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Theorem List for Intuitionistic Logic Explorer - 3201-3300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremin32 3201 A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∩ 𝐵)

Theoremin13 3202 A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
(𝐴 ∩ (𝐵𝐶)) = (𝐶 ∩ (𝐵𝐴))

Theoremin31 3203 A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
((𝐴𝐵) ∩ 𝐶) = ((𝐶𝐵) ∩ 𝐴)

Theoreminrot 3204 Rotate the intersection of 3 classes. (Contributed by NM, 27-Aug-2012.)
((𝐴𝐵) ∩ 𝐶) = ((𝐶𝐴) ∩ 𝐵)

Theoremin4 3205 Rearrangement of intersection of 4 classes. (Contributed by NM, 21-Apr-2001.)
((𝐴𝐵) ∩ (𝐶𝐷)) = ((𝐴𝐶) ∩ (𝐵𝐷))

Theoreminindi 3206 Intersection distributes over itself. (Contributed by NM, 6-May-1994.)
(𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))

Theoreminindir 3207 Intersection distributes over itself. (Contributed by NM, 17-Aug-2004.)
((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))

Theoremsseqin2 3208 A relationship between subclass and intersection. Similar to Exercise 9 of [TakeutiZaring] p. 18. (Contributed by NM, 17-May-1994.)
(𝐴𝐵 ↔ (𝐵𝐴) = 𝐴)

Theoreminss1 3209 The intersection of two classes is a subset of one of them. Part of Exercise 12 of [TakeutiZaring] p. 18. (Contributed by NM, 27-Apr-1994.)
(𝐴𝐵) ⊆ 𝐴

Theoreminss2 3210 The intersection of two classes is a subset of one of them. Part of Exercise 12 of [TakeutiZaring] p. 18. (Contributed by NM, 27-Apr-1994.)
(𝐴𝐵) ⊆ 𝐵

Theoremssin 3211 Subclass of intersection. Theorem 2.8(vii) of [Monk1] p. 26. (Contributed by NM, 15-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐵𝐴𝐶) ↔ 𝐴 ⊆ (𝐵𝐶))

Theoremssini 3212 An inference showing that a subclass of two classes is a subclass of their intersection. (Contributed by NM, 24-Nov-2003.)
𝐴𝐵    &   𝐴𝐶       𝐴 ⊆ (𝐵𝐶)

Theoremssind 3213 A deduction showing that a subclass of two classes is a subclass of their intersection. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝐴𝐵)    &   (𝜑𝐴𝐶)       (𝜑𝐴 ⊆ (𝐵𝐶))

Theoremssrin 3214 Add right intersection to subclass relation. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))

Theoremsslin 3215 Add left intersection to subclass relation. (Contributed by NM, 19-Oct-1999.)
(𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))

Theoremss2in 3216 Intersection of subclasses. (Contributed by NM, 5-May-2000.)
((𝐴𝐵𝐶𝐷) → (𝐴𝐶) ⊆ (𝐵𝐷))

Theoremssinss1 3217 Intersection preserves subclass relationship. (Contributed by NM, 14-Sep-1999.)
(𝐴𝐶 → (𝐴𝐵) ⊆ 𝐶)

Theoreminss 3218 Inclusion of an intersection of two classes. (Contributed by NM, 30-Oct-2014.)
((𝐴𝐶𝐵𝐶) → (𝐴𝐵) ⊆ 𝐶)

2.1.13.4  Combinations of difference, union, and intersection of two classes

Theoremunabs 3219 Absorption law for union. (Contributed by NM, 16-Apr-2006.)
(𝐴 ∪ (𝐴𝐵)) = 𝐴

Theoreminabs 3220 Absorption law for intersection. (Contributed by NM, 16-Apr-2006.)
(𝐴 ∩ (𝐴𝐵)) = 𝐴

Theoremdfss4st 3221* Subclass defined in terms of class difference. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(∀𝑥STAB 𝑥𝐴 → (𝐴𝐵 ↔ (𝐵 ∖ (𝐵𝐴)) = 𝐴))

Theoremssddif 3222 Double complement and subset. Similar to ddifss 3226 but inside a class 𝐵 instead of the universal class V. In classical logic the subset operation on the right hand side could be an equality (that is, 𝐴𝐵 ↔ (𝐵 ∖ (𝐵𝐴)) = 𝐴). (Contributed by Jim Kingdon, 24-Jul-2018.)
(𝐴𝐵𝐴 ⊆ (𝐵 ∖ (𝐵𝐴)))

Theoremunssdif 3223 Union of two classes and class difference. In classical logic this would be an equality. (Contributed by Jim Kingdon, 24-Jul-2018.)
(𝐴𝐵) ⊆ (V ∖ ((V ∖ 𝐴) ∖ 𝐵))

Theoreminssdif 3224 Intersection of two classes and class difference. In classical logic this would be an equality. (Contributed by Jim Kingdon, 24-Jul-2018.)
(𝐴𝐵) ⊆ (𝐴 ∖ (V ∖ 𝐵))

Theoremdifin 3225 Difference with intersection. Theorem 33 of [Suppes] p. 29. (Contributed by NM, 31-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)

Theoremddifss 3226 Double complement under universal class. In classical logic (or given an additional hypothesis, as in ddifnel 3120), this is equality rather than subset. (Contributed by Jim Kingdon, 24-Jul-2018.)
𝐴 ⊆ (V ∖ (V ∖ 𝐴))

Theoremunssin 3227 Union as a subset of class complement and intersection (De Morgan's law). One direction of the definition of union in [Mendelson] p. 231. This would be an equality, rather than subset, in classical logic. (Contributed by Jim Kingdon, 25-Jul-2018.)
(𝐴𝐵) ⊆ (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵)))

Theoreminssun 3228 Intersection in terms of class difference and union (De Morgan's law). Similar to Exercise 4.10(n) of [Mendelson] p. 231. This would be an equality, rather than subset, in classical logic. (Contributed by Jim Kingdon, 25-Jul-2018.)
(𝐴𝐵) ⊆ (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)))

Theoreminssddif 3229 Intersection of two classes and class difference. In classical logic, such as Exercise 4.10(q) of [Mendelson] p. 231, this is an equality rather than subset. (Contributed by Jim Kingdon, 26-Jul-2018.)
(𝐴𝐵) ⊆ (𝐴 ∖ (𝐴𝐵))

Theoreminvdif 3230 Intersection with universal complement. Remark in [Stoll] p. 20. (Contributed by NM, 17-Aug-2004.)
(𝐴 ∩ (V ∖ 𝐵)) = (𝐴𝐵)

Theoremindif 3231 Intersection with class difference. Theorem 34 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
(𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)

Theoremindif2 3232 Bring an intersection in and out of a class difference. (Contributed by Jeff Hankins, 15-Jul-2009.)
(𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶)

Theoremindif1 3233 Bring an intersection in and out of a class difference. (Contributed by Mario Carneiro, 15-May-2015.)
((𝐴𝐶) ∩ 𝐵) = ((𝐴𝐵) ∖ 𝐶)

Theoremindifcom 3234 Commutation law for intersection and difference. (Contributed by Scott Fenton, 18-Feb-2013.)
(𝐴 ∩ (𝐵𝐶)) = (𝐵 ∩ (𝐴𝐶))

Theoremindi 3235 Distributive law for intersection over union. Exercise 10 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))

Theoremundi 3236 Distributive law for union over intersection. Exercise 11 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))

Theoremindir 3237 Distributive law for intersection over union. Theorem 28 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))

Theoremundir 3238 Distributive law for union over intersection. Theorem 29 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))

Theoremuneqin 3239 Equality of union and intersection implies equality of their arguments. (Contributed by NM, 16-Apr-2006.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐵) = (𝐴𝐵) ↔ 𝐴 = 𝐵)

Theoremdifundi 3240 Distributive law for class difference. Theorem 39 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
(𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))

Theoremdifundir 3241 Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))

Theoremdifindiss 3242 Distributive law for class difference. In classical logic, for example, theorem 40 of [Suppes] p. 29, this is an equality instead of subset. (Contributed by Jim Kingdon, 26-Jul-2018.)
((𝐴𝐵) ∪ (𝐴𝐶)) ⊆ (𝐴 ∖ (𝐵𝐶))

Theoremdifindir 3243 Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))

Theoremindifdir 3244 Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011.)
((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∖ (𝐵𝐶))

Theoremdifdif2ss 3245 Set difference with a set difference. In classical logic this would be equality rather than subset. (Contributed by Jim Kingdon, 27-Jul-2018.)
((𝐴𝐵) ∪ (𝐴𝐶)) ⊆ (𝐴 ∖ (𝐵𝐶))

Theoremundm 3246 De Morgan's law for union. Theorem 5.2(13) of [Stoll] p. 19. (Contributed by NM, 18-Aug-2004.)
(V ∖ (𝐴𝐵)) = ((V ∖ 𝐴) ∩ (V ∖ 𝐵))

Theoremindmss 3247 De Morgan's law for intersection. In classical logic, this would be equality rather than subset, as in Theorem 5.2(13') of [Stoll] p. 19. (Contributed by Jim Kingdon, 27-Jul-2018.)
((V ∖ 𝐴) ∪ (V ∖ 𝐵)) ⊆ (V ∖ (𝐴𝐵))

Theoremdifun1 3248 A relationship involving double difference and union. (Contributed by NM, 29-Aug-2004.)
(𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶)

Theoremundif3ss 3249 A subset relationship involving class union and class difference. In classical logic, this would be equality rather than subset, as in the first equality of Exercise 13 of [TakeutiZaring] p. 22. (Contributed by Jim Kingdon, 28-Jul-2018.)
(𝐴 ∪ (𝐵𝐶)) ⊆ ((𝐴𝐵) ∖ (𝐶𝐴))

Theoremdifin2 3250 Represent a set difference as an intersection with a larger difference. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐴𝐶 → (𝐴𝐵) = ((𝐶𝐵) ∩ 𝐴))

Theoremdif32 3251 Swap second and third argument of double difference. (Contributed by NM, 18-Aug-2004.)
((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∖ 𝐵)

Theoremdifabs 3252 Absorption-like law for class difference: you can remove a class only once. (Contributed by FL, 2-Aug-2009.)
((𝐴𝐵) ∖ 𝐵) = (𝐴𝐵)

Theoremsymdif1 3253 Two ways to express symmetric difference. This theorem shows the equivalence of the definition of symmetric difference in [Stoll] p. 13 and the restated definition in Example 4.1 of [Stoll] p. 262. (Contributed by NM, 17-Aug-2004.)
((𝐴𝐵) ∪ (𝐵𝐴)) = ((𝐴𝐵) ∖ (𝐴𝐵))

2.1.13.5  Class abstractions with difference, union, and intersection of two classes

Theoremsymdifxor 3254* Expressing symmetric difference with exclusive-or or two differences. (Contributed by Jim Kingdon, 28-Jul-2018.)
((𝐴𝐵) ∪ (𝐵𝐴)) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}

Theoremunab 3255 Union of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
({𝑥𝜑} ∪ {𝑥𝜓}) = {𝑥 ∣ (𝜑𝜓)}

Theoreminab 3256 Intersection of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
({𝑥𝜑} ∩ {𝑥𝜓}) = {𝑥 ∣ (𝜑𝜓)}

Theoremdifab 3257 Difference of two class abstractions. (Contributed by NM, 23-Oct-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
({𝑥𝜑} ∖ {𝑥𝜓}) = {𝑥 ∣ (𝜑 ∧ ¬ 𝜓)}

Theoremnotab 3258 A class builder defined by a negation. (Contributed by FL, 18-Sep-2010.)
{𝑥 ∣ ¬ 𝜑} = (V ∖ {𝑥𝜑})

Theoremunrab 3259 Union of two restricted class abstractions. (Contributed by NM, 25-Mar-2004.)
({𝑥𝐴𝜑} ∪ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑𝜓)}

Theoreminrab 3260 Intersection of two restricted class abstractions. (Contributed by NM, 1-Sep-2006.)
({𝑥𝐴𝜑} ∩ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑𝜓)}

Theoreminrab2 3261* Intersection with a restricted class abstraction. (Contributed by NM, 19-Nov-2007.)
({𝑥𝐴𝜑} ∩ 𝐵) = {𝑥 ∈ (𝐴𝐵) ∣ 𝜑}

Theoremdifrab 3262 Difference of two restricted class abstractions. (Contributed by NM, 23-Oct-2004.)
({𝑥𝐴𝜑} ∖ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜓)}

Theoremdfrab2 3263* Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.)
{𝑥𝐴𝜑} = ({𝑥𝜑} ∩ 𝐴)

Theoremdfrab3 3264* Alternate definition of restricted class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
{𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝜑})

Theoremnotrab 3265* Complementation of restricted class abstractions. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝐴 ∖ {𝑥𝐴𝜑}) = {𝑥𝐴 ∣ ¬ 𝜑}

Theoremdfrab3ss 3266* Restricted class abstraction with a common superset. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Proof shortened by Mario Carneiro, 8-Nov-2015.)
(𝐴𝐵 → {𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝐵𝜑}))

Theoremrabun2 3267 Abstraction restricted to a union. (Contributed by Stefan O'Rear, 5-Feb-2015.)
{𝑥 ∈ (𝐴𝐵) ∣ 𝜑} = ({𝑥𝐴𝜑} ∪ {𝑥𝐵𝜑})

2.1.13.6  Restricted uniqueness with difference, union, and intersection

Theoremreuss2 3268* Transfer uniqueness to a smaller subclass. (Contributed by NM, 20-Oct-2005.)
(((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → ∃!𝑥𝐴 𝜑)

Theoremreuss 3269* Transfer uniqueness to a smaller subclass. (Contributed by NM, 21-Aug-1999.)
((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → ∃!𝑥𝐴 𝜑)

Theoremreuun1 3270* Transfer uniqueness to a smaller class. (Contributed by NM, 21-Oct-2005.)
((∃𝑥𝐴 𝜑 ∧ ∃!𝑥 ∈ (𝐴𝐵)(𝜑𝜓)) → ∃!𝑥𝐴 𝜑)

Theoremreuun2 3271* Transfer uniqueness to a smaller or larger class. (Contributed by NM, 21-Oct-2005.)
(¬ ∃𝑥𝐵 𝜑 → (∃!𝑥 ∈ (𝐴𝐵)𝜑 ↔ ∃!𝑥𝐴 𝜑))

Theoremreupick 3272* Restricted uniqueness "picks" a member of a subclass. (Contributed by NM, 21-Aug-1999.)
(((𝐴𝐵 ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) ∧ 𝜑) → (𝑥𝐴𝑥𝐵))

Theoremreupick3 3273* Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 19-Nov-2016.)
((∃!𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 (𝜑𝜓) ∧ 𝑥𝐴) → (𝜑𝜓))

Theoremreupick2 3274* Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 15-Dec-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
(((∀𝑥𝐴 (𝜓𝜑) ∧ ∃𝑥𝐴 𝜓 ∧ ∃!𝑥𝐴 𝜑) ∧ 𝑥𝐴) → (𝜑𝜓))

2.1.14  The empty set

Syntaxc0 3275 Extend class notation to include the empty set.
class

Definitiondf-nul 3276 Define the empty set. Special case of Exercise 4.10(o) of [Mendelson] p. 231. For a more traditional definition, but requiring a dummy variable, see dfnul2 3277. (Contributed by NM, 5-Aug-1993.)
∅ = (V ∖ V)

Theoremdfnul2 3277 Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.)
∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}

Theoremdfnul3 3278 Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.)
∅ = {𝑥𝐴 ∣ ¬ 𝑥𝐴}

Theoremnoel 3279 The empty set has no elements. Theorem 6.14 of [Quine] p. 44. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)
¬ 𝐴 ∈ ∅

Theoremn0i 3280 If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2629. (Contributed by NM, 31-Dec-1993.)
(𝐵𝐴 → ¬ 𝐴 = ∅)

Theoremne0i 3281 If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2629. (Contributed by NM, 31-Dec-1993.)
(𝐵𝐴𝐴 ≠ ∅)

Theoremvn0 3282 The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.)
V ≠ ∅

Theoremvn0m 3283 The universal class is inhabited. (Contributed by Jim Kingdon, 17-Dec-2018.)
𝑥 𝑥 ∈ V

Theoremn0rf 3284 An inhabited class is nonempty. Following the Definition of [Bauer], p. 483, we call a class 𝐴 nonempty if 𝐴 ≠ ∅ and inhabited if it has at least one element. In classical logic these two concepts are equivalent, for example see Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of n0r 3285 requires only that 𝑥 not be free in, rather than not occur in, 𝐴. (Contributed by Jim Kingdon, 31-Jul-2018.)
𝑥𝐴       (∃𝑥 𝑥𝐴𝐴 ≠ ∅)

Theoremn0r 3285* An inhabited class is nonempty. See n0rf 3284 for more discussion. (Contributed by Jim Kingdon, 31-Jul-2018.)
(∃𝑥 𝑥𝐴𝐴 ≠ ∅)

Theoremneq0r 3286* An inhabited class is nonempty. See n0rf 3284 for more discussion. (Contributed by Jim Kingdon, 31-Jul-2018.)
(∃𝑥 𝑥𝐴 → ¬ 𝐴 = ∅)

Theoremreximdva0m 3287* Restricted existence deduced from inhabited class. (Contributed by Jim Kingdon, 31-Jul-2018.)
((𝜑𝑥𝐴) → 𝜓)       ((𝜑 ∧ ∃𝑥 𝑥𝐴) → ∃𝑥𝐴 𝜓)

Theoremn0mmoeu 3288* A case of equivalence of "at most one" and "only one". If a class is inhabited, that class having at most one element is equivalent to it having only one element. (Contributed by Jim Kingdon, 31-Jul-2018.)
(∃𝑥 𝑥𝐴 → (∃*𝑥 𝑥𝐴 ↔ ∃!𝑥 𝑥𝐴))

Theoremrex0 3289 Vacuous existential quantification is false. (Contributed by NM, 15-Oct-2003.)
¬ ∃𝑥 ∈ ∅ 𝜑

Theoremeq0 3290* The empty set has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by NM, 29-Aug-1993.)
(𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)

Theoremeqv 3291* The universe contains every set. (Contributed by NM, 11-Sep-2006.)
(𝐴 = V ↔ ∀𝑥 𝑥𝐴)

Theoremnotm0 3292* A class is not inhabited if and only if it is empty. (Contributed by Jim Kingdon, 1-Jul-2022.)
(¬ ∃𝑥 𝑥𝐴𝐴 = ∅)

Theoremnel0 3293* From the general negation of membership in 𝐴, infer that 𝐴 is the empty set. (Contributed by BJ, 6-Oct-2018.)
¬ 𝑥𝐴       𝐴 = ∅

Theorem0el 3294* Membership of the empty set in another class. (Contributed by NM, 29-Jun-2004.)
(∅ ∈ 𝐴 ↔ ∃𝑥𝐴𝑦 ¬ 𝑦𝑥)

Theoremabvor0dc 3295* The class builder of a decidable proposition not containing the abstraction variable is either the universal class or the empty set. (Contributed by Jim Kingdon, 1-Aug-2018.)
(DECID 𝜑 → ({𝑥𝜑} = V ∨ {𝑥𝜑} = ∅))

Theoremabn0r 3296 Nonempty class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)
(∃𝑥𝜑 → {𝑥𝜑} ≠ ∅)

Theoremabn0m 3297* Inhabited class abstraction. (Contributed by Jim Kingdon, 8-Jul-2022.)
(∃𝑦 𝑦 ∈ {𝑥𝜑} ↔ ∃𝑥𝜑)

Theoremrabn0r 3298 Nonempty restricted class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)
(∃𝑥𝐴 𝜑 → {𝑥𝐴𝜑} ≠ ∅)

Theoremrabn0m 3299* Inhabited restricted class abstraction. (Contributed by Jim Kingdon, 18-Sep-2018.)
(∃𝑦 𝑦 ∈ {𝑥𝐴𝜑} ↔ ∃𝑥𝐴 𝜑)

Theoremrab0 3300 Any restricted class abstraction restricted to the empty set is empty. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
{𝑥 ∈ ∅ ∣ 𝜑} = ∅

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