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Theorem List for Intuitionistic Logic Explorer - 3101-3200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremun12 3101 A rearrangement of union. (Contributed by NM, 12-Aug-2004.)
(𝐴 ∪ (𝐵𝐶)) = (𝐵 ∪ (𝐴𝐶))

Theoremun23 3102 A rearrangement of union. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∪ 𝐵)

Theoremun4 3103 A rearrangement of the union of 4 classes. (Contributed by NM, 12-Aug-2004.)
((𝐴𝐵) ∪ (𝐶𝐷)) = ((𝐴𝐶) ∪ (𝐵𝐷))

Theoremunundi 3104 Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
(𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))

Theoremunundir 3105 Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))

Theoremssun1 3106 Subclass relationship for union of classes. Theorem 25 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
𝐴 ⊆ (𝐴𝐵)

Theoremssun2 3107 Subclass relationship for union of classes. (Contributed by NM, 30-Aug-1993.)
𝐴 ⊆ (𝐵𝐴)

Theoremssun3 3108 Subclass law for union of classes. (Contributed by NM, 5-Aug-1993.)
(𝐴𝐵𝐴 ⊆ (𝐵𝐶))

Theoremssun4 3109 Subclass law for union of classes. (Contributed by NM, 14-Aug-1994.)
(𝐴𝐵𝐴 ⊆ (𝐶𝐵))

Theoremelun1 3110 Membership law for union of classes. (Contributed by NM, 5-Aug-1993.)
(𝐴𝐵𝐴 ∈ (𝐵𝐶))

Theoremelun2 3111 Membership law for union of classes. (Contributed by NM, 30-Aug-1993.)
(𝐴𝐵𝐴 ∈ (𝐶𝐵))

Theoremunss1 3112 Subclass law for union of classes. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))

Theoremssequn1 3113 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 3114 Subclass law for union of classes. Exercise 7 of [TakeutiZaring] p. 18. (Contributed by NM, 14-Oct-1999.)
(𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))

Theoremunss12 3115 Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.)
((𝐴𝐵𝐶𝐷) → (𝐴𝐶) ⊆ (𝐵𝐷))

Theoremssequn2 3116 A relationship between subclass and union. (Contributed by NM, 13-Jun-1994.)
(𝐴𝐵 ↔ (𝐵𝐴) = 𝐵)

Theoremunss 3117 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 3118 An inference showing the union of two subclasses is a subclass. (Contributed by Raph Levien, 10-Dec-2002.)
𝐴𝐶    &   𝐵𝐶       (𝐴𝐵) ⊆ 𝐶

Theoremunssd 3119 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 3120 If (𝐴𝐵) is contained in 𝐶, so is 𝐴. One-way deduction form of unss 3117. Partial converse of unssd 3119. (Contributed by David Moews, 1-May-2017.)
(𝜑 → (𝐴𝐵) ⊆ 𝐶)       (𝜑𝐴𝐶)

Theoremunssbd 3121 If (𝐴𝐵) is contained in 𝐶, so is 𝐵. One-way deduction form of unss 3117. Partial converse of unssd 3119. (Contributed by David Moews, 1-May-2017.)
(𝜑 → (𝐴𝐵) ⊆ 𝐶)       (𝜑𝐵𝐶)

Theoremssun 3122 A condition that implies inclusion in the union of two classes. (Contributed by NM, 23-Nov-2003.)
((𝐴𝐵𝐴𝐶) → 𝐴 ⊆ (𝐵𝐶))

Theoremrexun 3123 Restricted existential quantification over union. (Contributed by Jeff Madsen, 5-Jan-2011.)
(∃𝑥 ∈ (𝐴𝐵)𝜑 ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐵 𝜑))

Theoremralunb 3124 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 3125 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 3126 Expansion of membership in an intersection of two classes. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 29-Apr-1994.)
(𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))

Theoremelin2 3127 Membership in a class defined as an intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝑋 = (𝐵𝐶)       (𝐴𝑋 ↔ (𝐴𝐵𝐴𝐶))

Theoremelin3 3128 Membership in a class defined as a ternary intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝑋 = ((𝐵𝐶) ∩ 𝐷)       (𝐴𝑋 ↔ (𝐴𝐵𝐴𝐶𝐴𝐷))

Theoremincom 3129 Commutative law for intersection of classes. Exercise 7 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.)
(𝐴𝐵) = (𝐵𝐴)

Theoremineqri 3130* Inference from membership to intersection. (Contributed by NM, 5-Aug-1993.)
((𝑥𝐴𝑥𝐵) ↔ 𝑥𝐶)       (𝐴𝐵) = 𝐶

Theoremineq1 3131 Equality theorem for intersection of two classes. (Contributed by NM, 14-Dec-1993.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))

Theoremineq2 3132 Equality theorem for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
(𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))

Theoremineq12 3133 Equality theorem for intersection of two classes. (Contributed by NM, 8-May-1994.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))

Theoremineq1i 3134 Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
𝐴 = 𝐵       (𝐴𝐶) = (𝐵𝐶)

Theoremineq2i 3135 Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
𝐴 = 𝐵       (𝐶𝐴) = (𝐶𝐵)

Theoremineq12i 3136 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 3137 Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶) = (𝐵𝐶))

Theoremineq2d 3138 Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴) = (𝐶𝐵))

Theoremineq12d 3139 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 3140 Equality deduction for intersection of two classes. (Contributed by NM, 7-Feb-2007.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜑𝜓) → (𝐴𝐶) = (𝐵𝐷))

Theoremdfss1 3141 A frequently-used variant of subclass definition df-ss 2931. (Contributed by NM, 10-Jan-2015.)
(𝐴𝐵 ↔ (𝐵𝐴) = 𝐴)

Theoremdfss5 3142 Another definition of subclasshood. Similar to df-ss 2931, dfss 2932, and dfss1 3141. (Contributed by David Moews, 1-May-2017.)
(𝐴𝐵𝐴 = (𝐵𝐴))

Theoremnfin 3143 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 3144 Distribute proper substitution through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.)
(𝐴𝐵𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))

Theoremrabbi2dva 3145* Deduction from a wff to a restricted class abstraction. (Contributed by NM, 14-Jan-2014.)
((𝜑𝑥𝐴) → (𝑥𝐵𝜓))       (𝜑 → (𝐴𝐵) = {𝑥𝐴𝜓})

Theoreminidm 3146 Idempotent law for intersection of classes. Theorem 15 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.)
(𝐴𝐴) = 𝐴

Theoreminass 3147 Associative law for intersection of classes. Exercise 9 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.)
((𝐴𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵𝐶))

Theoremin12 3148 A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.)
(𝐴 ∩ (𝐵𝐶)) = (𝐵 ∩ (𝐴𝐶))

Theoremin32 3149 A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∩ 𝐵)

Theoremin13 3150 A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
(𝐴 ∩ (𝐵𝐶)) = (𝐶 ∩ (𝐵𝐴))

Theoremin31 3151 A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
((𝐴𝐵) ∩ 𝐶) = ((𝐶𝐵) ∩ 𝐴)

Theoreminrot 3152 Rotate the intersection of 3 classes. (Contributed by NM, 27-Aug-2012.)
((𝐴𝐵) ∩ 𝐶) = ((𝐶𝐴) ∩ 𝐵)

Theoremin4 3153 Rearrangement of intersection of 4 classes. (Contributed by NM, 21-Apr-2001.)
((𝐴𝐵) ∩ (𝐶𝐷)) = ((𝐴𝐶) ∩ (𝐵𝐷))

Theoreminindi 3154 Intersection distributes over itself. (Contributed by NM, 6-May-1994.)
(𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))

Theoreminindir 3155 Intersection distributes over itself. (Contributed by NM, 17-Aug-2004.)
((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))

Theoremsseqin2 3156 A relationship between subclass and intersection. Similar to Exercise 9 of [TakeutiZaring] p. 18. (Contributed by NM, 17-May-1994.)
(𝐴𝐵 ↔ (𝐵𝐴) = 𝐴)

Theoreminss1 3157 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 3158 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 3159 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 3160 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 3161 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 3162 Add right intersection to subclass relation. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))

Theoremsslin 3163 Add left intersection to subclass relation. (Contributed by NM, 19-Oct-1999.)
(𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))

Theoremss2in 3164 Intersection of subclasses. (Contributed by NM, 5-May-2000.)
((𝐴𝐵𝐶𝐷) → (𝐴𝐶) ⊆ (𝐵𝐷))

Theoremssinss1 3165 Intersection preserves subclass relationship. (Contributed by NM, 14-Sep-1999.)
(𝐴𝐶 → (𝐴𝐵) ⊆ 𝐶)

Theoreminss 3166 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 3167 Absorption law for union. (Contributed by NM, 16-Apr-2006.)
(𝐴 ∪ (𝐴𝐵)) = 𝐴

Theoreminabs 3168 Absorption law for intersection. (Contributed by NM, 16-Apr-2006.)
(𝐴 ∩ (𝐴𝐵)) = 𝐴

Theoremnssinpss 3169 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 3170 Negation of subclass expressed in terms of proper subclass and union. (Contributed by NM, 15-Sep-2004.)
𝐴𝐵𝐵 ⊊ (𝐴𝐵))

Theoremssddif 3171 Double complement and subset. Similar to ddifss 3175 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 3172 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 3173 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 3174 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 3175 Double complement under universal class. In classical logic (or given an additional hypothesis, as in ddifnel 3075), this is equality rather than subset. (Contributed by Jim Kingdon, 24-Jul-2018.)
𝐴 ⊆ (V ∖ (V ∖ 𝐴))

Theoremunssin 3176 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 3177 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 3178 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 3179 Intersection with universal complement. Remark in [Stoll] p. 20. (Contributed by NM, 17-Aug-2004.)
(𝐴 ∩ (V ∖ 𝐵)) = (𝐴𝐵)

Theoremindif 3180 Intersection with class difference. Theorem 34 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
(𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)

Theoremindif2 3181 Bring an intersection in and out of a class difference. (Contributed by Jeff Hankins, 15-Jul-2009.)
(𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶)

Theoremindif1 3182 Bring an intersection in and out of a class difference. (Contributed by Mario Carneiro, 15-May-2015.)
((𝐴𝐶) ∩ 𝐵) = ((𝐴𝐵) ∖ 𝐶)

Theoremindifcom 3183 Commutation law for intersection and difference. (Contributed by Scott Fenton, 18-Feb-2013.)
(𝐴 ∩ (𝐵𝐶)) = (𝐵 ∩ (𝐴𝐶))

Theoremindi 3184 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 3185 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 3186 Distributive law for intersection over union. Theorem 28 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))

Theoremundir 3187 Distributive law for union over intersection. Theorem 29 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))

Theoremuneqin 3188 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 3189 Distributive law for class difference. Theorem 39 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.)
(𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))

Theoremdifundir 3190 Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))

Theoremdifindiss 3191 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 3192 Distributive law for class difference. (Contributed by NM, 17-Aug-2004.)
((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))

Theoremindifdir 3193 Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011.)
((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∖ (𝐵𝐶))

Theoremdifdif2ss 3194 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 3195 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 3196 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 3197 A relationship involving double difference and union. (Contributed by NM, 29-Aug-2004.)
(𝐴 ∖ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶)

Theoremundif3ss 3198 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 3199 Represent a set difference as an intersection with a larger difference. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐴𝐶 → (𝐴𝐵) = ((𝐶𝐵) ∩ 𝐴))

Theoremdif32 3200 Swap second and third argument of double difference. (Contributed by NM, 18-Aug-2004.)
((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∖ 𝐵)

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