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Theorem List for Intuitionistic Logic Explorer - 3801-3900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmpteq12f 3801 An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
((x A = 𝐶 x A B = 𝐷) → (x AB) = (x 𝐶𝐷))
 
Theoremmpteq12dva 3802* An equality inference for the maps to notation. (Contributed by Mario Carneiro, 26-Jan-2017.)
(φA = 𝐶)    &   ((φ x A) → B = 𝐷)       (φ → (x AB) = (x 𝐶𝐷))
 
Theoremmpteq12dv 3803* An equality inference for the maps to notation. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 16-Dec-2013.)
(φA = 𝐶)    &   (φB = 𝐷)       (φ → (x AB) = (x 𝐶𝐷))
 
Theoremmpteq12 3804* An equality theorem for the maps to notation. (Contributed by NM, 16-Dec-2013.)
((A = 𝐶 x A B = 𝐷) → (x AB) = (x 𝐶𝐷))
 
Theoremmpteq1 3805* An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
(A = B → (x A𝐶) = (x B𝐶))
 
Theoremmpteq1d 3806* An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 11-Jun-2016.)
(φA = B)       (φ → (x A𝐶) = (x B𝐶))
 
Theoremmpteq2ia 3807 An equality inference for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
(x AB = 𝐶)       (x AB) = (x A𝐶)
 
Theoremmpteq2i 3808 An equality inference for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
B = 𝐶       (x AB) = (x A𝐶)
 
Theoremmpteq12i 3809 An equality inference for the maps to notation. (Contributed by Scott Fenton, 27-Oct-2010.) (Revised by Mario Carneiro, 16-Dec-2013.)
A = 𝐶    &   B = 𝐷       (x AB) = (x 𝐶𝐷)
 
Theoremmpteq2da 3810 Slightly more general equality inference for the maps to notation. (Contributed by FL, 14-Sep-2013.) (Revised by Mario Carneiro, 16-Dec-2013.)
xφ    &   ((φ x A) → B = 𝐶)       (φ → (x AB) = (x A𝐶))
 
Theoremmpteq2dva 3811* Slightly more general equality inference for the maps to notation. (Contributed by Scott Fenton, 25-Apr-2012.)
((φ x A) → B = 𝐶)       (φ → (x AB) = (x A𝐶))
 
Theoremmpteq2dv 3812* An equality inference for the maps to notation. (Contributed by Mario Carneiro, 23-Aug-2014.)
(φB = 𝐶)       (φ → (x AB) = (x A𝐶))
 
Theoremnfmpt 3813* Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)
xA    &   xB       x(y AB)
 
Theoremnfmpt1 3814 Bound-variable hypothesis builder for the maps-to notation. (Contributed by FL, 17-Feb-2008.)
x(x AB)
 
Theoremcbvmpt 3815* Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 11-Sep-2011.)
yB    &   x𝐶    &   (x = yB = 𝐶)       (x AB) = (y A𝐶)
 
Theoremcbvmptv 3816* Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by Mario Carneiro, 19-Feb-2013.)
(x = yB = 𝐶)       (x AB) = (y A𝐶)
 
Theoremmptv 3817* Function with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.)
(x V ↦ B) = {⟨x, y⟩ ∣ y = B}
 
2.1.24  Transitive classes
 
Syntaxwtr 3818 Extend wff notation to include transitive classes. Notation from [TakeutiZaring] p. 35.
wff Tr A
 
Definitiondf-tr 3819 Define the transitive class predicate. Definition of [Enderton] p. 71 extended to arbitrary classes. For alternate definitions, see dftr2 3820 (which is suggestive of the word "transitive"), dftr3 3822, dftr4 3823, and dftr5 3821. The term "complete" is used instead of "transitive" in Definition 3 of [Suppes] p. 130. (Contributed by NM, 29-Aug-1993.)
(Tr A AA)
 
Theoremdftr2 3820* An alternate way of defining a transitive class. Exercise 7 of [TakeutiZaring] p. 40. (Contributed by NM, 24-Apr-1994.)
(Tr Axy((x y y A) → x A))
 
Theoremdftr5 3821* An alternate way of defining a transitive class. (Contributed by NM, 20-Mar-2004.)
(Tr Ax A y x y A)
 
Theoremdftr3 3822* An alternate way of defining a transitive class. Definition 7.1 of [TakeutiZaring] p. 35. (Contributed by NM, 29-Aug-1993.)
(Tr Ax A xA)
 
Theoremdftr4 3823 An alternate way of defining a transitive class. Definition of [Enderton] p. 71. (Contributed by NM, 29-Aug-1993.)
(Tr AA ⊆ 𝒫 A)
 
Theoremtreq 3824 Equality theorem for the transitive class predicate. (Contributed by NM, 17-Sep-1993.)
(A = B → (Tr A ↔ Tr B))
 
Theoremtrel 3825 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 𝐶 𝐶 A) → B A))
 
Theoremtrel3 3826 In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.)
(Tr A → ((B 𝐶 𝐶 𝐷 𝐷 A) → B A))
 
Theoremtrss 3827 An element of a transitive class is a subset of the class. (Contributed by NM, 7-Aug-1994.)
(Tr A → (B ABA))
 
Theoremtrin 3828 The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.)
((Tr A Tr B) → Tr (AB))
 
Theoremtr0 3829 The empty set is transitive. (Contributed by NM, 16-Sep-1993.)
Tr ∅
 
Theoremtrv 3830 The universe is transitive. (Contributed by NM, 14-Sep-2003.)
Tr V
 
Theoremtriun 3831* The indexed union of a class of transitive sets is transitive. (Contributed by Mario Carneiro, 16-Nov-2014.)
(x A Tr B → Tr x A B)
 
Theoremtruni 3832* 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.)
(x A Tr x → Tr A)
 
Theoremtrint 3833* The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. (Contributed by Scott Fenton, 25-Feb-2011.)
(x A Tr x → Tr A)
 
Theoremtrintssm 3834* If A is transitive and inhabited, then A is a subset of A. (Contributed by Jim Kingdon, 22-Aug-2018.)
((x x A Tr A) → AA)
 
Theoremtrint0m 3835* Any inhabited transitive class includes its intersection. Similar to Exercise 2 in [TakeutiZaring] p. 44. (Contributed by Jim Kingdon, 22-Aug-2018.)
((Tr A x x A) → AA)
 
2.2  IZF Set Theory - add the Axioms of Collection and Separation
 
2.2.1  Introduce the Axiom of Collection
 
Axiomax-coll 3836* Axiom of Collection. Axiom 7 of [Crosilla], p. "Axioms of CZF and IZF" (with unnecessary quantifier removed). It is similar to bnd 3889 but uses a freeness hypothesis in place of one of the distinct variable constraints. (Contributed by Jim Kingdon, 23-Aug-2018.)
𝑏φ       (x 𝑎 yφ𝑏x 𝑎 y 𝑏 φ)
 
Theoremrepizf 3837* 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 3836. It is identical to zfrep6 3838 except for the choice of a freeness hypothesis rather than a distinct variable constraint between 𝑏 and φ. (Contributed by Jim Kingdon, 23-Aug-2018.)
𝑏φ       (x 𝑎 ∃!yφ𝑏x 𝑎 y 𝑏 φ)
 
Theoremzfrep6 3838* A version of the Axiom of Replacement. Normally φ would have free variables x and y. Axiom 6 of [Kunen] p. 12. The Separation Scheme ax-sep 3839 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.)
(x z ∃!yφwx z y w φ)
 
2.2.2  Introduce the Axiom of Separation
 
Axiomax-sep 3839* 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 yφ condition replaced by a distinct variable constraint between y and φ).

The Separation Scheme is a weak form of Frege's Axiom of Comprehension, conditioning it (with x 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 2732. In some texts, this scheme is called "Aussonderung" or the Subset Axiom.

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

yx(x y ↔ (x z φ))
 
Theoremaxsep2 3840* A less restrictive version of the Separation Scheme ax-sep 3839, where variables x and z can both appear free in the wff φ, which can therefore be thought of as φ(x, z). This version was derived from the more restrictive ax-sep 3839 with no additional set theory axioms. (Contributed by NM, 10-Dec-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
yx(x y ↔ (x z φ))
 
Theoremzfauscl 3841* Separation Scheme (Aussonderung) using a class variable. To derive this from ax-sep 3839, we invoke the Axiom of Extensionality (indirectly via vtocl 2577), which is needed for the justification of class variable notation. (Contributed by NM, 5-Aug-1993.)
A V       yx(x y ↔ (x A φ))
 
Theorembm1.3ii 3842* Convert implication to equivalence using the Separation Scheme (Aussonderung) ax-sep 3839. Similar to Theorem 1.3ii of [BellMachover] p. 463. (Contributed by NM, 5-Aug-1993.)
xy(φy x)       xy(y xφ)
 
Theorema9evsep 3843* Derive a weakened version of ax-i9 1397, where x and y must be distinct, from Separation ax-sep 3839 and Extensionality ax-ext 1996. The theorem ¬ x¬ x = y also holds (ax9vsep 3844), but in intuitionistic logic xx = y is stronger. (Contributed by Jim Kingdon, 25-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
x x = y
 
Theoremax9vsep 3844* Derive a weakened version of ax-9 1398, where x and y must be distinct, from Separation ax-sep 3839 and Extensionality ax-ext 1996. In intuitionistic logic a9evsep 3843 is stronger and also holds. (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ x ¬ x = y
 
2.2.3  Derive the Null Set Axiom
 
Theoremzfnuleu 3845* Show the uniqueness of the empty set (using the Axiom of Extensionality via bm1.1 1999 to strengthen the hypothesis in the form of axnul 3846). (Contributed by NM, 22-Dec-2007.)
xy ¬ y x       ∃!xy ¬ y x
 
Theoremaxnul 3846* 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 3839. 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 3845).

This theorem should not be referenced by any proof. Instead, use ax-nul 3847 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.)

xy ¬ y x
 
Axiomax-nul 3847* The Null Set Axiom of IZF set theory. It was derived as axnul 3846 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.)
xy ¬ y x
 
Theorem0ex 3848 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 3847. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
V
 
Theoremcsbexga 3849 The existence of proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
((A 𝑉 x B 𝑊) → A / xB V)
 
Theoremcsbexa 3850 The existence of proper substitution into a class. (Contributed by NM, 7-Aug-2007.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
A V    &   B V       A / xB V
 
2.2.4  Theorems requiring subset and intersection existence
 
Theoremnalset 3851* No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.)
¬ xy y x
 
Theoremvprc 3852 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 V
 
Theoremnvel 3853 The universal class doesn't belong to any class. (Contributed by FL, 31-Dec-2006.)
¬ V A
 
Theoremvnex 3854 The universal class does not exist. (Contributed by NM, 4-Jul-2005.)
¬ x x = V
 
Theoreminex1 3855 Separation Scheme (Aussonderung) using class notation. Compare Exercise 4 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.)
A V       (AB) V
 
Theoreminex2 3856 Separation Scheme (Aussonderung) using class notation. (Contributed by NM, 27-Apr-1994.)
A V       (BA) V
 
Theoreminex1g 3857 Closed-form, generalized Separation Scheme. (Contributed by NM, 7-Apr-1995.)
(A 𝑉 → (AB) V)
 
Theoremssex 3858 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 3839 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.)
B V       (ABA V)
 
Theoremssexi 3859 The subset of a set is also a set. (Contributed by NM, 9-Sep-1993.)
B V    &   AB       A V
 
Theoremssexg 3860 The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22 (generalized). (Contributed by NM, 14-Aug-1994.)
((AB B 𝐶) → A V)
 
Theoremssexd 3861 A subclass of a set is a set. Deduction form of ssexg 3860. (Contributed by David Moews, 1-May-2017.)
(φB 𝐶)    &   (φAB)       (φA V)
 
Theoremdifexg 3862 Existence of a difference. (Contributed by NM, 26-May-1998.)
(A 𝑉 → (AB) V)
 
Theoremzfausab 3863* Separation Scheme (Aussonderung) in terms of a class abstraction. (Contributed by NM, 8-Jun-1994.)
A V       {x ∣ (x A φ)} V
 
Theoremrabexg 3864* Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 23-Oct-1999.)
(A 𝑉 → {x Aφ} V)
 
Theoremrabex 3865* Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 19-Jul-1996.)
A V       {x Aφ} V
 
Theoremelssabg 3866* Membership in a class abstraction involving a subset. Unlike elabg 2657, A does not have to be a set. (Contributed by NM, 29-Aug-2006.)
(x = A → (φψ))       (B 𝑉 → (A {x ∣ (xB φ)} ↔ (AB ψ)))
 
Theoreminteximm 3867* The intersection of an inhabited class exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
(x x A A V)
 
Theoremintexr 3868 If the intersection of a class exists, the class is non-empty. (Contributed by Jim Kingdon, 27-Aug-2018.)
( A V → A ≠ ∅)
 
Theoremintnexr 3869 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 V)
 
Theoremintexabim 3870 The intersection of an inhabited class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
(xφ {xφ} V)
 
Theoremintexrabim 3871 The intersection of an inhabited restricted class abstraction exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
(x A φ {x Aφ} V)
 
Theoremiinexgm 3872* 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.)
((x x A x A B 𝐶) → x A B V)
 
Theoreminuni 3873* The intersection of a union 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.)
( AB) = {xy A x = (yB)}
 
Theoremelpw2g 3874 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 7-Aug-2000.)
(B 𝑉 → (A 𝒫 BAB))
 
Theoremelpw2 3875 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 11-Oct-2007.)
B V       (A 𝒫 BAB)
 
Theorempwnss 3876 The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(A 𝑉 → ¬ 𝒫 AA)
 
Theorempwne 3877 No set equals its power set. The sethood antecedent is necessary; compare pwv 3543. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
(A 𝑉 → 𝒫 AA)
 
Theoremrepizf2lem 3878 Lemma for repizf2 3879. 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.)
(x w ∃*yφx {x wyφ}∃!yφ)
 
Theoremrepizf2 3879* Replacement. This version of replacement is stronger than repizf 3837 in the sense that φ 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 3837 with ax-sep 3839. Another variation would be x w∃*yφ → {yx(x w φ)} V but we don't have a proof of that yet. (Contributed by Jim Kingdon, 7-Sep-2018.)
zφ       (x w ∃*yφzx {x wyφ}y z φ)
 
2.2.5  Theorems requiring empty set existence
 
Theoremclass2seteq 3880* Equality theorem for classes and sets . (Contributed by NM, 13-Dec-2005.) (Proof shortened by Raph Levien, 30-Jun-2006.)
(A 𝑉 → {x AA V} = A)
 
Theorem0elpw 3881 Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.)
𝒫 A
 
Theorem0nep0 3882 The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.)
∅ ≠ {∅}
 
Theorem0inp0 3883 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 3884 The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.)
(A ∖ {∅}) = A
 
Theoremiin0imm 3885* An indexed intersection of the empty set, with an inhabited index set, is empty. (Contributed by Jim Kingdon, 29-Aug-2018.)
(y y A x A ∅ = ∅)
 
Theoremiin0r 3886* If an indexed intersection of the empty set is empty, the index set is non-empty. (Contributed by Jim Kingdon, 29-Aug-2018.)
( x A ∅ = ∅ → A ≠ ∅)
 
Theoremintv 3887 The intersection of the universal class is empty. (Contributed by NM, 11-Sep-2008.)
V = ∅
 
Theoremaxpweq 3888* Two equivalent ways to express the Power Set Axiom. Note that ax-pow 3891 is not used by the proof. (Contributed by NM, 22-Jun-2009.)
A V       (𝒫 A V ↔ xy(z(z yz A) → y x))
 
2.2.6  Collection principle
 
Theorembnd 3889* A very strong generalization of the Axiom of Replacement (compare zfrep6 3838). Its strength lies in the rather profound fact that φ(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 3836. (Contributed by NM, 17-Oct-2004.)
(x z yφwx z y w φ)
 
Theorembnd2 3890* A variant of the Boundedness Axiom bnd 3889 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 V       (x A y B φz(zB x A y z φ))
 
2.3  IZF Set Theory - add the Axioms of Power Sets and Pairing
 
2.3.1  Introduce the Axiom of Power Sets
 
Axiomax-pow 3891* 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 3842).

The variant axpow2 3893 uses explicit subset notation. A version using class notation is pwex 3896. (Contributed by NM, 5-Aug-1993.)

yz(w(w zw x) → z y)
 
Theoremzfpow 3892* Axiom of Power Sets expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)
xy(x(x yx z) → y x)
 
Theoremaxpow2 3893* A variant of the Axiom of Power Sets ax-pow 3891 using subset notation. Problem in {BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
yz(zxz y)
 
Theoremaxpow3 3894* A variant of the Axiom of Power Sets ax-pow 3891. 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.)
yz(zxz y)
 
Theoremel 3895* Every set is an element of some other set. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
y x y
 
Theorempwex 3896 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 V       𝒫 A V
 
Theorempwexg 3897 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 𝑉 → 𝒫 A V)
 
Theoremabssexg 3898* Existence of a class of subsets. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(A 𝑉 → {x ∣ (xA φ)} V)
 
TheoremsnexgOLD 3899 A singleton whose element exists is a set. The A V case of Theorem 7.12 of [Quine] p. 51, proved using only Extensionality, Power Set, and Separation. Replacement is not needed. This is a special case of snexg 3900 and new proofs should use snexg 3900 instead. (Contributed by Jim Kingdon, 26-Jan-2019.) (New usage is discouraged.) (Proof modification is discouraged.) TODO: replace its uses by uses of snexg 3900 and then remove it.
(A V → {A} V)
 
Theoremsnexg 3900 A singleton whose element exists is a set. The A 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 𝑉 → {A} V)
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