-like proposition which provides at most one value of  y for each  x in a set  w, we can change "at most one" to "exactly one" by restricting the values of  x to those values for which the proposition provides a value of  y. (Contributed by Jim Kingdon, 7-Sep-2018.)
 |-  ( A. x  e.  w  E* y ph  <->  A. x  e.  { x  e.  w  |  E. y ph } E! y ph )
 
Theoremrepizf2 4056* Replacement. This version of replacement is stronger than repizf 4014 in the sense that  ph does not need to map all values of  x in  w to a value of  y. The resulting set contains those elements for which there is a value of  y and in that sense, this theorem combines repizf 4014 with ax-sep 4016. Another variation would be  A. x  e.  w E* y ph  ->  { y  |  E. x ( x  e.  w  /\  ph ) }  e.  _V but we don't have a proof of that yet. (Contributed by Jim Kingdon, 7-Sep-2018.)
 |- 
 F/ z ph   =>    |-  ( A. x  e.  w  E* y ph  ->  E. z A. x  e.  { x  e.  w  |  E. y ph } E. y  e.  z  ph )
 
2.2.5  Theorems requiring empty set existence
 
Theoremclass2seteq 4057* Equality theorem for classes and sets . (Contributed by NM, 13-Dec-2005.) (Proof shortened by Raph Levien, 30-Jun-2006.)
 |-  ( A  e.  V  ->  { x  e.  A  |  A  e.  _V }  =  A )
 
Theorem0elpw 4058 Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.)
 |-  (/)  e.  ~P A
 
Theorem0nep0 4059 The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.)
 |-  (/)  =/=  { (/) }
 
Theorem0inp0 4060 Something cannot be equal to both the null set and the power set of the null set. (Contributed by NM, 30-Sep-2003.)
 |-  ( A  =  (/)  ->  -.  A  =  { (/) } )
 
Theoremunidif0 4061 The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.)
 |- 
 U. ( A  \  { (/) } )  = 
 U. A
 
Theoremiin0imm 4062* An indexed intersection of the empty set, with an inhabited index set, is empty. (Contributed by Jim Kingdon, 29-Aug-2018.)
 |-  ( E. y  y  e.  A  ->  |^|_ x  e.  A  (/)  =  (/) )
 
Theoremiin0r 4063* If an indexed intersection of the empty set is empty, the index set is nonempty. (Contributed by Jim Kingdon, 29-Aug-2018.)
 |-  ( |^|_ x  e.  A  (/) 
 =  (/)  ->  A  =/=  (/) )
 
Theoremintv 4064 The intersection of the universal class is empty. (Contributed by NM, 11-Sep-2008.)
 |- 
 |^| _V  =  (/)
 
Theoremaxpweq 4065* Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4068 is not used by the proof. (Contributed by NM, 22-Jun-2009.)
 |-  A  e.  _V   =>    |-  ( ~P A  e.  _V  <->  E. x A. y
 ( A. z ( z  e.  y  ->  z  e.  A )  ->  y  e.  x ) )
 
2.2.6  Collection principle
 
Theorembnd 4066* A very strong generalization of the Axiom of Replacement (compare zfrep6 4015). Its strength lies in the rather profound fact that  ph ( x ,  y ) does not have to be a "function-like" wff, as it does in the standard Axiom of Replacement. This theorem is sometimes called the Boundedness Axiom. In the context of IZF, it is just a slight variation of ax-coll 4013. (Contributed by NM, 17-Oct-2004.)
 |-  ( A. x  e.  z  E. y ph  ->  E. w A. x  e.  z  E. y  e.  w  ph )
 
Theorembnd2 4067* A variant of the Boundedness Axiom bnd 4066 that picks a subset  z out of a possibly proper class 
B in which a property is true. (Contributed by NM, 4-Feb-2004.)
 |-  A  e.  _V   =>    |-  ( A. x  e.  A  E. y  e.  B  ph  ->  E. z
 ( z  C_  B  /\  A. x  e.  A  E. y  e.  z  ph ) )
 
2.3  IZF Set Theory - add the Axioms of Power Sets and Pairing
 
2.3.1  Introduce the Axiom of Power Sets
 
Axiomax-pow 4068* Axiom of Power Sets. An axiom of Intuitionistic Zermelo-Fraenkel set theory. It states that a set 
y exists that includes the power set of a given set  x i.e. contains every subset of  x. This is Axiom 8 of [Crosilla] p. "Axioms of CZF and IZF" except (a) unnecessary quantifiers are removed, and (b) Crosilla has a biconditional rather than an implication (but the two are equivalent by bm1.3ii 4019).

The variant axpow2 4070 uses explicit subset notation. A version using class notation is pwex 4077. (Contributed by NM, 5-Aug-1993.)

 |- 
 E. y A. z
 ( A. w ( w  e.  z  ->  w  e.  x )  ->  z  e.  y )
 
Theoremzfpow 4069* Axiom of Power Sets expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)
 |- 
 E. x A. y
 ( A. x ( x  e.  y  ->  x  e.  z )  ->  y  e.  x )
 
Theoremaxpow2 4070* A variant of the Axiom of Power Sets ax-pow 4068 using subset notation. Problem in {BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
 |- 
 E. y A. z
 ( z  C_  x  ->  z  e.  y )
 
Theoremaxpow3 4071* A variant of the Axiom of Power Sets ax-pow 4068. For any set  x, there exists a set  y whose members are exactly the subsets of  x i.e. the power set of  x. Axiom Pow of [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
 |- 
 E. y A. z
 ( z  C_  x  <->  z  e.  y )
 
Theoremel 4072* Every set is an element of some other set. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |- 
 E. y  x  e.  y
 
Theoremvpwex 4073 Power set axiom: the powerclass of a set is a set. Axiom 4 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) Revised to prove pwexg 4074 from vpwex 4073. (Revised by BJ, 10-Aug-2022.)
 |- 
 ~P x  e.  _V
 
Theorempwexg 4074 Power set axiom expressed in class notation, with the sethood requirement as an antecedent. (Contributed by NM, 30-Oct-2003.)
 |-  ( A  e.  V  ->  ~P A  e.  _V )
 
Theorempwexd 4075 Deduction version of the power set axiom. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
 |-  ( ph  ->  A  e.  V )   =>    |-  ( ph  ->  ~P A  e.  _V )
 
Theoremabssexg 4076* Existence of a class of subsets. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  ( A  e.  V  ->  { x  |  ( x  C_  A  /\  ph ) }  e.  _V )
 
Theorempwex 4077 Power set axiom expressed in class notation. (Contributed by NM, 21-Jun-1993.)
 |-  A  e.  _V   =>    |-  ~P A  e.  _V
 
Theoremsnexg 4078 A singleton whose element exists is a set. The  A  e.  _V case of Theorem 7.12 of [Quine] p. 51, proved using only Extensionality, Power Set, and Separation. Replacement is not needed. (Contributed by Jim Kingdon, 1-Sep-2018.)
 |-  ( A  e.  V  ->  { A }  e.  _V )
 
Theoremsnex 4079 A singleton whose element exists is a set. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 24-May-2019.)
 |-  A  e.  _V   =>    |-  { A }  e.  _V
 
Theoremsnexprc 4080 A singleton whose element is a proper class is a set. The  -.  A  e.  _V case of Theorem 7.12 of [Quine] p. 51, proved using only Extensionality, Power Set, and Separation. Replacement is not needed. (Contributed by Jim Kingdon, 1-Sep-2018.)
 |-  ( -.  A  e.  _V 
 ->  { A }  e.  _V )
 
Theoremnotnotsnex 4081 A singleton is never a proper class. (Contributed by Mario Carneiro and Jim Kingdon, 3-Jul-2022.)
 |- 
 -.  -.  { A }  e.  _V
 
Theoremp0ex 4082 The power set of the empty set (the ordinal 1) is a set. (Contributed by NM, 23-Dec-1993.)
 |- 
 { (/) }  e.  _V
 
Theorempp0ex 4083  { (/) ,  { (/)
} } (the ordinal 2) is a set. (Contributed by NM, 5-Aug-1993.)
 |- 
 { (/) ,  { (/) } }  e.  _V
 
Theoremord3ex 4084 The ordinal number 3 is a set, proved without the Axiom of Union. (Contributed by NM, 2-May-2009.)
 |- 
 { (/) ,  { (/) } ,  { (/) ,  { (/) } } }  e.  _V
 
Theoremdtruarb 4085* At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). This theorem asserts the existence of two sets which do not equal each other; compare with dtruex 4444 in which we are given a set  y and go from there to a set  x which is not equal to it. (Contributed by Jim Kingdon, 2-Sep-2018.)
 |- 
 E. x E. y  -.  x  =  y
 
Theorempwuni 4086 A class is a subclass of the power class of its union. Exercise 6(b) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.)
 |-  A  C_  ~P U. A
 
Theoremundifexmid 4087* Union of complementary parts producing the whole and excluded middle. Although special cases such as undifss 3413 and undifdcss 6779 are provable, the full statement implies excluded middle as shown here. (Contributed by Jim Kingdon, 16-Jun-2022.)
 |-  ( x  C_  y  <->  ( x  u.  ( y 
 \  x ) )  =  y )   =>    |-  ( ph  \/  -.  ph )
 
2.3.2  A notation for excluded middle
 
Syntaxwem 4088 Formula for an abbreviation of excluded middle.
 wff EXMID
 
Definitiondf-exmid 4089 The expression EXMID will be used as a readable shorthand for any form of the law of the excluded middle; this is a useful shorthand largely because it hides statements of the form "for any proposition" in a system which can only quantify over sets, not propositions.

To see how this compares with other ways of expressing excluded middle, compare undifexmid 4087 with exmidundif 4099. The former may be more recognizable as excluded middle because it is in terms of propositions, and the proof may be easier to follow for much the same reason (it just has to show  ph and  -.  ph in the the relevant parts of the proof). The latter, however, has the key advantage of being able to prove both directions of the biconditional. To state that excluded middle implies a proposition is hard to do gracefully without EXMID, because there is no way to write a hypothesis  ph  \/  -.  ph for an arbitrary proposition; instead the hypothesis would need to be the particular instance of excluded middle which that proof needs. Or to say it another way, EXMID implies DECID  ph by exmidexmid 4090 but there is no good way to express the converse.

This definition and how we use it is easiest to understand (and most appropriate to assign the name "excluded middle" to) if we assume ax-sep 4016, in which case EXMID means that all propositions are decidable (see exmidexmid 4090 and notice that it relies on ax-sep 4016). If we instead work with ax-bdsep 13009, EXMID as defined here means that all bounded propositions are decidable.

(Contributed by Mario Carneiro and Jim Kingdon, 18-Jun-2022.)

 |-  (EXMID  <->  A. x ( x  C_  { (/) }  -> DECID  (/)  e.  x ) )
 
Theoremexmidexmid 4090 EXMID implies that an arbitrary proposition is decidable. That is, EXMID captures the usual meaning of excluded middle when stated in terms of propositions.

To get other propositional statements which are equivalent to excluded middle, combine this with notnotrdc 813, peircedc 884, or condc 823.

(Contributed by Jim Kingdon, 18-Jun-2022.)

 |-  (EXMID 
 -> DECID  ph )
 
Theoremexmid01 4091 Excluded middle is equivalent to saying any subset of  { (/)
} is either  (/) or  { (/) }. (Contributed by BJ and Jim Kingdon, 18-Jun-2022.)
 |-  (EXMID  <->  A. x ( x  C_  { (/) }  ->  ( x  =  (/)  \/  x  =  { (/) } ) ) )
 
Theorempwntru 4092 A slight strengthening of pwtrufal 13119. (Contributed by Mario Carneiro and Jim Kingdon, 12-Sep-2023.)
 |-  ( ( A  C_  { (/) }  /\  A  =/=  { (/) } )  ->  A  =  (/) )
 
Theoremexmid1dc 4093* A convenience theorem for proving that something implies EXMID. Think of this as an alternative to using a proposition, as in proofs like undifexmid 4087 or ordtriexmid 4407. In this context  x  =  { (/) } can be thought of as "x is true". (Contributed by Jim Kingdon, 21-Nov-2023.)
 |-  ( ( ph  /\  x  C_ 
 { (/) } )  -> DECID  x  =  { (/) } )   =>    |-  ( ph  -> EXMID )
 
Theoremexmidn0m 4094* Excluded middle is equivalent to any set being empty or inhabited. (Contributed by Jim Kingdon, 5-Mar-2023.)
 |-  (EXMID  <->  A. x ( x  =  (/)  \/  E. y  y  e.  x ) )
 
Theoremexmidsssn 4095* Excluded middle is equivalent to the biconditionalized version of sssnr 3650 for sets. (Contributed by Jim Kingdon, 5-Mar-2023.)
 |-  (EXMID  <->  A. x A. y ( x  C_  { y } 
 <->  ( x  =  (/)  \/  x  =  { y } ) ) )
 
Theoremexmidsssnc 4096* Excluded middle in terms of subsets of a singleton. This is similar to exmid01 4091 but lets you choose any set as the element of the singleton rather than just  (/). It is similar to exmidsssn 4095 but for a particular set  B rather than all sets. (Contributed by Jim Kingdon, 29-Jul-2023.)
 |-  ( B  e.  V  ->  (EXMID  <->  A. x ( x  C_  { B }  ->  ( x  =  (/)  \/  x  =  { B } )
 ) ) )
 
Theoremexmid0el 4097 Excluded middle is equivalent to decidability of  (/) being an element of an arbitrary set. (Contributed by Jim Kingdon, 18-Jun-2022.)
 |-  (EXMID  <->  A. xDECID  (/)  e.  x )
 
Theoremexmidel 4098* Excluded middle is equivalent to decidability of membership for two arbitrary sets. (Contributed by Jim Kingdon, 18-Jun-2022.)
 |-  (EXMID  <->  A. x A. yDECID  x  e.  y )
 
Theoremexmidundif 4099* Excluded middle is equivalent to every subset having a complement. That is, the union of a subset and its relative complement being the whole set. Although special cases such as undifss 3413 and undifdcss 6779 are provable, the full statement is equivalent to excluded middle as shown here. (Contributed by Jim Kingdon, 18-Jun-2022.)
 |-  (EXMID  <->  A. x A. y ( x  C_  y  <->  ( x  u.  ( y  \  x ) )  =  y ) )
 
Theoremexmidundifim 4100* Excluded middle is equivalent to every subset having a complement. Variation of exmidundif 4099 with an implication rather than a biconditional. (Contributed by Jim Kingdon, 16-Feb-2023.)
 |-  (EXMID  <->  A. x A. y ( x  C_  y  ->  ( x  u.  ( y 
 \  x ) )  =  y ) )
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