Type  Label  Description 
Statement 

Theorem  ssun2 3101 
Subclass relationship for union of classes. (Contributed by NM,
30Aug1993.)

⊢ A ⊆
(B ∪ A) 

Theorem  ssun3 3102 
Subclass law for union of classes. (Contributed by NM, 5Aug1993.)

⊢ (A ⊆
B → A ⊆ (B
∪ 𝐶)) 

Theorem  ssun4 3103 
Subclass law for union of classes. (Contributed by NM, 14Aug1994.)

⊢ (A ⊆
B → A ⊆ (𝐶 ∪ B)) 

Theorem  elun1 3104 
Membership law for union of classes. (Contributed by NM, 5Aug1993.)

⊢ (A ∈ B →
A ∈
(B ∪ 𝐶)) 

Theorem  elun2 3105 
Membership law for union of classes. (Contributed by NM, 30Aug1993.)

⊢ (A ∈ B →
A ∈
(𝐶 ∪ B)) 

Theorem  unss1 3106 
Subclass law for union of classes. (Contributed by NM, 14Oct1999.)
(Proof shortened by Andrew Salmon, 26Jun2011.)

⊢ (A ⊆
B → (A ∪ 𝐶) ⊆ (B ∪ 𝐶)) 

Theorem  ssequn1 3107 
A relationship between subclass and union. Theorem 26 of [Suppes]
p. 27. (Contributed by NM, 30Aug1993.) (Proof shortened by Andrew
Salmon, 26Jun2011.)

⊢ (A ⊆
B ↔ (A ∪ B) =
B) 

Theorem  unss2 3108 
Subclass law for union of classes. Exercise 7 of [TakeutiZaring] p. 18.
(Contributed by NM, 14Oct1999.)

⊢ (A ⊆
B → (𝐶 ∪ A) ⊆ (𝐶 ∪ B)) 

Theorem  unss12 3109 
Subclass law for union of classes. (Contributed by NM, 2Jun2004.)

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

Theorem  ssequn2 3110 
A relationship between subclass and union. (Contributed by NM,
13Jun1994.)

⊢ (A ⊆
B ↔ (B ∪ A) =
B) 

Theorem  unss 3111 
The union of two subclasses is a subclass. Theorem 27 of [Suppes] p. 27
and its converse. (Contributed by NM, 11Jun2004.)

⊢ ((A ⊆
𝐶 ∧ B ⊆
𝐶) ↔ (A ∪ B)
⊆ 𝐶) 

Theorem  unssi 3112 
An inference showing the union of two subclasses is a subclass.
(Contributed by Raph Levien, 10Dec2002.)

⊢ A ⊆
𝐶 & ⊢ B ⊆ 𝐶 ⇒ ⊢ (A ∪ B)
⊆ 𝐶 

Theorem  unssd 3113 
A deduction showing the union of two subclasses is a subclass.
(Contributed by Jonathan BenNaim, 3Jun2011.)

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

Theorem  unssad 3114 
If (A ∪ B) is contained in 𝐶, so is A. Oneway
deduction form of unss 3111. Partial converse of unssd 3113. (Contributed
by David Moews, 1May2017.)

⊢ (φ
→ (A ∪ B) ⊆ 𝐶) ⇒ ⊢ (φ → A ⊆ 𝐶) 

Theorem  unssbd 3115 
If (A ∪ B) is contained in 𝐶, so is B. Oneway
deduction form of unss 3111. Partial converse of unssd 3113. (Contributed
by David Moews, 1May2017.)

⊢ (φ
→ (A ∪ B) ⊆ 𝐶) ⇒ ⊢ (φ → B ⊆ 𝐶) 

Theorem  ssun 3116 
A condition that implies inclusion in the union of two classes.
(Contributed by NM, 23Nov2003.)

⊢ ((A ⊆
B ∨
A ⊆ 𝐶) → A ⊆ (B
∪ 𝐶)) 

Theorem  rexun 3117 
Restricted existential quantification over union. (Contributed by Jeff
Madsen, 5Jan2011.)

⊢ (∃x ∈ (A ∪ B)φ ↔ (∃x ∈ A φ ∨ ∃x ∈ B φ)) 

Theorem  ralunb 3118 
Restricted quantification over a union. (Contributed by Scott Fenton,
12Apr2011.) (Proof shortened by Andrew Salmon, 29Jun2011.)

⊢ (∀x ∈ (A ∪ B)φ ↔ (∀x ∈ A φ ∧ ∀x ∈ B φ)) 

Theorem  ralun 3119 
Restricted quantification over union. (Contributed by Jeff Madsen,
2Sep2009.)

⊢ ((∀x ∈ A φ ∧ ∀x ∈ B φ) →
∀x
∈ (A
∪ B)φ) 

2.1.13.3 The intersection of two
classes


Theorem  elin 3120 
Expansion of membership in an intersection of two classes. Theorem 12
of [Suppes] p. 25. (Contributed by NM,
29Apr1994.)

⊢ (A ∈ (B ∩
𝐶) ↔ (A ∈ B ∧ A ∈ 𝐶)) 

Theorem  elin2 3121 
Membership in a class defined as an intersection. (Contributed by
Stefan O'Rear, 29Mar2015.)

⊢ 𝑋 = (B
∩ 𝐶) ⇒ ⊢ (A ∈ 𝑋 ↔ (A ∈ B ∧ A ∈ 𝐶)) 

Theorem  elin3 3122 
Membership in a class defined as a ternary intersection. (Contributed
by Stefan O'Rear, 29Mar2015.)

⊢ 𝑋 = ((B
∩ 𝐶) ∩ 𝐷)
⇒ ⊢ (A ∈ 𝑋 ↔ (A ∈ B ∧ A ∈ 𝐶 ∧ A ∈ 𝐷)) 

Theorem  incom 3123 
Commutative law for intersection of classes. Exercise 7 of
[TakeutiZaring] p. 17.
(Contributed by NM, 5Aug1993.)

⊢ (A ∩
B) = (B ∩ A) 

Theorem  ineqri 3124* 
Inference from membership to intersection. (Contributed by NM,
5Aug1993.)

⊢ ((x ∈ A ∧ x ∈ B) ↔
x ∈
𝐶) ⇒ ⊢ (A ∩ B) =
𝐶 

Theorem  ineq1 3125 
Equality theorem for intersection of two classes. (Contributed by NM,
14Dec1993.)

⊢ (A =
B → (A ∩ 𝐶) = (B
∩ 𝐶)) 

Theorem  ineq2 3126 
Equality theorem for intersection of two classes. (Contributed by NM,
26Dec1993.)

⊢ (A =
B → (𝐶 ∩ A) = (𝐶 ∩ B)) 

Theorem  ineq12 3127 
Equality theorem for intersection of two classes. (Contributed by NM,
8May1994.)

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

Theorem  ineq1i 3128 
Equality inference for intersection of two classes. (Contributed by NM,
26Dec1993.)

⊢ A =
B ⇒ ⊢ (A ∩ 𝐶) = (B
∩ 𝐶) 

Theorem  ineq2i 3129 
Equality inference for intersection of two classes. (Contributed by NM,
26Dec1993.)

⊢ A =
B ⇒ ⊢ (𝐶 ∩ A) = (𝐶 ∩ B) 

Theorem  ineq12i 3130 
Equality inference for intersection of two classes. (Contributed by
NM, 24Jun2004.) (Proof shortened by Eric Schmidt, 26Jan2007.)

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

Theorem  ineq1d 3131 
Equality deduction for intersection of two classes. (Contributed by NM,
10Apr1994.)

⊢ (φ
→ A = B) ⇒ ⊢ (φ → (A ∩ 𝐶) = (B
∩ 𝐶)) 

Theorem  ineq2d 3132 
Equality deduction for intersection of two classes. (Contributed by NM,
10Apr1994.)

⊢ (φ
→ A = B) ⇒ ⊢ (φ → (𝐶 ∩ A) = (𝐶 ∩ B)) 

Theorem  ineq12d 3133 
Equality deduction for intersection of two classes. (Contributed by
NM, 24Jun2004.) (Proof shortened by Andrew Salmon, 26Jun2011.)

⊢ (φ
→ A = B)
& ⊢ (φ
→ 𝐶 = 𝐷)
⇒ ⊢ (φ → (A ∩ 𝐶) = (B
∩ 𝐷)) 

Theorem  ineqan12d 3134 
Equality deduction for intersection of two classes. (Contributed by
NM, 7Feb2007.)

⊢ (φ
→ A = B)
& ⊢ (ψ
→ 𝐶 = 𝐷)
⇒ ⊢ ((φ ∧ ψ) → (A ∩ 𝐶) = (B
∩ 𝐷)) 

Theorem  dfss1 3135 
A frequentlyused variant of subclass definition dfss 2925. (Contributed
by NM, 10Jan2015.)

⊢ (A ⊆
B ↔ (B ∩ A) =
A) 

Theorem  dfss5 3136 
Another definition of subclasshood. Similar to dfss 2925, dfss 2926, and
dfss1 3135. (Contributed by David Moews, 1May2017.)

⊢ (A ⊆
B ↔ A = (B ∩
A)) 

Theorem  nfin 3137 
Boundvariable hypothesis builder for the intersection of classes.
(Contributed by NM, 15Sep2003.) (Revised by Mario Carneiro,
14Oct2016.)

⊢ ℲxA & ⊢
ℲxB ⇒ ⊢ Ⅎx(A ∩
B) 

Theorem  csbing 3138 
Distribute proper substitution through an intersection relation.
(Contributed by Alan Sare, 22Jul2012.)

⊢ (A ∈ B →
⦋A / x⦌(𝐶 ∩ 𝐷) = (⦋A / x⦌𝐶 ∩ ⦋A / x⦌𝐷)) 

Theorem  rabbi2dva 3139* 
Deduction from a wff to a restricted class abstraction. (Contributed by
NM, 14Jan2014.)

⊢ ((φ
∧ x ∈ A) →
(x ∈
B ↔ ψ)) ⇒ ⊢ (φ → (A ∩ B) =
{x ∈
A ∣ ψ}) 

Theorem  inidm 3140 
Idempotent law for intersection of classes. Theorem 15 of [Suppes]
p. 26. (Contributed by NM, 5Aug1993.)

⊢ (A ∩
A) = A 

Theorem  inass 3141 
Associative law for intersection of classes. Exercise 9 of
[TakeutiZaring] p. 17.
(Contributed by NM, 3May1994.)

⊢ ((A ∩
B) ∩ 𝐶) = (A
∩ (B ∩ 𝐶)) 

Theorem  in12 3142 
A rearrangement of intersection. (Contributed by NM, 21Apr2001.)

⊢ (A ∩
(B ∩ 𝐶)) = (B
∩ (A ∩ 𝐶)) 

Theorem  in32 3143 
A rearrangement of intersection. (Contributed by NM, 21Apr2001.)
(Proof shortened by Andrew Salmon, 26Jun2011.)

⊢ ((A ∩
B) ∩ 𝐶) = ((A
∩ 𝐶) ∩ B) 

Theorem  in13 3144 
A rearrangement of intersection. (Contributed by NM, 27Aug2012.)

⊢ (A ∩
(B ∩ 𝐶)) = (𝐶 ∩ (B ∩ A)) 

Theorem  in31 3145 
A rearrangement of intersection. (Contributed by NM, 27Aug2012.)

⊢ ((A ∩
B) ∩ 𝐶) = ((𝐶 ∩ B) ∩ A) 

Theorem  inrot 3146 
Rotate the intersection of 3 classes. (Contributed by NM,
27Aug2012.)

⊢ ((A ∩
B) ∩ 𝐶) = ((𝐶 ∩ A) ∩ B) 

Theorem  in4 3147 
Rearrangement of intersection of 4 classes. (Contributed by NM,
21Apr2001.)

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

Theorem  inindi 3148 
Intersection distributes over itself. (Contributed by NM, 6May1994.)

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

Theorem  inindir 3149 
Intersection distributes over itself. (Contributed by NM,
17Aug2004.)

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

Theorem  sseqin2 3150 
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 3151 
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 3152 
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 3153 
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 3154 
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 3155 
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 3156 
Add right intersection to subclass relation. (Contributed by NM,
16Aug1994.) (Proof shortened by Andrew Salmon, 26Jun2011.)

⊢ (A ⊆
B → (A ∩ 𝐶) ⊆ (B ∩ 𝐶)) 

Theorem  sslin 3157 
Add left intersection to subclass relation. (Contributed by NM,
19Oct1999.)

⊢ (A ⊆
B → (𝐶 ∩ A) ⊆ (𝐶 ∩ B)) 

Theorem  ss2in 3158 
Intersection of subclasses. (Contributed by NM, 5May2000.)

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

Theorem  ssinss1 3159 
Intersection preserves subclass relationship. (Contributed by NM,
14Sep1999.)

⊢ (A ⊆
𝐶 → (A ∩ B)
⊆ 𝐶) 

Theorem  inss 3160 
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 3161 
Absorption law for union. (Contributed by NM, 16Apr2006.)

⊢ (A ∪
(A ∩ B)) = A 

Theorem  inabs 3162 
Absorption law for intersection. (Contributed by NM, 16Apr2006.)

⊢ (A ∩
(A ∪ B)) = A 

Theorem  nssinpss 3163 
Negation of subclass expressed in terms of intersection and proper
subclass. (Contributed by NM, 30Jun2004.) (Proof shortened by Andrew
Salmon, 26Jun2011.)

⊢ (¬ A
⊆ B ↔ (A ∩ B)
⊊ A) 

Theorem  nsspssun 3164 
Negation of subclass expressed in terms of proper subclass and union.
(Contributed by NM, 15Sep2004.)

⊢ (¬ A
⊆ B ↔ B ⊊ (A
∪ B)) 

Theorem  ssddif 3165 
Double complement and subset. Similar to ddifss 3169 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 3166 
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 3167 
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 3168 
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 3169 
Double complement under universal class. In classical logic (or given an
additional hypothesis, as in ddifnel 3069), this is equality rather than
subset. (Contributed by Jim Kingdon, 24Jul2018.)

⊢ A ⊆
(V ∖ (V ∖ A)) 

Theorem  unssin 3170 
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 3171 
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 3172 
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 3173 
Intersection with universal complement. Remark in [Stoll] p. 20.
(Contributed by NM, 17Aug2004.)

⊢ (A ∩ (V
∖ B)) = (A ∖ B) 

Theorem  indif 3174 
Intersection with class difference. Theorem 34 of [Suppes] p. 29.
(Contributed by NM, 17Aug2004.)

⊢ (A ∩
(A ∖ B)) = (A
∖ B) 

Theorem  indif2 3175 
Bring an intersection in and out of a class difference. (Contributed by
Jeff Hankins, 15Jul2009.)

⊢ (A ∩
(B ∖ 𝐶)) = ((A ∩ B)
∖ 𝐶) 

Theorem  indif1 3176 
Bring an intersection in and out of a class difference. (Contributed by
Mario Carneiro, 15May2015.)

⊢ ((A ∖
𝐶) ∩ B) = ((A ∩
B) ∖ 𝐶) 

Theorem  indifcom 3177 
Commutation law for intersection and difference. (Contributed by Scott
Fenton, 18Feb2013.)

⊢ (A ∩
(B ∖ 𝐶)) = (B
∩ (A ∖ 𝐶)) 

Theorem  indi 3178 
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 3179 
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 3180 
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 3181 
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 3182 
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 3183 
Distributive law for class difference. Theorem 39 of [Suppes] p. 29.
(Contributed by NM, 17Aug2004.)

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

Theorem  difundir 3184 
Distributive law for class difference. (Contributed by NM,
17Aug2004.)

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

Theorem  difindiss 3185 
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 3186 
Distributive law for class difference. (Contributed by NM,
17Aug2004.)

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

Theorem  indifdir 3187 
Distribute intersection over difference. (Contributed by Scott Fenton,
14Apr2011.)

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

Theorem  difdif2ss 3188 
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 3189 
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 3190 
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 3191 
A relationship involving double difference and union. (Contributed by NM,
29Aug2004.)

⊢ (A ∖
(B ∪ 𝐶)) = ((A ∖ B)
∖ 𝐶) 

Theorem  undif3ss 3192 
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 3193 
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 3194 
Swap second and third argument of double difference. (Contributed by NM,
18Aug2004.)

⊢ ((A ∖
B) ∖ 𝐶) = ((A
∖ 𝐶) ∖
B) 

Theorem  difabs 3195 
Absorptionlike law for class difference: you can remove a class only
once. (Contributed by FL, 2Aug2009.)

⊢ ((A ∖
B) ∖ B) = (A ∖
B) 

Theorem  symdif1 3196 
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 3197* 
Expressing symmetric difference with exclusiveor or two differences.
(Contributed by Jim Kingdon, 28Jul2018.)

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

Theorem  unab 3198 
Union of two class abstractions. (Contributed by NM, 29Sep2002.)
(Proof shortened by Andrew Salmon, 26Jun2011.)

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

Theorem  inab 3199 
Intersection of two class abstractions. (Contributed by NM,
29Sep2002.) (Proof shortened by Andrew Salmon, 26Jun2011.)

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

Theorem  difab 3200 
Difference of two class abstractions. (Contributed by NM,
23Oct2004.) (Proof shortened by Andrew Salmon, 26Jun2011.)

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