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Theorem List for Intuitionistic Logic Explorer - 3101-3200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelin3 3101 Membership in a class defined as a ternary intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝑋 = ((B𝐶) ∩ 𝐷)       (A 𝑋 ↔ (A B A 𝐶 A 𝐷))
 
Theoremincom 3102 Commutative law for intersection of classes. Exercise 7 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.)
(AB) = (BA)
 
Theoremineqri 3103* Inference from membership to intersection. (Contributed by NM, 5-Aug-1993.)
((x A x B) ↔ x 𝐶)       (AB) = 𝐶
 
Theoremineq1 3104 Equality theorem for intersection of two classes. (Contributed by NM, 14-Dec-1993.)
(A = B → (A𝐶) = (B𝐶))
 
Theoremineq2 3105 Equality theorem for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
(A = B → (𝐶A) = (𝐶B))
 
Theoremineq12 3106 Equality theorem for intersection of two classes. (Contributed by NM, 8-May-1994.)
((A = B 𝐶 = 𝐷) → (A𝐶) = (B𝐷))
 
Theoremineq1i 3107 Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
A = B       (A𝐶) = (B𝐶)
 
Theoremineq2i 3108 Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
A = B       (𝐶A) = (𝐶B)
 
Theoremineq12i 3109 Equality inference for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
A = B    &   𝐶 = 𝐷       (A𝐶) = (B𝐷)
 
Theoremineq1d 3110 Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)
(φA = B)       (φ → (A𝐶) = (B𝐶))
 
Theoremineq2d 3111 Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)
(φA = B)       (φ → (𝐶A) = (𝐶B))
 
Theoremineq12d 3112 Equality deduction for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(φA = B)    &   (φ𝐶 = 𝐷)       (φ → (A𝐶) = (B𝐷))
 
Theoremineqan12d 3113 Equality deduction for intersection of two classes. (Contributed by NM, 7-Feb-2007.)
(φA = B)    &   (ψ𝐶 = 𝐷)       ((φ ψ) → (A𝐶) = (B𝐷))
 
Theoremdfss1 3114 A frequently-used variant of subclass definition df-ss 2904. (Contributed by NM, 10-Jan-2015.)
(AB ↔ (BA) = A)
 
Theoremdfss5 3115 Another definition of subclasshood. Similar to df-ss 2904, dfss 2905, and dfss1 3114. (Contributed by David Moews, 1-May-2017.)
(ABA = (BA))
 
Theoremnfin 3116 Bound-variable hypothesis builder for the intersection of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)
xA    &   xB       x(AB)
 
Theoremcsbing 3117 Distribute proper substitution through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.)
(A BA / x(𝐶𝐷) = (A / x𝐶A / x𝐷))
 
Theoremrabbi2dva 3118* Deduction from a wff to a restricted class abstraction. (Contributed by NM, 14-Jan-2014.)
((φ x A) → (x Bψ))       (φ → (AB) = {x Aψ})
 
Theoreminidm 3119 Idempotent law for intersection of classes. Theorem 15 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.)
(AA) = A
 
Theoreminass 3120 Associative law for intersection of classes. Exercise 9 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.)
((AB) ∩ 𝐶) = (A ∩ (B𝐶))
 
Theoremin12 3121 A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.)
(A ∩ (B𝐶)) = (B ∩ (A𝐶))
 
Theoremin32 3122 A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((AB) ∩ 𝐶) = ((A𝐶) ∩ B)
 
Theoremin13 3123 A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
(A ∩ (B𝐶)) = (𝐶 ∩ (BA))
 
Theoremin31 3124 A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
((AB) ∩ 𝐶) = ((𝐶B) ∩ A)
 
Theoreminrot 3125 Rotate the intersection of 3 classes. (Contributed by NM, 27-Aug-2012.)
((AB) ∩ 𝐶) = ((𝐶A) ∩ B)
 
Theoremin4 3126 Rearrangement of intersection of 4 classes. (Contributed by NM, 21-Apr-2001.)
((AB) ∩ (𝐶𝐷)) = ((A𝐶) ∩ (B𝐷))
 
Theoreminindi 3127 Intersection distributes over itself. (Contributed by NM, 6-May-1994.)
(A ∩ (B𝐶)) = ((AB) ∩ (A𝐶))
 
Theoreminindir 3128 Intersection distributes over itself. (Contributed by NM, 17-Aug-2004.)
((AB) ∩ 𝐶) = ((A𝐶) ∩ (B𝐶))
 
Theoremsseqin2 3129 A relationship between subclass and intersection. Similar to Exercise 9 of [TakeutiZaring] p. 18. (Contributed by NM, 17-May-1994.)
(AB ↔ (BA) = A)
 
Theoreminss1 3130 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 3131 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 3132 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.)
((AB A𝐶) ↔ A ⊆ (B𝐶))
 
Theoremssini 3133 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 3134 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 3135 Add right intersection to subclass relation. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(AB → (A𝐶) ⊆ (B𝐶))
 
Theoremsslin 3136 Add left intersection to subclass relation. (Contributed by NM, 19-Oct-1999.)
(AB → (𝐶A) ⊆ (𝐶B))
 
Theoremss2in 3137 Intersection of subclasses. (Contributed by NM, 5-May-2000.)
((AB 𝐶𝐷) → (A𝐶) ⊆ (B𝐷))
 
Theoremssinss1 3138 Intersection preserves subclass relationship. (Contributed by NM, 14-Sep-1999.)
(A𝐶 → (AB) ⊆ 𝐶)
 
Theoreminss 3139 Inclusion of an intersection of two classes. (Contributed by NM, 30-Oct-2014.)
((A𝐶 B𝐶) → (AB) ⊆ 𝐶)
 
2.1.13.4  Combinations of difference, union, and intersection of two classes
 
Theoremunabs 3140 Absorption law for union. (Contributed by NM, 16-Apr-2006.)
(A ∪ (AB)) = A
 
Theoreminabs 3141 Absorption law for intersection. (Contributed by NM, 16-Apr-2006.)
(A ∩ (AB)) = A
 
Theoremnssinpss 3142 Negation of subclass expressed in terms of intersection and proper subclass. (Contributed by NM, 30-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
AB ↔ (AB) ⊊ A)
 
Theoremnsspssun 3143 Negation of subclass expressed in terms of proper subclass and union. (Contributed by NM, 15-Sep-2004.)
ABB ⊊ (AB))
 
Theoremssddif 3144 Double complement and subset. Similar to ddifss 3148 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 3145 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 3146 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 3147 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 3148 Double complement under universal class. In classical logic (or given an additional hypothesis, as in ddifnel 3048), this is equality rather than subset. (Contributed by Jim Kingdon, 24-Jul-2018.)
A ⊆ (V ∖ (V ∖ A))
 
Theoremunssin 3149 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 3150 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 3151 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 3152 Intersection with universal complement. Remark in [Stoll] p. 20. (Contributed by NM, 17-Aug-2004.)
(A ∩ (V ∖ B)) = (AB)
 
Theoremindif 3153 Intersection with class difference. Theorem 34 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
(A ∩ (AB)) = (AB)
 
Theoremindif2 3154 Bring an intersection in and out of a class difference. (Contributed by Jeff Hankins, 15-Jul-2009.)
(A ∩ (B𝐶)) = ((AB) ∖ 𝐶)
 
Theoremindif1 3155 Bring an intersection in and out of a class difference. (Contributed by Mario Carneiro, 15-May-2015.)
((A𝐶) ∩ B) = ((AB) ∖ 𝐶)
 
Theoremindifcom 3156 Commutation law for intersection and difference. (Contributed by Scott Fenton, 18-Feb-2013.)
(A ∩ (B𝐶)) = (B ∩ (A𝐶))
 
Theoremindi 3157 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 3158 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 3159 Distributive law for intersection over union. Theorem 28 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
((AB) ∩ 𝐶) = ((A𝐶) ∪ (B𝐶))
 
Theoremundir 3160 Distributive law for union over intersection. Theorem 29 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
((AB) ∪ 𝐶) = ((A𝐶) ∩ (B𝐶))
 
Theoremuneqin 3161 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 3162 Distributive law for class difference. Theorem 39 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
(A ∖ (B𝐶)) = ((AB) ∩ (A𝐶))
 
Theoremdifundir 3163 Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
((AB) ∖ 𝐶) = ((A𝐶) ∪ (B𝐶))
 
Theoremdifindiss 3164 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 3165 Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
((AB) ∖ 𝐶) = ((A𝐶) ∩ (B𝐶))
 
Theoremindifdir 3166 Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011.)
((AB) ∩ 𝐶) = ((A𝐶) ∖ (B𝐶))
 
Theoremdifdif2ss 3167 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 3168 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 3169 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 3170 A relationship involving double difference and union. (Contributed by NM, 29-Aug-2004.)
(A ∖ (B𝐶)) = ((AB) ∖ 𝐶)
 
Theoremundif3ss 3171 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 3172 Represent a set difference as an intersection with a larger difference. (Contributed by Jeff Madsen, 2-Sep-2009.)
(A𝐶 → (AB) = ((𝐶B) ∩ A))
 
Theoremdif32 3173 Swap second and third argument of double difference. (Contributed by NM, 18-Aug-2004.)
((AB) ∖ 𝐶) = ((A𝐶) ∖ B)
 
Theoremdifabs 3174 Absorption-like law for class difference: you can remove a class only once. (Contributed by FL, 2-Aug-2009.)
((AB) ∖ B) = (AB)
 
Theoremsymdif1 3175 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 3176* Expressing symmetric difference with exclusive-or or two differences. (Contributed by Jim Kingdon, 28-Jul-2018.)
((AB) ∪ (BA)) = {x ∣ (x Ax B)}
 
Theoremunab 3177 Union of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
({xφ} ∪ {xψ}) = {x ∣ (φ ψ)}
 
Theoreminab 3178 Intersection of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
({xφ} ∩ {xψ}) = {x ∣ (φ ψ)}
 
Theoremdifab 3179 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 3180 A class builder defined by a negation. (Contributed by FL, 18-Sep-2010.)
{x ∣ ¬ φ} = (V ∖ {xφ})
 
Theoremunrab 3181 Union of two restricted class abstractions. (Contributed by NM, 25-Mar-2004.)
({x Aφ} ∪ {x Aψ}) = {x A ∣ (φ ψ)}
 
Theoreminrab 3182 Intersection of two restricted class abstractions. (Contributed by NM, 1-Sep-2006.)
({x Aφ} ∩ {x Aψ}) = {x A ∣ (φ ψ)}
 
Theoreminrab2 3183* Intersection with a restricted class abstraction. (Contributed by NM, 19-Nov-2007.)
({x Aφ} ∩ B) = {x (AB) ∣ φ}
 
Theoremdifrab 3184 Difference of two restricted class abstractions. (Contributed by NM, 23-Oct-2004.)
({x Aφ} ∖ {x Aψ}) = {x A ∣ (φ ¬ ψ)}
 
Theoremdfrab2 3185* Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.)
{x Aφ} = ({xφ} ∩ A)
 
Theoremdfrab3 3186* Alternate definition of restricted class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
{x Aφ} = (A ∩ {xφ})
 
Theoremnotrab 3187* Complementation of restricted class abstractions. (Contributed by Mario Carneiro, 3-Sep-2015.)
(A ∖ {x Aφ}) = {x A ∣ ¬ φ}
 
Theoremdfrab3ss 3188* Restricted class abstraction with a common superset. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Proof shortened by Mario Carneiro, 8-Nov-2015.)
(AB → {x Aφ} = (A ∩ {x Bφ}))
 
Theoremrabun2 3189 Abstraction restricted to a union. (Contributed by Stefan O'Rear, 5-Feb-2015.)
{x (AB) ∣ φ} = ({x Aφ} ∪ {x Bφ})
 
2.1.13.6  Restricted uniqueness with difference, union, and intersection
 
Theoremreuss2 3190* Transfer uniqueness to a smaller subclass. (Contributed by NM, 20-Oct-2005.)
(((AB x A (φψ)) (x A φ ∃!x B ψ)) → ∃!x A φ)
 
Theoremreuss 3191* Transfer uniqueness to a smaller subclass. (Contributed by NM, 21-Aug-1999.)
((AB x A φ ∃!x B φ) → ∃!x A φ)
 
Theoremreuun1 3192* Transfer uniqueness to a smaller class. (Contributed by NM, 21-Oct-2005.)
((x A φ ∃!x (AB)(φ ψ)) → ∃!x A φ)
 
Theoremreuun2 3193* Transfer uniqueness to a smaller or larger class. (Contributed by NM, 21-Oct-2005.)
x B φ → (∃!x (AB)φ∃!x A φ))
 
Theoremreupick 3194* Restricted uniqueness "picks" a member of a subclass. (Contributed by NM, 21-Aug-1999.)
(((AB (x A φ ∃!x B φ)) φ) → (x Ax B))
 
Theoremreupick3 3195* Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 19-Nov-2016.)
((∃!x A φ x A (φ ψ) x A) → (φψ))
 
Theoremreupick2 3196* Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 15-Dec-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
(((x A (ψφ) x A ψ ∃!x A φ) x A) → (φψ))
 
2.1.14  The empty set
 
Syntaxc0 3197 Extend class notation to include the empty set.
class
 
Definitiondf-nul 3198 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 3199. (Contributed by NM, 5-Aug-1993.)
∅ = (V ∖ V)
 
Theoremdfnul2 3199 Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.)
∅ = {x ∣ ¬ x = x}
 
Theoremdfnul3 3200 Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.)
∅ = {x A ∣ ¬ x A}
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