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

Theoremnotab 3201 A class builder defined by a negation. (Contributed by FL, 18-Sep-2010.)
{x ∣ ¬ φ} = (V ∖ {xφ})

Theoremunrab 3202 Union of two restricted class abstractions. (Contributed by NM, 25-Mar-2004.)
({x Aφ} ∪ {x Aψ}) = {x A ∣ (φ ψ)}

Theoreminrab 3203 Intersection of two restricted class abstractions. (Contributed by NM, 1-Sep-2006.)
({x Aφ} ∩ {x Aψ}) = {x A ∣ (φ ψ)}

Theoreminrab2 3204* Intersection with a restricted class abstraction. (Contributed by NM, 19-Nov-2007.)
({x Aφ} ∩ B) = {x (AB) ∣ φ}

Theoremdifrab 3205 Difference of two restricted class abstractions. (Contributed by NM, 23-Oct-2004.)
({x Aφ} ∖ {x Aψ}) = {x A ∣ (φ ¬ ψ)}

Theoremdfrab2 3206* Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.)
{x Aφ} = ({xφ} ∩ A)

Theoremdfrab3 3207* Alternate definition of restricted class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
{x Aφ} = (A ∩ {xφ})

Theoremnotrab 3208* Complementation of restricted class abstractions. (Contributed by Mario Carneiro, 3-Sep-2015.)
(A ∖ {x Aφ}) = {x A ∣ ¬ φ}

Theoremdfrab3ss 3209* 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 3210 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 3211* 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 3212* Transfer uniqueness to a smaller subclass. (Contributed by NM, 21-Aug-1999.)
((AB x A φ ∃!x B φ) → ∃!x A φ)

Theoremreuun1 3213* Transfer uniqueness to a smaller class. (Contributed by NM, 21-Oct-2005.)
((x A φ ∃!x (AB)(φ ψ)) → ∃!x A φ)

Theoremreuun2 3214* Transfer uniqueness to a smaller or larger class. (Contributed by NM, 21-Oct-2005.)
x B φ → (∃!x (AB)φ∃!x A φ))

Theoremreupick 3215* 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 3216* 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 3217* 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 3218 Extend class notation to include the empty set.
class

Definitiondf-nul 3219 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 3220. (Contributed by NM, 5-Aug-1993.)
∅ = (V ∖ V)

Theoremdfnul2 3220 Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.)
∅ = {x ∣ ¬ x = x}

Theoremdfnul3 3221 Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.)
∅ = {x A ∣ ¬ x A}

Theoremnoel 3222 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 3223 If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2564. (Contributed by NM, 31-Dec-1993.)
(B A → ¬ A = ∅)

Theoremne0i 3224 If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2564. (Contributed by NM, 31-Dec-1993.)
(B AA ≠ ∅)

Theoremvn0 3225 The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.)
V ≠ ∅

Theoremvn0m 3226 The universal class is inhabited. (Contributed by Jim Kingdon, 17-Dec-2018.)
x x V

Theoremn0rf 3227 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 3228 requires only that x not be free in, rather than not occur in, A. (Contributed by Jim Kingdon, 31-Jul-2018.)
xA       (x x AA ≠ ∅)

Theoremn0r 3228* An inhabited class is nonempty. See n0rf 3227 for more discussion. (Contributed by Jim Kingdon, 31-Jul-2018.)
(x x AA ≠ ∅)

Theoremneq0r 3229* An inhabited class is nonempty. See n0rf 3227 for more discussion. (Contributed by Jim Kingdon, 31-Jul-2018.)
(x x A → ¬ A = ∅)

Theoremreximdva0m 3230* Restricted existence deduced from inhabited class. (Contributed by Jim Kingdon, 31-Jul-2018.)
((φ x A) → ψ)       ((φ x x A) → x A ψ)

Theoremn0mmoeu 3231* 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 x A → (∃*x x A∃!x x A))

Theoremrex0 3232 Vacuous existential quantification is false. (Contributed by NM, 15-Oct-2003.)
¬ x φ

Theoremeq0 3233* The empty set has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by NM, 29-Aug-1993.)
(A = ∅ ↔ x ¬ x A)

Theoremeqv 3234* The universe contains every set. (Contributed by NM, 11-Sep-2006.)
(A = V ↔ x x A)

Theorem0el 3235* Membership of the empty set in another class. (Contributed by NM, 29-Jun-2004.)
(∅ Ax A y ¬ y x)

Theoremabvor0dc 3236* 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 3237 Nonempty class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)
(xφ → {xφ} ≠ ∅)

Theoremrabn0r 3238 Non-empty restricted class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)
(x A φ → {x Aφ} ≠ ∅)

Theoremrabn0m 3239* Inhabited restricted class abstraction. (Contributed by Jim Kingdon, 18-Sep-2018.)
(y y {x Aφ} ↔ x A φ)

Theoremrab0 3240 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} = ∅

Theoremrabeq0 3241 Condition for a restricted class abstraction to be empty. (Contributed by Jeff Madsen, 7-Jun-2010.)
({x Aφ} = ∅ ↔ x A ¬ φ)

Theoremabeq0 3242 Condition for a class abstraction to be empty. (Contributed by Jim Kingdon, 12-Aug-2018.)
({xφ} = ∅ ↔ x ¬ φ)

Theoremrabxmdc 3243* Law of excluded middle given decidability, in terms of restricted class abstractions. (Contributed by Jim Kingdon, 2-Aug-2018.)
(xDECID φA = ({x Aφ} ∪ {x A ∣ ¬ φ}))

Theoremrabnc 3244* Law of noncontradiction, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)
({x Aφ} ∩ {x A ∣ ¬ φ}) = ∅

Theoremun0 3245 The union of a class with the empty set is itself. Theorem 24 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
(A ∪ ∅) = A

Theoremin0 3246 The intersection of a class with the empty set is the empty set. Theorem 16 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.)
(A ∩ ∅) = ∅

Theoreminv1 3247 The intersection of a class with the universal class is itself. Exercise 4.10(k) of [Mendelson] p. 231. (Contributed by NM, 17-May-1998.)
(A ∩ V) = A

Theoremunv 3248 The union of a class with the universal class is the universal class. Exercise 4.10(l) of [Mendelson] p. 231. (Contributed by NM, 17-May-1998.)
(A ∪ V) = V

Theorem0ss 3249 The null set is a subset of any class. Part of Exercise 1 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.)
∅ ⊆ A

Theoremss0b 3250 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23 and its converse. (Contributed by NM, 17-Sep-2003.)
(A ⊆ ∅ ↔ A = ∅)

Theoremss0 3251 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23. (Contributed by NM, 13-Aug-1994.)
(A ⊆ ∅ → A = ∅)

Theoremsseq0 3252 A subclass of an empty class is empty. (Contributed by NM, 7-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((AB B = ∅) → A = ∅)

Theoremssn0 3253 A class with a nonempty subclass is nonempty. (Contributed by NM, 17-Feb-2007.)
((AB A ≠ ∅) → B ≠ ∅)

Theoremabf 3254 A class builder with a false argument is empty. (Contributed by NM, 20-Jan-2012.)
¬ φ       {xφ} = ∅

Theoremeq0rdv 3255* Deduction rule for equality to the empty set. (Contributed by NM, 11-Jul-2014.)
(φ → ¬ x A)       (φA = ∅)

Theoremcsbprc 3256 The proper substitution of a proper class for a set into a class results in the empty set. (Contributed by NM, 17-Aug-2018.)
A V → A / xB = ∅)

Theoremun00 3257 Two classes are empty iff their union is empty. (Contributed by NM, 11-Aug-2004.)
((A = ∅ B = ∅) ↔ (AB) = ∅)

Theoremvss 3258 Only the universal class has the universal class as a subclass. (Contributed by NM, 17-Sep-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(V ⊆ AA = V)

Theorem0pss 3259 The null set is a proper subset of any non-empty set. (Contributed by NM, 27-Feb-1996.)
(∅ ⊊ AA ≠ ∅)

Theoremnpss0 3260 No set is a proper subset of the empty set. (Contributed by NM, 17-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
¬ A ⊊ ∅

Theorempssv 3261 Any non-universal class is a proper subclass of the universal class. (Contributed by NM, 17-May-1998.)
(A ⊊ V ↔ ¬ A = V)

Theoremdisj 3262* Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.)
((AB) = ∅ ↔ x A ¬ x B)

Theoremdisjr 3263* Two ways of saying that two classes are disjoint. (Contributed by Jeff Madsen, 19-Jun-2011.)
((AB) = ∅ ↔ x B ¬ x A)

Theoremdisj1 3264* Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 19-Aug-1993.)
((AB) = ∅ ↔ x(x A → ¬ x B))

Theoremreldisj 3265 Two ways of saying that two classes are disjoint, using the complement of B relative to a universe 𝐶. (Contributed by NM, 15-Feb-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(A𝐶 → ((AB) = ∅ ↔ A ⊆ (𝐶B)))

Theoremdisj3 3266 Two ways of saying that two classes are disjoint. (Contributed by NM, 19-May-1998.)
((AB) = ∅ ↔ A = (AB))

Theoremdisjne 3267 Members of disjoint sets are not equal. (Contributed by NM, 28-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(((AB) = ∅ 𝐶 A 𝐷 B) → 𝐶𝐷)

Theoremdisjel 3268 A set can't belong to both members of disjoint classes. (Contributed by NM, 28-Feb-2015.)
(((AB) = ∅ 𝐶 A) → ¬ 𝐶 B)

Theoremdisj2 3269 Two ways of saying that two classes are disjoint. (Contributed by NM, 17-May-1998.)
((AB) = ∅ ↔ A ⊆ (V ∖ B))

Theoremdisj4im 3270 A consequence of two classes being disjoint. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 2-Aug-2018.)
((AB) = ∅ → ¬ (AB) ⊊ A)

Theoremssdisj 3271 Intersection with a subclass of a disjoint class. (Contributed by FL, 24-Jan-2007.)
((AB (B𝐶) = ∅) → (A𝐶) = ∅)

Theoremdisjpss 3272 A class is a proper subset of its union with a disjoint nonempty class. (Contributed by NM, 15-Sep-2004.)
(((AB) = ∅ B ≠ ∅) → A ⊊ (AB))

Theoremundisj1 3273 The union of disjoint classes is disjoint. (Contributed by NM, 26-Sep-2004.)
(((A𝐶) = ∅ (B𝐶) = ∅) ↔ ((AB) ∩ 𝐶) = ∅)

Theoremundisj2 3274 The union of disjoint classes is disjoint. (Contributed by NM, 13-Sep-2004.)
(((AB) = ∅ (A𝐶) = ∅) ↔ (A ∩ (B𝐶)) = ∅)

Theoremssindif0im 3275 Subclass implies empty intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.)
(AB → (A ∩ (V ∖ B)) = ∅)

Theoreminelcm 3276 The intersection of classes with a common member is nonempty. (Contributed by NM, 7-Apr-1994.)
((A B A 𝐶) → (B𝐶) ≠ ∅)

Theoremminel 3277 A minimum element of a class has no elements in common with the class. (Contributed by NM, 22-Jun-1994.)
((A B (𝐶B) = ∅) → ¬ A 𝐶)

Theoremundif4 3278 Distribute union over difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((A𝐶) = ∅ → (A ∪ (B𝐶)) = ((AB) ∖ 𝐶))

Theoremdisjssun 3279 Subset relation for disjoint classes. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((AB) = ∅ → (A ⊆ (B𝐶) ↔ A𝐶))

Theoremssdif0im 3280 Subclass implies empty difference. One direction of Exercise 7 of [TakeutiZaring] p. 22. In classical logic this would be an equivalence. (Contributed by Jim Kingdon, 2-Aug-2018.)
(AB → (AB) = ∅)

Theoremvdif0im 3281 Universal class equality in terms of empty difference. (Contributed by Jim Kingdon, 3-Aug-2018.)
(A = V → (V ∖ A) = ∅)

Theoremdifrab0eqim 3282* If the difference between the restricting class of a restricted class abstraction and the restricted class abstraction is empty, the restricting class is equal to this restricted class abstraction. (Contributed by Jim Kingdon, 3-Aug-2018.)
(𝑉 = {x 𝑉φ} → (𝑉 ∖ {x 𝑉φ}) = ∅)

Theoremssnelpss 3283 A subclass missing a member is a proper subclass. (Contributed by NM, 12-Jan-2002.)
(AB → ((𝐶 B ¬ 𝐶 A) → AB))

Theoremssnelpssd 3284 Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 3283. (Contributed by David Moews, 1-May-2017.)
(φAB)    &   (φ𝐶 B)    &   (φ → ¬ 𝐶 A)       (φAB)

Theoreminssdif0im 3285 Intersection, subclass, and difference relationship. In classical logic the converse would also hold. (Contributed by Jim Kingdon, 3-Aug-2018.)
((AB) ⊆ 𝐶 → (A ∩ (B𝐶)) = ∅)

Theoremdifid 3286 The difference between a class and itself is the empty set. Proposition 5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28. (Contributed by NM, 22-Apr-2004.)
(AA) = ∅

TheoremdifidALT 3287 The difference between a class and itself is the empty set. Proposition 5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28. Alternate proof of difid 3286. (Contributed by David Abernethy, 17-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(AA) = ∅

Theoremdif0 3288 The difference between a class and the empty set. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.)
(A ∖ ∅) = A

Theorem0dif 3289 The difference between the empty set and a class. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.)
(∅ ∖ A) = ∅

Theoremdisjdif 3290 A class and its relative complement are disjoint. Theorem 38 of [Suppes] p. 29. (Contributed by NM, 24-Mar-1998.)
(A ∩ (BA)) = ∅

Theoremdifin0 3291 The difference of a class from its intersection is empty. Theorem 37 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((AB) ∖ B) = ∅

Theoremundif1ss 3292 Absorption of difference by union. In classical logic, as Theorem 35 of [Suppes] p. 29, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)
((AB) ∪ B) ⊆ (AB)

Theoremundif2ss 3293 Absorption of difference by union. In classical logic, as in Part of proof of Corollary 6K of [Enderton] p. 144, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)
(A ∪ (BA)) ⊆ (AB)

Theoremundifabs 3294 Absorption of difference by union. (Contributed by NM, 18-Aug-2013.)
(A ∪ (AB)) = A

Theoreminundifss 3295 The intersection and class difference of a class with another class are contained in the original class. In classical logic we'd be able to make a stronger statement: that everything in the original class is in the intersection or the difference (that is, this theorem would be equality rather than subset). (Contributed by Jim Kingdon, 4-Aug-2018.)
((AB) ∪ (AB)) ⊆ A

Theoremdifun2 3296 Absorption of union by difference. Theorem 36 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)
((AB) ∖ B) = (AB)

Theoremundifss 3297 Union of complementary parts into whole. (Contributed by Jim Kingdon, 4-Aug-2018.)
(AB ↔ (A ∪ (BA)) ⊆ B)

Theoremssdifin0 3298 A subset of a difference does not intersect the subtrahend. (Contributed by Jeff Hankins, 1-Sep-2013.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)
(A ⊆ (B𝐶) → (A𝐶) = ∅)

Theoremssdifeq0 3299 A class is a subclass of itself subtracted from another iff it is the empty set. (Contributed by Steve Rodriguez, 20-Nov-2015.)
(A ⊆ (BA) ↔ A = ∅)

Theoremssundifim 3300 A consequence of inclusion in the union of two classes. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 4-Aug-2018.)
(A ⊆ (B𝐶) → (AB) ⊆ 𝐶)

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