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Theorem List for Intuitionistic Logic Explorer - 3201-3300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremunass 3201 Associative law for union of classes. Exercise 8 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐵) ∪ 𝐶) = (𝐴 ∪ (𝐵𝐶))
 
Theoremun12 3202 A rearrangement of union. (Contributed by NM, 12-Aug-2004.)
(𝐴 ∪ (𝐵𝐶)) = (𝐵 ∪ (𝐴𝐶))
 
Theoremun23 3203 A rearrangement of union. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∪ 𝐵)
 
Theoremun4 3204 A rearrangement of the union of 4 classes. (Contributed by NM, 12-Aug-2004.)
((𝐴𝐵) ∪ (𝐶𝐷)) = ((𝐴𝐶) ∪ (𝐵𝐷))
 
Theoremunundi 3205 Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
(𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
 
Theoremunundir 3206 Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
 
Theoremssun1 3207 Subclass relationship for union of classes. Theorem 25 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
𝐴 ⊆ (𝐴𝐵)
 
Theoremssun2 3208 Subclass relationship for union of classes. (Contributed by NM, 30-Aug-1993.)
𝐴 ⊆ (𝐵𝐴)
 
Theoremssun3 3209 Subclass law for union of classes. (Contributed by NM, 5-Aug-1993.)
(𝐴𝐵𝐴 ⊆ (𝐵𝐶))
 
Theoremssun4 3210 Subclass law for union of classes. (Contributed by NM, 14-Aug-1994.)
(𝐴𝐵𝐴 ⊆ (𝐶𝐵))
 
Theoremelun1 3211 Membership law for union of classes. (Contributed by NM, 5-Aug-1993.)
(𝐴𝐵𝐴 ∈ (𝐵𝐶))
 
Theoremelun2 3212 Membership law for union of classes. (Contributed by NM, 30-Aug-1993.)
(𝐴𝐵𝐴 ∈ (𝐶𝐵))
 
Theoremunss1 3213 Subclass law for union of classes. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
 
Theoremssequn1 3214 A relationship between subclass and union. Theorem 26 of [Suppes] p. 27. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴𝐵 ↔ (𝐴𝐵) = 𝐵)
 
Theoremunss2 3215 Subclass law for union of classes. Exercise 7 of [TakeutiZaring] p. 18. (Contributed by NM, 14-Oct-1999.)
(𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
 
Theoremunss12 3216 Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.)
((𝐴𝐵𝐶𝐷) → (𝐴𝐶) ⊆ (𝐵𝐷))
 
Theoremssequn2 3217 A relationship between subclass and union. (Contributed by NM, 13-Jun-1994.)
(𝐴𝐵 ↔ (𝐵𝐴) = 𝐵)
 
Theoremunss 3218 The union of two subclasses is a subclass. Theorem 27 of [Suppes] p. 27 and its converse. (Contributed by NM, 11-Jun-2004.)
((𝐴𝐶𝐵𝐶) ↔ (𝐴𝐵) ⊆ 𝐶)
 
Theoremunssi 3219 An inference showing the union of two subclasses is a subclass. (Contributed by Raph Levien, 10-Dec-2002.)
𝐴𝐶    &   𝐵𝐶       (𝐴𝐵) ⊆ 𝐶
 
Theoremunssd 3220 A deduction showing the union of two subclasses is a subclass. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝐴𝐶)    &   (𝜑𝐵𝐶)       (𝜑 → (𝐴𝐵) ⊆ 𝐶)
 
Theoremunssad 3221 If (𝐴𝐵) is contained in 𝐶, so is 𝐴. One-way deduction form of unss 3218. Partial converse of unssd 3220. (Contributed by David Moews, 1-May-2017.)
(𝜑 → (𝐴𝐵) ⊆ 𝐶)       (𝜑𝐴𝐶)
 
Theoremunssbd 3222 If (𝐴𝐵) is contained in 𝐶, so is 𝐵. One-way deduction form of unss 3218. Partial converse of unssd 3220. (Contributed by David Moews, 1-May-2017.)
(𝜑 → (𝐴𝐵) ⊆ 𝐶)       (𝜑𝐵𝐶)
 
Theoremssun 3223 A condition that implies inclusion in the union of two classes. (Contributed by NM, 23-Nov-2003.)
((𝐴𝐵𝐴𝐶) → 𝐴 ⊆ (𝐵𝐶))
 
Theoremrexun 3224 Restricted existential quantification over union. (Contributed by Jeff Madsen, 5-Jan-2011.)
(∃𝑥 ∈ (𝐴𝐵)𝜑 ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐵 𝜑))
 
Theoremralunb 3225 Restricted quantification over a union. (Contributed by Scott Fenton, 12-Apr-2011.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(∀𝑥 ∈ (𝐴𝐵)𝜑 ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐵 𝜑))
 
Theoremralun 3226 Restricted quantification over union. (Contributed by Jeff Madsen, 2-Sep-2009.)
((∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐵 𝜑) → ∀𝑥 ∈ (𝐴𝐵)𝜑)
 
2.1.13.3  The intersection of two classes
 
Theoremelin 3227 Expansion of membership in an intersection of two classes. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 29-Apr-1994.)
(𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))
 
Theoremelini 3228 Membership in an intersection of two classes. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
𝐴𝐵    &   𝐴𝐶       𝐴 ∈ (𝐵𝐶)
 
Theoremelind 3229 Deduce membership in an intersection of two classes. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝑋𝐴)    &   (𝜑𝑋𝐵)       (𝜑𝑋 ∈ (𝐴𝐵))
 
Theoremelinel1 3230 Membership in an intersection implies membership in the first set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ (𝐵𝐶) → 𝐴𝐵)
 
Theoremelinel2 3231 Membership in an intersection implies membership in the second set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ (𝐵𝐶) → 𝐴𝐶)
 
Theoremelin2 3232 Membership in a class defined as an intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝑋 = (𝐵𝐶)       (𝐴𝑋 ↔ (𝐴𝐵𝐴𝐶))
 
Theoremelin1d 3233 Elementhood in the first set of an intersection - deduction version. (Contributed by Thierry Arnoux, 3-May-2020.)
(𝜑𝑋 ∈ (𝐴𝐵))       (𝜑𝑋𝐴)
 
Theoremelin2d 3234 Elementhood in the first set of an intersection - deduction version. (Contributed by Thierry Arnoux, 3-May-2020.)
(𝜑𝑋 ∈ (𝐴𝐵))       (𝜑𝑋𝐵)
 
Theoremelin3 3235 Membership in a class defined as a ternary intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝑋 = ((𝐵𝐶) ∩ 𝐷)       (𝐴𝑋 ↔ (𝐴𝐵𝐴𝐶𝐴𝐷))
 
Theoremincom 3236 Commutative law for intersection of classes. Exercise 7 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.)
(𝐴𝐵) = (𝐵𝐴)
 
Theoremineqri 3237* Inference from membership to intersection. (Contributed by NM, 5-Aug-1993.)
((𝑥𝐴𝑥𝐵) ↔ 𝑥𝐶)       (𝐴𝐵) = 𝐶
 
Theoremineq1 3238 Equality theorem for intersection of two classes. (Contributed by NM, 14-Dec-1993.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
 
Theoremineq2 3239 Equality theorem for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
(𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
 
Theoremineq12 3240 Equality theorem for intersection of two classes. (Contributed by NM, 8-May-1994.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))
 
Theoremineq1i 3241 Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
𝐴 = 𝐵       (𝐴𝐶) = (𝐵𝐶)
 
Theoremineq2i 3242 Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
𝐴 = 𝐵       (𝐶𝐴) = (𝐶𝐵)
 
Theoremineq12i 3243 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 3244 Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶) = (𝐵𝐶))
 
Theoremineq2d 3245 Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴) = (𝐶𝐵))
 
Theoremineq12d 3246 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 3247 Equality deduction for intersection of two classes. (Contributed by NM, 7-Feb-2007.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜑𝜓) → (𝐴𝐶) = (𝐵𝐷))
 
Theoremdfss1 3248 A frequently-used variant of subclass definition df-ss 3052. (Contributed by NM, 10-Jan-2015.)
(𝐴𝐵 ↔ (𝐵𝐴) = 𝐴)
 
Theoremdfss5 3249 Another definition of subclasshood. Similar to df-ss 3052, dfss 3053, and dfss1 3248. (Contributed by David Moews, 1-May-2017.)
(𝐴𝐵𝐴 = (𝐵𝐴))
 
Theoremnfin 3250 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 3251 Distribute proper substitution through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.)
(𝐴𝐵𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))
 
Theoremrabbi2dva 3252* Deduction from a wff to a restricted class abstraction. (Contributed by NM, 14-Jan-2014.)
((𝜑𝑥𝐴) → (𝑥𝐵𝜓))       (𝜑 → (𝐴𝐵) = {𝑥𝐴𝜓})
 
Theoreminidm 3253 Idempotent law for intersection of classes. Theorem 15 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.)
(𝐴𝐴) = 𝐴
 
Theoreminass 3254 Associative law for intersection of classes. Exercise 9 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.)
((𝐴𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵𝐶))
 
Theoremin12 3255 A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.)
(𝐴 ∩ (𝐵𝐶)) = (𝐵 ∩ (𝐴𝐶))
 
Theoremin32 3256 A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∩ 𝐵)
 
Theoremin13 3257 A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
(𝐴 ∩ (𝐵𝐶)) = (𝐶 ∩ (𝐵𝐴))
 
Theoremin31 3258 A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
((𝐴𝐵) ∩ 𝐶) = ((𝐶𝐵) ∩ 𝐴)
 
Theoreminrot 3259 Rotate the intersection of 3 classes. (Contributed by NM, 27-Aug-2012.)
((𝐴𝐵) ∩ 𝐶) = ((𝐶𝐴) ∩ 𝐵)
 
Theoremin4 3260 Rearrangement of intersection of 4 classes. (Contributed by NM, 21-Apr-2001.)
((𝐴𝐵) ∩ (𝐶𝐷)) = ((𝐴𝐶) ∩ (𝐵𝐷))
 
Theoreminindi 3261 Intersection distributes over itself. (Contributed by NM, 6-May-1994.)
(𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))
 
Theoreminindir 3262 Intersection distributes over itself. (Contributed by NM, 17-Aug-2004.)
((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))
 
Theoremsseqin2 3263 A relationship between subclass and intersection. Similar to Exercise 9 of [TakeutiZaring] p. 18. (Contributed by NM, 17-May-1994.)
(𝐴𝐵 ↔ (𝐵𝐴) = 𝐴)
 
Theoreminss1 3264 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 3265 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 3266 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.)
((𝐴𝐵𝐴𝐶) ↔ 𝐴 ⊆ (𝐵𝐶))
 
Theoremssini 3267 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 3268 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 3269 Add right intersection to subclass relation. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
 
Theoremsslin 3270 Add left intersection to subclass relation. (Contributed by NM, 19-Oct-1999.)
(𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
 
Theoremssrind 3271 Add right intersection to subclass relation. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
(𝜑𝐴𝐵)       (𝜑 → (𝐴𝐶) ⊆ (𝐵𝐶))
 
Theoremss2in 3272 Intersection of subclasses. (Contributed by NM, 5-May-2000.)
((𝐴𝐵𝐶𝐷) → (𝐴𝐶) ⊆ (𝐵𝐷))
 
Theoremssinss1 3273 Intersection preserves subclass relationship. (Contributed by NM, 14-Sep-1999.)
(𝐴𝐶 → (𝐴𝐵) ⊆ 𝐶)
 
Theoreminss 3274 Inclusion of an intersection of two classes. (Contributed by NM, 30-Oct-2014.)
((𝐴𝐶𝐵𝐶) → (𝐴𝐵) ⊆ 𝐶)
 
2.1.13.4  Combinations of difference, union, and intersection of two classes
 
Theoremunabs 3275 Absorption law for union. (Contributed by NM, 16-Apr-2006.)
(𝐴 ∪ (𝐴𝐵)) = 𝐴
 
Theoreminabs 3276 Absorption law for intersection. (Contributed by NM, 16-Apr-2006.)
(𝐴 ∩ (𝐴𝐵)) = 𝐴
 
Theoremdfss4st 3277* Subclass defined in terms of class difference. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(∀𝑥STAB 𝑥𝐴 → (𝐴𝐵 ↔ (𝐵 ∖ (𝐵𝐴)) = 𝐴))
 
Theoremssddif 3278 Double complement and subset. Similar to ddifss 3282 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 3279 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 3280 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 3281 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 3282 Double complement under universal class. In classical logic (or given an additional hypothesis, as in ddifnel 3175), this is equality rather than subset. (Contributed by Jim Kingdon, 24-Jul-2018.)
𝐴 ⊆ (V ∖ (V ∖ 𝐴))
 
Theoremunssin 3283 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 3284 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 3285 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 3286 Intersection with universal complement. Remark in [Stoll] p. 20. (Contributed by NM, 17-Aug-2004.)
(𝐴 ∩ (V ∖ 𝐵)) = (𝐴𝐵)
 
Theoremindif 3287 Intersection with class difference. Theorem 34 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
(𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)
 
Theoremindif2 3288 Bring an intersection in and out of a class difference. (Contributed by Jeff Hankins, 15-Jul-2009.)
(𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶)
 
Theoremindif1 3289 Bring an intersection in and out of a class difference. (Contributed by Mario Carneiro, 15-May-2015.)
((𝐴𝐶) ∩ 𝐵) = ((𝐴𝐵) ∖ 𝐶)
 
Theoremindifcom 3290 Commutation law for intersection and difference. (Contributed by Scott Fenton, 18-Feb-2013.)
(𝐴 ∩ (𝐵𝐶)) = (𝐵 ∩ (𝐴𝐶))
 
Theoremindi 3291 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 3292 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 3293 Distributive law for intersection over union. Theorem 28 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
 
Theoremundir 3294 Distributive law for union over intersection. Theorem 29 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))
 
Theoremuneqin 3295 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 3296 Distributive law for class difference. Theorem 39 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
(𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))
 
Theoremdifundir 3297 Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
 
Theoremdifindiss 3298 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 3299 Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))
 
Theoremindifdir 3300 Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011.)
((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∖ (𝐵𝐶))
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