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Theorem List for Intuitionistic Logic Explorer - 3901-4000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtreq 3901 Equality theorem for the transitive class predicate. (Contributed by NM, 17-Sep-1993.)
 |-  ( A  =  B  ->  ( Tr  A  <->  Tr  B ) )
 
Theoremtrel 3902 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 3903 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 3904 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 3905 The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.)
 |-  ( ( Tr  A  /\  Tr  B )  ->  Tr  ( A  i^i  B ) )
 
Theoremtr0 3906 The empty set is transitive. (Contributed by NM, 16-Sep-1993.)
 |- 
 Tr  (/)
 
Theoremtrv 3907 The universe is transitive. (Contributed by NM, 14-Sep-2003.)
 |- 
 Tr  _V
 
Theoremtriun 3908* 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 3909* 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 3910* 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 3911* 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 )
 
TheoremtrintssmOLD 3912* Obsolete version of trintssm 3911 as of 30-Oct-2021. (Contributed by Jim Kingdon, 22-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( E. x  x  e.  A  /\  Tr  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 3913* Axiom of Collection. Axiom 7 of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed). It is similar to bnd 3966 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 3914* 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 3913. It is identical to zfrep6 3915 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 3915* 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 3916 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 3916* 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 2823. 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 3917* A less restrictive version of the Separation Scheme ax-sep 3916, 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 3916 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 3918* Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 3916, we invoke the Axiom of Extensionality (indirectly via vtocl 2662), 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 3919* Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 3916. 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 3920* Derive a weakened version of ax-i9 1464, where  x and  y must be distinct, from Separation ax-sep 3916 and Extensionality ax-ext 2065. The theorem  -.  A. x -.  x  =  y also holds (ax9vsep 3921), 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 3921* Derive a weakened version of ax-9 1465, where  x and  y must be distinct, from Separation ax-sep 3916 and Extensionality ax-ext 2065. In intuitionistic logic a9evsep 3920 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 3922* Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 2068 to strengthen the hypothesis in the form of axnul 3923). (Contributed by NM, 22-Dec-2007.)
 |- 
 E. x A. y  -.  y  e.  x   =>    |-  E! x A. y  -.  y  e.  x
 
Theoremaxnul 3923* 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 3916. 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 3922).

This theorem should not be referenced by any proof. Instead, use ax-nul 3924 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 3924* The Null Set Axiom of IZF set theory. It was derived as axnul 3923 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 3925 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 3924. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  (/)  e.  _V
 
Theoremcsbexga 3926 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 3927 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 3928* No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.)
 |- 
 -.  E. x A. y  y  e.  x
 
Theoremvprc 3929 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 3930 The universal class doesn't belong to any class. (Contributed by FL, 31-Dec-2006.)
 |- 
 -.  _V  e.  A
 
Theoremvnex 3931 The universal class does not exist. (Contributed by NM, 4-Jul-2005.)
 |- 
 -.  E. x  x  =  _V
 
Theoreminex1 3932 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 3933 Separation Scheme (Aussonderung) using class notation. (Contributed by NM, 27-Apr-1994.)
 |-  A  e.  _V   =>    |-  ( B  i^i  A )  e.  _V
 
Theoreminex1g 3934 Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.)
 |-  ( A  e.  V  ->  ( A  i^i  B )  e.  _V )
 
Theoremssex 3935 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 3916 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.)
 |-  B  e.  _V   =>    |-  ( A  C_  B  ->  A  e.  _V )
 
Theoremssexi 3936 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 3937 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 3938 A subclass of a set is a set. Deduction form of ssexg 3937. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  B  e.  C )   &    |-  ( ph  ->  A 
 C_  B )   =>    |-  ( ph  ->  A  e.  _V )
 
Theoremdifexg 3939 Existence of a difference. (Contributed by NM, 26-May-1998.)
 |-  ( A  e.  V  ->  ( A  \  B )  e.  _V )
 
Theoremzfausab 3940* 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 3941* 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 3942* 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 3943* Membership in a class abstraction involving a subset. Unlike elabg 2747,  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 3944* The intersection of an inhabited class exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
 |-  ( E. x  x  e.  A  ->  |^| A  e.  _V )
 
Theoremintexr 3945 If the intersection of a class exists, the class is non-empty. (Contributed by Jim Kingdon, 27-Aug-2018.)
 |-  ( |^| A  e.  _V 
 ->  A  =/=  (/) )
 
Theoremintnexr 3946 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 3947 The intersection of an inhabited class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
 |-  ( E. x ph  -> 
 |^| { x  |  ph }  e.  _V )
 
Theoremintexrabim 3948 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 3949* 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 3950* 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 3951 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 3952 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 3953 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 3954 No set equals its power set. The sethood antecedent is necessary; compare pwv 3620. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
 |-  ( A  e.  V  ->  ~P A  =/=  A )
 
Theoremrepizf2lem 3955 Lemma for repizf2 3956. 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 3956* Replacement. This version of replacement is stronger than repizf 3914 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 3914 with ax-sep 3916. 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 3957* 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 3958 Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.)
 |-  (/)  e.  ~P A
 
Theorem0nep0 3959 The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.)
 |-  (/)  =/=  { (/) }
 
Theorem0inp0 3960 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 3961 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 3962* 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 3963* If an indexed intersection of the empty set is empty, the index set is non-empty. (Contributed by Jim Kingdon, 29-Aug-2018.)
 |-  ( |^|_ x  e.  A  (/) 
 =  (/)  ->  A  =/=  (/) )
 
Theoremintv 3964 The intersection of the universal class is empty. (Contributed by NM, 11-Sep-2008.)
 |- 
 |^| _V  =  (/)
 
Theoremaxpweq 3965* Two equivalent ways to express the Power Set Axiom. Note that ax-pow 3968 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 3966* A very strong generalization of the Axiom of Replacement (compare zfrep6 3915). 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 3913. (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 3967* A variant of the Boundedness Axiom bnd 3966 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 3968* 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 3919).

The variant axpow2 3970 uses explicit subset notation. A version using class notation is pwex 3973. (Contributed by NM, 5-Aug-1993.)

 |- 
 E. y A. z
 ( A. w ( w  e.  z  ->  w  e.  x )  ->  z  e.  y )
 
Theoremzfpow 3969* 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 3970* A variant of the Axiom of Power Sets ax-pow 3968 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 3971* A variant of the Axiom of Power Sets ax-pow 3968. 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 3972* 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
 
Theorempwex 3973 Power set axiom expressed in class notation. Axiom 4 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
 |-  A  e.  _V   =>    |-  ~P A  e.  _V
 
Theorempwexg 3974 Power set axiom expressed in class notation, with the sethood requirement as an antecedent. Axiom 4 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Oct-2003.)
 |-  ( A  e.  V  ->  ~P A  e.  _V )
 
Theoremabssexg 3975* 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 )
 
Theoremsnexg 3976 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 3977 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 3978 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 )
 
Theoremp0ex 3979 The power set of the empty set (the ordinal 1) is a set. (Contributed by NM, 23-Dec-1993.)
 |- 
 { (/) }  e.  _V
 
Theorempp0ex 3980  { (/) ,  { (/)
} } (the ordinal 2) is a set. (Contributed by NM, 5-Aug-1993.)
 |- 
 { (/) ,  { (/) } }  e.  _V
 
Theoremord3ex 3981 The ordinal number 3 is a set, proved without the Axiom of Union. (Contributed by NM, 2-May-2009.)
 |- 
 { (/) ,  { (/) } ,  { (/) ,  { (/) } } }  e.  _V
 
Theoremdtruarb 3982* 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 4330 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 3983 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 3984* Union of complementary parts producing the whole and excluded middle. Although special cases such as undifss 3339 and undifdcss 6467 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 3985 Formula for an abbreviation of excluded middle.
 wff EXMID
 
Definitiondf-exmid 3986 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 3984 with exmidundif 3991. 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 3987 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 3916, in which case EXMID means that all propositions are decidable (see exmidexmid 3987 and notice that it relies on ax-sep 3916). If we instead work with ax-bdsep 10942, 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 3987 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 785, peircedc 854, or condc 783.

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

 |-  (EXMID 
 -> DECID  ph )
 
Theoremexmid01 3988 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  =  { (/) } ) ) )
 
Theoremexmid0el 3989 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 3990* 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 3991* 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 3339 and undifdcss 6467 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 ) )
 
2.3.3  Axiom of Pairing
 
Axiomax-pr 3992* The Axiom of Pairing of IZF set theory. Axiom 2 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 3919). (Contributed by NM, 14-Nov-2006.)
 |- 
 E. z A. w ( ( w  =  x  \/  w  =  y )  ->  w  e.  z )
 
Theoremzfpair2 3993 Derive the abbreviated version of the Axiom of Pairing from ax-pr 3992. (Contributed by NM, 14-Nov-2006.)
 |- 
 { x ,  y }  e.  _V
 
Theoremprexg 3994 The Axiom of Pairing using class variables. Theorem 7.13 of [Quine] p. 51, but restricted to classes which exist. For proper classes, see prprc 3520, prprc1 3518, and prprc2 3519. (Contributed by Jim Kingdon, 16-Sep-2018.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  { A ,  B }  e.  _V )
 
Theoremsnelpwi 3995 A singleton of a set belongs to the power class of a class containing the set. (Contributed by Alan Sare, 25-Aug-2011.)
 |-  ( A  e.  B  ->  { A }  e.  ~P B )
 
Theoremsnelpw 3996 A singleton of a set belongs to the power class of a class containing the set. (Contributed by NM, 1-Apr-1998.)
 |-  A  e.  _V   =>    |-  ( A  e.  B 
 <->  { A }  e.  ~P B )
 
Theoremprelpwi 3997 A pair of two sets belongs to the power class of a class containing those two sets. (Contributed by Thierry Arnoux, 10-Mar-2017.)
 |-  ( ( A  e.  C  /\  B  e.  C )  ->  { A ,  B }  e.  ~P C )
 
Theoremrext 3998* A theorem similar to extensionality, requiring the existence of a singleton. Exercise 8 of [TakeutiZaring] p. 16. (Contributed by NM, 10-Aug-1993.)
 |-  ( A. z ( x  e.  z  ->  y  e.  z )  ->  x  =  y )
 
Theoremsspwb 3999 Classes are subclasses if and only if their power classes are subclasses. Exercise 18 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.)
 |-  ( A  C_  B  <->  ~P A  C_  ~P B )
 
Theoremunipw 4000 A class equals the union of its power class. Exercise 6(a) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.) (Proof shortened by Alan Sare, 28-Dec-2008.)
 |- 
 U. ~P A  =  A
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