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

Theoremdifabs 3201 Absorption-like law for class difference: you can remove a class only once. (Contributed by FL, 2-Aug-2009.)
((𝐴𝐵) ∖ 𝐵) = (𝐴𝐵)

Theoremsymdif1 3202 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 3203* Expressing symmetric difference with exclusive-or or two differences. (Contributed by Jim Kingdon, 28-Jul-2018.)
((𝐴𝐵) ∪ (𝐵𝐴)) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}

Theoremunab 3204 Union of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
({𝑥𝜑} ∪ {𝑥𝜓}) = {𝑥 ∣ (𝜑𝜓)}

Theoreminab 3205 Intersection of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
({𝑥𝜑} ∩ {𝑥𝜓}) = {𝑥 ∣ (𝜑𝜓)}

Theoremdifab 3206 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 3207 A class builder defined by a negation. (Contributed by FL, 18-Sep-2010.)
{𝑥 ∣ ¬ 𝜑} = (V ∖ {𝑥𝜑})

Theoremunrab 3208 Union of two restricted class abstractions. (Contributed by NM, 25-Mar-2004.)
({𝑥𝐴𝜑} ∪ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑𝜓)}

Theoreminrab 3209 Intersection of two restricted class abstractions. (Contributed by NM, 1-Sep-2006.)
({𝑥𝐴𝜑} ∩ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑𝜓)}

Theoreminrab2 3210* Intersection with a restricted class abstraction. (Contributed by NM, 19-Nov-2007.)
({𝑥𝐴𝜑} ∩ 𝐵) = {𝑥 ∈ (𝐴𝐵) ∣ 𝜑}

Theoremdifrab 3211 Difference of two restricted class abstractions. (Contributed by NM, 23-Oct-2004.)
({𝑥𝐴𝜑} ∖ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑 ∧ ¬ 𝜓)}

Theoremdfrab2 3212* Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.)
{𝑥𝐴𝜑} = ({𝑥𝜑} ∩ 𝐴)

Theoremdfrab3 3213* Alternate definition of restricted class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
{𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝜑})

Theoremnotrab 3214* Complementation of restricted class abstractions. (Contributed by Mario Carneiro, 3-Sep-2015.)
(𝐴 ∖ {𝑥𝐴𝜑}) = {𝑥𝐴 ∣ ¬ 𝜑}

Theoremdfrab3ss 3215* 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 3216 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 3217* Transfer uniqueness to a smaller subclass. (Contributed by NM, 20-Oct-2005.)
(((𝐴𝐵 ∧ ∀𝑥𝐴 (𝜑𝜓)) ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜓)) → ∃!𝑥𝐴 𝜑)

Theoremreuss 3218* Transfer uniqueness to a smaller subclass. (Contributed by NM, 21-Aug-1999.)
((𝐴𝐵 ∧ ∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) → ∃!𝑥𝐴 𝜑)

Theoremreuun1 3219* Transfer uniqueness to a smaller class. (Contributed by NM, 21-Oct-2005.)
((∃𝑥𝐴 𝜑 ∧ ∃!𝑥 ∈ (𝐴𝐵)(𝜑𝜓)) → ∃!𝑥𝐴 𝜑)

Theoremreuun2 3220* Transfer uniqueness to a smaller or larger class. (Contributed by NM, 21-Oct-2005.)
(¬ ∃𝑥𝐵 𝜑 → (∃!𝑥 ∈ (𝐴𝐵)𝜑 ↔ ∃!𝑥𝐴 𝜑))

Theoremreupick 3221* Restricted uniqueness "picks" a member of a subclass. (Contributed by NM, 21-Aug-1999.)
(((𝐴𝐵 ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) ∧ 𝜑) → (𝑥𝐴𝑥𝐵))

Theoremreupick3 3222* Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 19-Nov-2016.)
((∃!𝑥𝐴 𝜑 ∧ ∃𝑥𝐴 (𝜑𝜓) ∧ 𝑥𝐴) → (𝜑𝜓))

Theoremreupick2 3223* 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 3224 Extend class notation to include the empty set.
class

Definitiondf-nul 3225 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 3226. (Contributed by NM, 5-Aug-1993.)
∅ = (V ∖ V)

Theoremdfnul2 3226 Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.)
∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}

Theoremdfnul3 3227 Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.)
∅ = {𝑥𝐴 ∣ ¬ 𝑥𝐴}

Theoremnoel 3228 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 3229 If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2570. (Contributed by NM, 31-Dec-1993.)
(𝐵𝐴 → ¬ 𝐴 = ∅)

Theoremne0i 3230 If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2570. (Contributed by NM, 31-Dec-1993.)
(𝐵𝐴𝐴 ≠ ∅)

Theoremvn0 3231 The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.)
V ≠ ∅

Theoremvn0m 3232 The universal class is inhabited. (Contributed by Jim Kingdon, 17-Dec-2018.)
𝑥 𝑥 ∈ V

Theoremn0rf 3233 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 3234 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 3234* An inhabited class is nonempty. See n0rf 3233 for more discussion. (Contributed by Jim Kingdon, 31-Jul-2018.)
(∃𝑥 𝑥𝐴𝐴 ≠ ∅)

Theoremneq0r 3235* An inhabited class is nonempty. See n0rf 3233 for more discussion. (Contributed by Jim Kingdon, 31-Jul-2018.)
(∃𝑥 𝑥𝐴 → ¬ 𝐴 = ∅)

Theoremreximdva0m 3236* Restricted existence deduced from inhabited class. (Contributed by Jim Kingdon, 31-Jul-2018.)
((𝜑𝑥𝐴) → 𝜓)       ((𝜑 ∧ ∃𝑥 𝑥𝐴) → ∃𝑥𝐴 𝜓)

Theoremn0mmoeu 3237* 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 3238 Vacuous existential quantification is false. (Contributed by NM, 15-Oct-2003.)
¬ ∃𝑥 ∈ ∅ 𝜑

Theoremeq0 3239* The empty set has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by NM, 29-Aug-1993.)
(𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)

Theoremeqv 3240* The universe contains every set. (Contributed by NM, 11-Sep-2006.)
(𝐴 = V ↔ ∀𝑥 𝑥𝐴)

Theorem0el 3241* Membership of the empty set in another class. (Contributed by NM, 29-Jun-2004.)
(∅ ∈ 𝐴 ↔ ∃𝑥𝐴𝑦 ¬ 𝑦𝑥)

Theoremabvor0dc 3242* 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 3243 Nonempty class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)
(∃𝑥𝜑 → {𝑥𝜑} ≠ ∅)

Theoremrabn0r 3244 Non-empty restricted class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)
(∃𝑥𝐴 𝜑 → {𝑥𝐴𝜑} ≠ ∅)

Theoremrabn0m 3245* Inhabited restricted class abstraction. (Contributed by Jim Kingdon, 18-Sep-2018.)
(∃𝑦 𝑦 ∈ {𝑥𝐴𝜑} ↔ ∃𝑥𝐴 𝜑)

Theoremrab0 3246 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 3247 Condition for a restricted class abstraction to be empty. (Contributed by Jeff Madsen, 7-Jun-2010.)
({𝑥𝐴𝜑} = ∅ ↔ ∀𝑥𝐴 ¬ 𝜑)

Theoremabeq0 3248 Condition for a class abstraction to be empty. (Contributed by Jim Kingdon, 12-Aug-2018.)
({𝑥𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)

Theoremrabxmdc 3249* Law of excluded middle given decidability, in terms of restricted class abstractions. (Contributed by Jim Kingdon, 2-Aug-2018.)
(∀𝑥DECID 𝜑𝐴 = ({𝑥𝐴𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑}))

Theoremrabnc 3250* Law of noncontradiction, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)
({𝑥𝐴𝜑} ∩ {𝑥𝐴 ∣ ¬ 𝜑}) = ∅

Theoremun0 3251 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 3252 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 3253 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 3254 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 3255 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 3256 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 3257 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23. (Contributed by NM, 13-Aug-1994.)
(𝐴 ⊆ ∅ → 𝐴 = ∅)

Theoremsseq0 3258 A subclass of an empty class is empty. (Contributed by NM, 7-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐵𝐵 = ∅) → 𝐴 = ∅)

Theoremssn0 3259 A class with a nonempty subclass is nonempty. (Contributed by NM, 17-Feb-2007.)
((𝐴𝐵𝐴 ≠ ∅) → 𝐵 ≠ ∅)

Theoremabf 3260 A class builder with a false argument is empty. (Contributed by NM, 20-Jan-2012.)
¬ 𝜑       {𝑥𝜑} = ∅

Theoremeq0rdv 3261* Deduction rule for equality to the empty set. (Contributed by NM, 11-Jul-2014.)
(𝜑 → ¬ 𝑥𝐴)       (𝜑𝐴 = ∅)

Theoremcsbprc 3262 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 3263 Two classes are empty iff their union is empty. (Contributed by NM, 11-Aug-2004.)
((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (𝐴𝐵) = ∅)

Theoremvss 3264 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 3265 The null set is a proper subset of any non-empty set. (Contributed by NM, 27-Feb-1996.)
(∅ ⊊ 𝐴𝐴 ≠ ∅)

Theoremnpss0 3266 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 3267 Any non-universal class is a proper subclass of the universal class. (Contributed by NM, 17-May-1998.)
(𝐴 ⊊ V ↔ ¬ 𝐴 = V)

Theoremdisj 3268* Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.)
((𝐴𝐵) = ∅ ↔ ∀𝑥𝐴 ¬ 𝑥𝐵)

Theoremdisjr 3269* Two ways of saying that two classes are disjoint. (Contributed by Jeff Madsen, 19-Jun-2011.)
((𝐴𝐵) = ∅ ↔ ∀𝑥𝐵 ¬ 𝑥𝐴)

Theoremdisj1 3270* Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 19-Aug-1993.)
((𝐴𝐵) = ∅ ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐵))

Theoremreldisj 3271 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 3272 Two ways of saying that two classes are disjoint. (Contributed by NM, 19-May-1998.)
((𝐴𝐵) = ∅ ↔ 𝐴 = (𝐴𝐵))

Theoremdisjne 3273 Members of disjoint sets are not equal. (Contributed by NM, 28-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(((𝐴𝐵) = ∅ ∧ 𝐶𝐴𝐷𝐵) → 𝐶𝐷)

Theoremdisjel 3274 A set can't belong to both members of disjoint classes. (Contributed by NM, 28-Feb-2015.)
(((𝐴𝐵) = ∅ ∧ 𝐶𝐴) → ¬ 𝐶𝐵)

Theoremdisj2 3275 Two ways of saying that two classes are disjoint. (Contributed by NM, 17-May-1998.)
((𝐴𝐵) = ∅ ↔ 𝐴 ⊆ (V ∖ 𝐵))

Theoremdisj4im 3276 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 3277 Intersection with a subclass of a disjoint class. (Contributed by FL, 24-Jan-2007.)
((𝐴𝐵 ∧ (𝐵𝐶) = ∅) → (𝐴𝐶) = ∅)

Theoremdisjpss 3278 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 3279 The union of disjoint classes is disjoint. (Contributed by NM, 26-Sep-2004.)
(((𝐴𝐶) = ∅ ∧ (𝐵𝐶) = ∅) ↔ ((𝐴𝐵) ∩ 𝐶) = ∅)

Theoremundisj2 3280 The union of disjoint classes is disjoint. (Contributed by NM, 13-Sep-2004.)
(((𝐴𝐵) = ∅ ∧ (𝐴𝐶) = ∅) ↔ (𝐴 ∩ (𝐵𝐶)) = ∅)

Theoremssindif0im 3281 Subclass implies empty intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.)
(𝐴𝐵 → (𝐴 ∩ (V ∖ 𝐵)) = ∅)

Theoreminelcm 3282 The intersection of classes with a common member is nonempty. (Contributed by NM, 7-Apr-1994.)
((𝐴𝐵𝐴𝐶) → (𝐵𝐶) ≠ ∅)

Theoremminel 3283 A minimum element of a class has no elements in common with the class. (Contributed by NM, 22-Jun-1994.)
((𝐴𝐵 ∧ (𝐶𝐵) = ∅) → ¬ 𝐴𝐶)

Theoremundif4 3284 Distribute union over difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐶) = ∅ → (𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶))

Theoremdisjssun 3285 Subset relation for disjoint classes. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐵) = ∅ → (𝐴 ⊆ (𝐵𝐶) ↔ 𝐴𝐶))

Theoremssdif0im 3286 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 3287 Universal class equality in terms of empty difference. (Contributed by Jim Kingdon, 3-Aug-2018.)
(𝐴 = V → (V ∖ 𝐴) = ∅)

Theoremdifrab0eqim 3288* 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 = {𝑥𝑉𝜑} → (𝑉 ∖ {𝑥𝑉𝜑}) = ∅)

Theoremssnelpss 3289 A subclass missing a member is a proper subclass. (Contributed by NM, 12-Jan-2002.)
(𝐴𝐵 → ((𝐶𝐵 ∧ ¬ 𝐶𝐴) → 𝐴𝐵))

Theoremssnelpssd 3290 Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 3289. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)    &   (𝜑𝐶𝐵)    &   (𝜑 → ¬ 𝐶𝐴)       (𝜑𝐴𝐵)

Theoreminssdif0im 3291 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 3292 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 3293 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 3292. (Contributed by David Abernethy, 17-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐴) = ∅

Theoremdif0 3294 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 3295 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 3296 A class and its relative complement are disjoint. Theorem 38 of [Suppes] p. 29. (Contributed by NM, 24-Mar-1998.)
(𝐴 ∩ (𝐵𝐴)) = ∅

Theoremdifin0 3297 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 3298 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 3299 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 3300 Absorption of difference by union. (Contributed by NM, 18-Aug-2013.)
(𝐴 ∪ (𝐴𝐵)) = 𝐴

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