Type  Label  Description 
Statement 

Theorem  in32 3201 
A rearrangement of intersection. (Contributed by NM, 21Apr2001.)
(Proof shortened by Andrew Salmon, 26Jun2011.)

⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∩ 𝐵) 

Theorem  in13 3202 
A rearrangement of intersection. (Contributed by NM, 27Aug2012.)

⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐶 ∩ (𝐵 ∩ 𝐴)) 

Theorem  in31 3203 
A rearrangement of intersection. (Contributed by NM, 27Aug2012.)

⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐶 ∩ 𝐵) ∩ 𝐴) 

Theorem  inrot 3204 
Rotate the intersection of 3 classes. (Contributed by NM,
27Aug2012.)

⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐶 ∩ 𝐴) ∩ 𝐵) 

Theorem  in4 3205 
Rearrangement of intersection of 4 classes. (Contributed by NM,
21Apr2001.)

⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 ∩ 𝐷)) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐷)) 

Theorem  inindi 3206 
Intersection distributes over itself. (Contributed by NM, 6May1994.)

⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = ((𝐴 ∩ 𝐵) ∩ (𝐴 ∩ 𝐶)) 

Theorem  inindir 3207 
Intersection distributes over itself. (Contributed by NM,
17Aug2004.)

⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐶)) 

Theorem  sseqin2 3208 
A relationship between subclass and intersection. Similar to Exercise 9
of [TakeutiZaring] p. 18.
(Contributed by NM, 17May1994.)

⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴) 

Theorem  inss1 3209 
The intersection of two classes is a subset of one of them. Part of
Exercise 12 of [TakeutiZaring] p.
18. (Contributed by NM,
27Apr1994.)

⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 

Theorem  inss2 3210 
The intersection of two classes is a subset of one of them. Part of
Exercise 12 of [TakeutiZaring] p.
18. (Contributed by NM,
27Apr1994.)

⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 

Theorem  ssin 3211 
Subclass of intersection. Theorem 2.8(vii) of [Monk1] p. 26.
(Contributed by NM, 15Jun2004.) (Proof shortened by Andrew Salmon,
26Jun2011.)

⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶) ↔ 𝐴 ⊆ (𝐵 ∩ 𝐶)) 

Theorem  ssini 3212 
An inference showing that a subclass of two classes is a subclass of
their intersection. (Contributed by NM, 24Nov2003.)

⊢ 𝐴 ⊆ 𝐵
& ⊢ 𝐴 ⊆ 𝐶 ⇒ ⊢ 𝐴 ⊆ (𝐵 ∩ 𝐶) 

Theorem  ssind 3213 
A deduction showing that a subclass of two classes is a subclass of
their intersection. (Contributed by Jonathan BenNaim, 3Jun2011.)

⊢ (𝜑 → 𝐴 ⊆ 𝐵)
& ⊢ (𝜑 → 𝐴 ⊆ 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ⊆ (𝐵 ∩ 𝐶)) 

Theorem  ssrin 3214 
Add right intersection to subclass relation. (Contributed by NM,
16Aug1994.) (Proof shortened by Andrew Salmon, 26Jun2011.)

⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐶)) 

Theorem  sslin 3215 
Add left intersection to subclass relation. (Contributed by NM,
19Oct1999.)

⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∩ 𝐴) ⊆ (𝐶 ∩ 𝐵)) 

Theorem  ss2in 3216 
Intersection of subclasses. (Contributed by NM, 5May2000.)

⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐷)) 

Theorem  ssinss1 3217 
Intersection preserves subclass relationship. (Contributed by NM,
14Sep1999.)

⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∩ 𝐵) ⊆ 𝐶) 

Theorem  inss 3218 
Inclusion of an intersection of two classes. (Contributed by NM,
30Oct2014.)

⊢ ((𝐴 ⊆ 𝐶 ∨ 𝐵 ⊆ 𝐶) → (𝐴 ∩ 𝐵) ⊆ 𝐶) 

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


Theorem  unabs 3219 
Absorption law for union. (Contributed by NM, 16Apr2006.)

⊢ (𝐴 ∪ (𝐴 ∩ 𝐵)) = 𝐴 

Theorem  inabs 3220 
Absorption law for intersection. (Contributed by NM, 16Apr2006.)

⊢ (𝐴 ∩ (𝐴 ∪ 𝐵)) = 𝐴 

Theorem  dfss4st 3221* 
Subclass defined in terms of class difference. (Contributed by NM,
22Mar1998.) (Proof shortened by Andrew Salmon, 26Jun2011.)

⊢ (∀𝑥STAB 𝑥 ∈ 𝐴 → (𝐴 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝐴)) = 𝐴)) 

Theorem  ssddif 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,
𝐴
⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝐴)) = 𝐴). (Contributed by Jim Kingdon,
24Jul2018.)

⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ (𝐵 ∖ (𝐵 ∖ 𝐴))) 

Theorem  unssdif 3223 
Union of two classes and class difference. In classical logic this
would be an equality. (Contributed by Jim Kingdon, 24Jul2018.)

⊢ (𝐴 ∪ 𝐵) ⊆ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) 

Theorem  inssdif 3224 
Intersection of two classes and class difference. In classical logic
this would be an equality. (Contributed by Jim Kingdon,
24Jul2018.)

⊢ (𝐴 ∩ 𝐵) ⊆ (𝐴 ∖ (V ∖ 𝐵)) 

Theorem  difin 3225 
Difference with intersection. Theorem 33 of [Suppes] p. 29.
(Contributed by NM, 31Mar1998.) (Proof shortened by Andrew Salmon,
26Jun2011.)

⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵) 

Theorem  ddifss 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, 24Jul2018.)

⊢ 𝐴 ⊆ (V ∖ (V ∖ 𝐴)) 

Theorem  unssin 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, 25Jul2018.)

⊢ (𝐴 ∪ 𝐵) ⊆ (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵))) 

Theorem  inssun 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, 25Jul2018.)

⊢ (𝐴 ∩ 𝐵) ⊆ (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))) 

Theorem  inssddif 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, 26Jul2018.)

⊢ (𝐴 ∩ 𝐵) ⊆ (𝐴 ∖ (𝐴 ∖ 𝐵)) 

Theorem  invdif 3230 
Intersection with universal complement. Remark in [Stoll] p. 20.
(Contributed by NM, 17Aug2004.)

⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ 𝐵) 

Theorem  indif 3231 
Intersection with class difference. Theorem 34 of [Suppes] p. 29.
(Contributed by NM, 17Aug2004.)

⊢ (𝐴 ∩ (𝐴 ∖ 𝐵)) = (𝐴 ∖ 𝐵) 

Theorem  indif2 3232 
Bring an intersection in and out of a class difference. (Contributed by
Jeff Hankins, 15Jul2009.)

⊢ (𝐴 ∩ (𝐵 ∖ 𝐶)) = ((𝐴 ∩ 𝐵) ∖ 𝐶) 

Theorem  indif1 3233 
Bring an intersection in and out of a class difference. (Contributed by
Mario Carneiro, 15May2015.)

⊢ ((𝐴 ∖ 𝐶) ∩ 𝐵) = ((𝐴 ∩ 𝐵) ∖ 𝐶) 

Theorem  indifcom 3234 
Commutation law for intersection and difference. (Contributed by Scott
Fenton, 18Feb2013.)

⊢ (𝐴 ∩ (𝐵 ∖ 𝐶)) = (𝐵 ∩ (𝐴 ∖ 𝐶)) 

Theorem  indi 3235 
Distributive law for intersection over union. Exercise 10 of
[TakeutiZaring] p. 17.
(Contributed by NM, 30Sep2002.) (Proof
shortened by Andrew Salmon, 26Jun2011.)

⊢ (𝐴 ∩ (𝐵 ∪ 𝐶)) = ((𝐴 ∩ 𝐵) ∪ (𝐴 ∩ 𝐶)) 

Theorem  undi 3236 
Distributive law for union over intersection. Exercise 11 of
[TakeutiZaring] p. 17.
(Contributed by NM, 30Sep2002.) (Proof
shortened by Andrew Salmon, 26Jun2011.)

⊢ (𝐴 ∪ (𝐵 ∩ 𝐶)) = ((𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶)) 

Theorem  indir 3237 
Distributive law for intersection over union. Theorem 28 of [Suppes]
p. 27. (Contributed by NM, 30Sep2002.)

⊢ ((𝐴 ∪ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∪ (𝐵 ∩ 𝐶)) 

Theorem  undir 3238 
Distributive law for union over intersection. Theorem 29 of [Suppes]
p. 27. (Contributed by NM, 30Sep2002.)

⊢ ((𝐴 ∩ 𝐵) ∪ 𝐶) = ((𝐴 ∪ 𝐶) ∩ (𝐵 ∪ 𝐶)) 

Theorem  uneqin 3239 
Equality of union and intersection implies equality of their arguments.
(Contributed by NM, 16Apr2006.) (Proof shortened by Andrew Salmon,
26Jun2011.)

⊢ ((𝐴 ∪ 𝐵) = (𝐴 ∩ 𝐵) ↔ 𝐴 = 𝐵) 

Theorem  difundi 3240 
Distributive law for class difference. Theorem 39 of [Suppes] p. 29.
(Contributed by NM, 17Aug2004.)

⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = ((𝐴 ∖ 𝐵) ∩ (𝐴 ∖ 𝐶)) 

Theorem  difundir 3241 
Distributive law for class difference. (Contributed by NM,
17Aug2004.)

⊢ ((𝐴 ∪ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∪ (𝐵 ∖ 𝐶)) 

Theorem  difindiss 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, 26Jul2018.)

⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ 𝐶)) ⊆ (𝐴 ∖ (𝐵 ∩ 𝐶)) 

Theorem  difindir 3243 
Distributive law for class difference. (Contributed by NM,
17Aug2004.)

⊢ ((𝐴 ∩ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∩ (𝐵 ∖ 𝐶)) 

Theorem  indifdir 3244 
Distribute intersection over difference. (Contributed by Scott Fenton,
14Apr2011.)

⊢ ((𝐴 ∖ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∖ (𝐵 ∩ 𝐶)) 

Theorem  difdif2ss 3245 
Set difference with a set difference. In classical logic this would be
equality rather than subset. (Contributed by Jim Kingdon,
27Jul2018.)

⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶)) ⊆ (𝐴 ∖ (𝐵 ∖ 𝐶)) 

Theorem  undm 3246 
De Morgan's law for union. Theorem 5.2(13) of [Stoll] p. 19.
(Contributed by NM, 18Aug2004.)

⊢ (V ∖ (𝐴 ∪ 𝐵)) = ((V ∖ 𝐴) ∩ (V ∖ 𝐵)) 

Theorem  indmss 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, 27Jul2018.)

⊢ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)) ⊆ (V ∖ (𝐴 ∩ 𝐵)) 

Theorem  difun1 3248 
A relationship involving double difference and union. (Contributed by NM,
29Aug2004.)

⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = ((𝐴 ∖ 𝐵) ∖ 𝐶) 

Theorem  undif3ss 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, 28Jul2018.)

⊢ (𝐴 ∪ (𝐵 ∖ 𝐶)) ⊆ ((𝐴 ∪ 𝐵) ∖ (𝐶 ∖ 𝐴)) 

Theorem  difin2 3250 
Represent a set difference as an intersection with a larger difference.
(Contributed by Jeff Madsen, 2Sep2009.)

⊢ (𝐴 ⊆ 𝐶 → (𝐴 ∖ 𝐵) = ((𝐶 ∖ 𝐵) ∩ 𝐴)) 

Theorem  dif32 3251 
Swap second and third argument of double difference. (Contributed by NM,
18Aug2004.)

⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∖ 𝐵) 

Theorem  difabs 3252 
Absorptionlike law for class difference: you can remove a class only
once. (Contributed by FL, 2Aug2009.)

⊢ ((𝐴 ∖ 𝐵) ∖ 𝐵) = (𝐴 ∖ 𝐵) 

Theorem  symdif1 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, 17Aug2004.)

⊢ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) = ((𝐴 ∪ 𝐵) ∖ (𝐴 ∩ 𝐵)) 

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


Theorem  symdifxor 3254* 
Expressing symmetric difference with exclusiveor or two differences.
(Contributed by Jim Kingdon, 28Jul2018.)

⊢ ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) = {𝑥 ∣ (𝑥 ∈ 𝐴 ⊻ 𝑥 ∈ 𝐵)} 

Theorem  unab 3255 
Union of two class abstractions. (Contributed by NM, 29Sep2002.)
(Proof shortened by Andrew Salmon, 26Jun2011.)

⊢ ({𝑥 ∣ 𝜑} ∪ {𝑥 ∣ 𝜓}) = {𝑥 ∣ (𝜑 ∨ 𝜓)} 

Theorem  inab 3256 
Intersection of two class abstractions. (Contributed by NM,
29Sep2002.) (Proof shortened by Andrew Salmon, 26Jun2011.)

⊢ ({𝑥 ∣ 𝜑} ∩ {𝑥 ∣ 𝜓}) = {𝑥 ∣ (𝜑 ∧ 𝜓)} 

Theorem  difab 3257 
Difference of two class abstractions. (Contributed by NM, 23Oct2004.)
(Proof shortened by Andrew Salmon, 26Jun2011.)

⊢ ({𝑥 ∣ 𝜑} ∖ {𝑥 ∣ 𝜓}) = {𝑥 ∣ (𝜑 ∧ ¬ 𝜓)} 

Theorem  notab 3258 
A class builder defined by a negation. (Contributed by FL,
18Sep2010.)

⊢ {𝑥 ∣ ¬ 𝜑} = (V ∖ {𝑥 ∣ 𝜑}) 

Theorem  unrab 3259 
Union of two restricted class abstractions. (Contributed by NM,
25Mar2004.)

⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ 𝜓}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ 𝜓)} 

Theorem  inrab 3260 
Intersection of two restricted class abstractions. (Contributed by NM,
1Sep2006.)

⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ {𝑥 ∈ 𝐴 ∣ 𝜓}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ 𝜓)} 

Theorem  inrab2 3261* 
Intersection with a restricted class abstraction. (Contributed by NM,
19Nov2007.)

⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∩ 𝐵) = {𝑥 ∈ (𝐴 ∩ 𝐵) ∣ 𝜑} 

Theorem  difrab 3262 
Difference of two restricted class abstractions. (Contributed by NM,
23Oct2004.)

⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∖ {𝑥 ∈ 𝐴 ∣ 𝜓}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∧ ¬ 𝜓)} 

Theorem  dfrab2 3263* 
Alternate definition of restricted class abstraction. (Contributed by
NM, 20Sep2003.)

⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = ({𝑥 ∣ 𝜑} ∩ 𝐴) 

Theorem  dfrab3 3264* 
Alternate definition of restricted class abstraction. (Contributed by
Mario Carneiro, 8Sep2013.)

⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑥 ∣ 𝜑}) 

Theorem  notrab 3265* 
Complementation of restricted class abstractions. (Contributed by Mario
Carneiro, 3Sep2015.)

⊢ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝜑}) = {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} 

Theorem  dfrab3ss 3266* 
Restricted class abstraction with a common superset. (Contributed by
Stefan O'Rear, 12Sep2015.) (Proof shortened by Mario Carneiro,
8Nov2015.)

⊢ (𝐴 ⊆ 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜑} = (𝐴 ∩ {𝑥 ∈ 𝐵 ∣ 𝜑})) 

Theorem  rabun2 3267 
Abstraction restricted to a union. (Contributed by Stefan O'Rear,
5Feb2015.)

⊢ {𝑥 ∈ (𝐴 ∪ 𝐵) ∣ 𝜑} = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐵 ∣ 𝜑}) 

2.1.13.6 Restricted uniqueness with difference,
union, and intersection


Theorem  reuss2 3268* 
Transfer uniqueness to a smaller subclass. (Contributed by NM,
20Oct2005.)

⊢ (((𝐴 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)) ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜓)) → ∃!𝑥 ∈ 𝐴 𝜑) 

Theorem  reuss 3269* 
Transfer uniqueness to a smaller subclass. (Contributed by NM,
21Aug1999.)

⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑) → ∃!𝑥 ∈ 𝐴 𝜑) 

Theorem  reuun1 3270* 
Transfer uniqueness to a smaller class. (Contributed by NM,
21Oct2005.)

⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ (𝐴 ∪ 𝐵)(𝜑 ∨ 𝜓)) → ∃!𝑥 ∈ 𝐴 𝜑) 

Theorem  reuun2 3271* 
Transfer uniqueness to a smaller or larger class. (Contributed by NM,
21Oct2005.)

⊢ (¬ ∃𝑥 ∈ 𝐵 𝜑 → (∃!𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ ∃!𝑥 ∈ 𝐴 𝜑)) 

Theorem  reupick 3272* 
Restricted uniqueness "picks" a member of a subclass. (Contributed
by
NM, 21Aug1999.)

⊢ (((𝐴 ⊆ 𝐵 ∧ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃!𝑥 ∈ 𝐵 𝜑)) ∧ 𝜑) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) 

Theorem  reupick3 3273* 
Restricted uniqueness "picks" a member of a subclass. (Contributed
by
Mario Carneiro, 19Nov2016.)

⊢ ((∃!𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ∧ 𝑥 ∈ 𝐴) → (𝜑 → 𝜓)) 

Theorem  reupick2 3274* 
Restricted uniqueness "picks" a member of a subclass. (Contributed
by
Mario Carneiro, 15Dec2013.) (Proof shortened by Mario Carneiro,
19Nov2016.)

⊢ (((∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 𝜓 ∧ ∃!𝑥 ∈ 𝐴 𝜑) ∧ 𝑥 ∈ 𝐴) → (𝜑 ↔ 𝜓)) 

2.1.14 The empty set


Syntax  c0 3275 
Extend class notation to include the empty set.

class ∅ 

Definition  dfnul 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, 5Aug1993.)

⊢ ∅ = (V ∖ V) 

Theorem  dfnul2 3277 
Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring]
p. 20. (Contributed by NM, 26Dec1996.)

⊢ ∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥} 

Theorem  dfnul3 3278 
Alternate definition of the empty set. (Contributed by NM,
25Mar2004.)

⊢ ∅ = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴} 

Theorem  noel 3279 
The empty set has no elements. Theorem 6.14 of [Quine] p. 44.
(Contributed by NM, 5Aug1993.) (Proof shortened by Mario Carneiro,
1Sep2015.)

⊢ ¬ 𝐴 ∈ ∅ 

Theorem  n0i 3280 
If a set has elements, it is not empty. A set with elements is also
inhabited, see elex2 2629. (Contributed by NM, 31Dec1993.)

⊢ (𝐵 ∈ 𝐴 → ¬ 𝐴 = ∅) 

Theorem  ne0i 3281 
If a set has elements, it is not empty. A set with elements is also
inhabited, see elex2 2629. (Contributed by NM, 31Dec1993.)

⊢ (𝐵 ∈ 𝐴 → 𝐴 ≠ ∅) 

Theorem  vn0 3282 
The universal class is not equal to the empty set. (Contributed by NM,
11Sep2008.)

⊢ V ≠ ∅ 

Theorem  vn0m 3283 
The universal class is inhabited. (Contributed by Jim Kingdon,
17Dec2018.)

⊢ ∃𝑥 𝑥 ∈ V 

Theorem  n0rf 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,
31Jul2018.)

⊢ Ⅎ𝑥𝐴 ⇒ ⊢ (∃𝑥 𝑥 ∈ 𝐴 → 𝐴 ≠ ∅) 

Theorem  n0r 3285* 
An inhabited class is nonempty. See n0rf 3284 for more discussion.
(Contributed by Jim Kingdon, 31Jul2018.)

⊢ (∃𝑥 𝑥 ∈ 𝐴 → 𝐴 ≠ ∅) 

Theorem  neq0r 3286* 
An inhabited class is nonempty. See n0rf 3284 for more discussion.
(Contributed by Jim Kingdon, 31Jul2018.)

⊢ (∃𝑥 𝑥 ∈ 𝐴 → ¬ 𝐴 = ∅) 

Theorem  reximdva0m 3287* 
Restricted existence deduced from inhabited class. (Contributed by Jim
Kingdon, 31Jul2018.)

⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓) ⇒ ⊢ ((𝜑 ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 𝜓) 

Theorem  n0mmoeu 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, 31Jul2018.)

⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥 𝑥 ∈ 𝐴)) 

Theorem  rex0 3289 
Vacuous existential quantification is false. (Contributed by NM,
15Oct2003.)

⊢ ¬ ∃𝑥 ∈ ∅ 𝜑 

Theorem  eq0 3290* 
The empty set has no elements. Theorem 2 of [Suppes] p. 22.
(Contributed by NM, 29Aug1993.)

⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) 

Theorem  eqv 3291* 
The universe contains every set. (Contributed by NM, 11Sep2006.)

⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴) 

Theorem  notm0 3292* 
A class is not inhabited if and only if it is empty. (Contributed by
Jim Kingdon, 1Jul2022.)

⊢ (¬ ∃𝑥 𝑥 ∈ 𝐴 ↔ 𝐴 = ∅) 

Theorem  nel0 3293* 
From the general negation of membership in 𝐴, infer that 𝐴 is
the empty set. (Contributed by BJ, 6Oct2018.)

⊢ ¬ 𝑥 ∈ 𝐴 ⇒ ⊢ 𝐴 = ∅ 

Theorem  0el 3294* 
Membership of the empty set in another class. (Contributed by NM,
29Jun2004.)

⊢ (∅ ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ¬ 𝑦 ∈ 𝑥) 

Theorem  abvor0dc 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, 1Aug2018.)

⊢ (DECID 𝜑 → ({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅)) 

Theorem  abn0r 3296 
Nonempty class abstraction. (Contributed by Jim Kingdon, 1Aug2018.)

⊢ (∃𝑥𝜑 → {𝑥 ∣ 𝜑} ≠ ∅) 

Theorem  abn0m 3297* 
Inhabited class abstraction. (Contributed by Jim Kingdon,
8Jul2022.)

⊢ (∃𝑦 𝑦 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥𝜑) 

Theorem  rabn0r 3298 
Nonempty restricted class abstraction. (Contributed by Jim Kingdon,
1Aug2018.)

⊢ (∃𝑥 ∈ 𝐴 𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜑} ≠ ∅) 

Theorem  rabn0m 3299* 
Inhabited restricted class abstraction. (Contributed by Jim Kingdon,
18Sep2018.)

⊢ (∃𝑦 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐴 𝜑) 

Theorem  rab0 3300 
Any restricted class abstraction restricted to the empty set is empty.
(Contributed by NM, 15Oct2003.) (Proof shortened by Andrew Salmon,
26Jun2011.)

⊢ {𝑥 ∈ ∅ ∣ 𝜑} = ∅ 