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Theorem List for Intuitionistic Logic Explorer - 3101-3200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremssdifss 3101 Preservation of a subclass relationship by class difference. (Contributed by NM, 15-Feb-2007.)
(𝐴𝐵 → (𝐴𝐶) ⊆ 𝐵)
 
Theoremddifnel 3102* Double complement under universal class. The hypothesis is one way of expressing the idea that membership in 𝐴 is decidable. Exercise 4.10(s) of [Mendelson] p. 231, but with an additional hypothesis. For a version without a hypothesis, but which only states that 𝐴 is a subset of V ∖ (V ∖ 𝐴), see ddifss 3202. (Contributed by Jim Kingdon, 21-Jul-2018.)
𝑥 ∈ (V ∖ 𝐴) → 𝑥𝐴)       (V ∖ (V ∖ 𝐴)) = 𝐴
 
Theoremssconb 3103 Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)
((𝐴𝐶𝐵𝐶) → (𝐴 ⊆ (𝐶𝐵) ↔ 𝐵 ⊆ (𝐶𝐴)))
 
Theoremsscon 3104 Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22. (Contributed by NM, 22-Mar-1998.)
(𝐴𝐵 → (𝐶𝐵) ⊆ (𝐶𝐴))
 
Theoremssdif 3105 Difference law for subsets. (Contributed by NM, 28-May-1998.)
(𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
 
Theoremssdifd 3106 If 𝐴 is contained in 𝐵, then (𝐴𝐶) is contained in (𝐵𝐶). Deduction form of ssdif 3105. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)       (𝜑 → (𝐴𝐶) ⊆ (𝐵𝐶))
 
Theoremsscond 3107 If 𝐴 is contained in 𝐵, then (𝐶𝐵) is contained in (𝐶𝐴). Deduction form of sscon 3104. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)       (𝜑 → (𝐶𝐵) ⊆ (𝐶𝐴))
 
Theoremssdifssd 3108 If 𝐴 is contained in 𝐵, then (𝐴𝐶) is also contained in 𝐵. Deduction form of ssdifss 3101. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)       (𝜑 → (𝐴𝐶) ⊆ 𝐵)
 
Theoremssdif2d 3109 If 𝐴 is contained in 𝐵 and 𝐶 is contained in 𝐷, then (𝐴𝐷) is contained in (𝐵𝐶). Deduction form. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)    &   (𝜑𝐶𝐷)       (𝜑 → (𝐴𝐷) ⊆ (𝐵𝐶))
 
Theoremraldifb 3110 Restricted universal quantification on a class difference in terms of an implication. (Contributed by Alexander van der Vekens, 3-Jan-2018.)
(∀𝑥𝐴 (𝑥𝐵𝜑) ↔ ∀𝑥 ∈ (𝐴𝐵)𝜑)
 
2.1.13.2  The union of two classes
 
Theoremelun 3111 Expansion of membership in class union. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 7-Aug-1994.)
(𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))
 
Theoremuneqri 3112* Inference from membership to union. (Contributed by NM, 5-Aug-1993.)
((𝑥𝐴𝑥𝐵) ↔ 𝑥𝐶)       (𝐴𝐵) = 𝐶
 
Theoremunidm 3113 Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
(𝐴𝐴) = 𝐴
 
Theoremuncom 3114 Commutative law for union of classes. Exercise 6 of [TakeutiZaring] p. 17. (Contributed by NM, 25-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴𝐵) = (𝐵𝐴)
 
Theoremequncom 3115 If a class equals the union of two other classes, then it equals the union of those two classes commuted. (Contributed by Alan Sare, 18-Feb-2012.)
(𝐴 = (𝐵𝐶) ↔ 𝐴 = (𝐶𝐵))
 
Theoremequncomi 3116 Inference form of equncom 3115. (Contributed by Alan Sare, 18-Feb-2012.)
𝐴 = (𝐵𝐶)       𝐴 = (𝐶𝐵)
 
Theoremuneq1 3117 Equality theorem for union of two classes. (Contributed by NM, 5-Aug-1993.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
 
Theoremuneq2 3118 Equality theorem for the union of two classes. (Contributed by NM, 5-Aug-1993.)
(𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
 
Theoremuneq12 3119 Equality theorem for union of two classes. (Contributed by NM, 29-Mar-1998.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))
 
Theoremuneq1i 3120 Inference adding union to the right in a class equality. (Contributed by NM, 30-Aug-1993.)
𝐴 = 𝐵       (𝐴𝐶) = (𝐵𝐶)
 
Theoremuneq2i 3121 Inference adding union to the left in a class equality. (Contributed by NM, 30-Aug-1993.)
𝐴 = 𝐵       (𝐶𝐴) = (𝐶𝐵)
 
Theoremuneq12i 3122 Equality inference for union of two classes. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐶) = (𝐵𝐷)
 
Theoremuneq1d 3123 Deduction adding union to the right in a class equality. (Contributed by NM, 29-Mar-1998.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶) = (𝐵𝐶))
 
Theoremuneq2d 3124 Deduction adding union to the left in a class equality. (Contributed by NM, 29-Mar-1998.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴) = (𝐶𝐵))
 
Theoremuneq12d 3125 Equality deduction for union of two classes. (Contributed by NM, 29-Sep-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶) = (𝐵𝐷))
 
Theoremnfun 3126 Bound-variable hypothesis builder for the union of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)
𝑥𝐴    &   𝑥𝐵       𝑥(𝐴𝐵)
 
Theoremunass 3127 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 3128 A rearrangement of union. (Contributed by NM, 12-Aug-2004.)
(𝐴 ∪ (𝐵𝐶)) = (𝐵 ∪ (𝐴𝐶))
 
Theoremun23 3129 A rearrangement of union. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∪ 𝐵)
 
Theoremun4 3130 A rearrangement of the union of 4 classes. (Contributed by NM, 12-Aug-2004.)
((𝐴𝐵) ∪ (𝐶𝐷)) = ((𝐴𝐶) ∪ (𝐵𝐷))
 
Theoremunundi 3131 Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
(𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
 
Theoremunundir 3132 Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
 
Theoremssun1 3133 Subclass relationship for union of classes. Theorem 25 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
𝐴 ⊆ (𝐴𝐵)
 
Theoremssun2 3134 Subclass relationship for union of classes. (Contributed by NM, 30-Aug-1993.)
𝐴 ⊆ (𝐵𝐴)
 
Theoremssun3 3135 Subclass law for union of classes. (Contributed by NM, 5-Aug-1993.)
(𝐴𝐵𝐴 ⊆ (𝐵𝐶))
 
Theoremssun4 3136 Subclass law for union of classes. (Contributed by NM, 14-Aug-1994.)
(𝐴𝐵𝐴 ⊆ (𝐶𝐵))
 
Theoremelun1 3137 Membership law for union of classes. (Contributed by NM, 5-Aug-1993.)
(𝐴𝐵𝐴 ∈ (𝐵𝐶))
 
Theoremelun2 3138 Membership law for union of classes. (Contributed by NM, 30-Aug-1993.)
(𝐴𝐵𝐴 ∈ (𝐶𝐵))
 
Theoremunss1 3139 Subclass law for union of classes. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
 
Theoremssequn1 3140 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 3141 Subclass law for union of classes. Exercise 7 of [TakeutiZaring] p. 18. (Contributed by NM, 14-Oct-1999.)
(𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
 
Theoremunss12 3142 Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.)
((𝐴𝐵𝐶𝐷) → (𝐴𝐶) ⊆ (𝐵𝐷))
 
Theoremssequn2 3143 A relationship between subclass and union. (Contributed by NM, 13-Jun-1994.)
(𝐴𝐵 ↔ (𝐵𝐴) = 𝐵)
 
Theoremunss 3144 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 3145 An inference showing the union of two subclasses is a subclass. (Contributed by Raph Levien, 10-Dec-2002.)
𝐴𝐶    &   𝐵𝐶       (𝐴𝐵) ⊆ 𝐶
 
Theoremunssd 3146 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 3147 If (𝐴𝐵) is contained in 𝐶, so is 𝐴. One-way deduction form of unss 3144. Partial converse of unssd 3146. (Contributed by David Moews, 1-May-2017.)
(𝜑 → (𝐴𝐵) ⊆ 𝐶)       (𝜑𝐴𝐶)
 
Theoremunssbd 3148 If (𝐴𝐵) is contained in 𝐶, so is 𝐵. One-way deduction form of unss 3144. Partial converse of unssd 3146. (Contributed by David Moews, 1-May-2017.)
(𝜑 → (𝐴𝐵) ⊆ 𝐶)       (𝜑𝐵𝐶)
 
Theoremssun 3149 A condition that implies inclusion in the union of two classes. (Contributed by NM, 23-Nov-2003.)
((𝐴𝐵𝐴𝐶) → 𝐴 ⊆ (𝐵𝐶))
 
Theoremrexun 3150 Restricted existential quantification over union. (Contributed by Jeff Madsen, 5-Jan-2011.)
(∃𝑥 ∈ (𝐴𝐵)𝜑 ↔ (∃𝑥𝐴 𝜑 ∨ ∃𝑥𝐵 𝜑))
 
Theoremralunb 3151 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 3152 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 3153 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 3154 Membership in a class defined as an intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝑋 = (𝐵𝐶)       (𝐴𝑋 ↔ (𝐴𝐵𝐴𝐶))
 
Theoremelin3 3155 Membership in a class defined as a ternary intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝑋 = ((𝐵𝐶) ∩ 𝐷)       (𝐴𝑋 ↔ (𝐴𝐵𝐴𝐶𝐴𝐷))
 
Theoremincom 3156 Commutative law for intersection of classes. Exercise 7 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.)
(𝐴𝐵) = (𝐵𝐴)
 
Theoremineqri 3157* Inference from membership to intersection. (Contributed by NM, 5-Aug-1993.)
((𝑥𝐴𝑥𝐵) ↔ 𝑥𝐶)       (𝐴𝐵) = 𝐶
 
Theoremineq1 3158 Equality theorem for intersection of two classes. (Contributed by NM, 14-Dec-1993.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
 
Theoremineq2 3159 Equality theorem for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
(𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
 
Theoremineq12 3160 Equality theorem for intersection of two classes. (Contributed by NM, 8-May-1994.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))
 
Theoremineq1i 3161 Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
𝐴 = 𝐵       (𝐴𝐶) = (𝐵𝐶)
 
Theoremineq2i 3162 Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
𝐴 = 𝐵       (𝐶𝐴) = (𝐶𝐵)
 
Theoremineq12i 3163 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 3164 Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶) = (𝐵𝐶))
 
Theoremineq2d 3165 Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴) = (𝐶𝐵))
 
Theoremineq12d 3166 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 3167 Equality deduction for intersection of two classes. (Contributed by NM, 7-Feb-2007.)
(𝜑𝐴 = 𝐵)    &   (𝜓𝐶 = 𝐷)       ((𝜑𝜓) → (𝐴𝐶) = (𝐵𝐷))
 
Theoremdfss1 3168 A frequently-used variant of subclass definition df-ss 2958. (Contributed by NM, 10-Jan-2015.)
(𝐴𝐵 ↔ (𝐵𝐴) = 𝐴)
 
Theoremdfss5 3169 Another definition of subclasshood. Similar to df-ss 2958, dfss 2959, and dfss1 3168. (Contributed by David Moews, 1-May-2017.)
(𝐴𝐵𝐴 = (𝐵𝐴))
 
Theoremnfin 3170 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 3171 Distribute proper substitution through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.)
(𝐴𝐵𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))
 
Theoremrabbi2dva 3172* Deduction from a wff to a restricted class abstraction. (Contributed by NM, 14-Jan-2014.)
((𝜑𝑥𝐴) → (𝑥𝐵𝜓))       (𝜑 → (𝐴𝐵) = {𝑥𝐴𝜓})
 
Theoreminidm 3173 Idempotent law for intersection of classes. Theorem 15 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.)
(𝐴𝐴) = 𝐴
 
Theoreminass 3174 Associative law for intersection of classes. Exercise 9 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.)
((𝐴𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵𝐶))
 
Theoremin12 3175 A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.)
(𝐴 ∩ (𝐵𝐶)) = (𝐵 ∩ (𝐴𝐶))
 
Theoremin32 3176 A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∩ 𝐵)
 
Theoremin13 3177 A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
(𝐴 ∩ (𝐵𝐶)) = (𝐶 ∩ (𝐵𝐴))
 
Theoremin31 3178 A rearrangement of intersection. (Contributed by NM, 27-Aug-2012.)
((𝐴𝐵) ∩ 𝐶) = ((𝐶𝐵) ∩ 𝐴)
 
Theoreminrot 3179 Rotate the intersection of 3 classes. (Contributed by NM, 27-Aug-2012.)
((𝐴𝐵) ∩ 𝐶) = ((𝐶𝐴) ∩ 𝐵)
 
Theoremin4 3180 Rearrangement of intersection of 4 classes. (Contributed by NM, 21-Apr-2001.)
((𝐴𝐵) ∩ (𝐶𝐷)) = ((𝐴𝐶) ∩ (𝐵𝐷))
 
Theoreminindi 3181 Intersection distributes over itself. (Contributed by NM, 6-May-1994.)
(𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))
 
Theoreminindir 3182 Intersection distributes over itself. (Contributed by NM, 17-Aug-2004.)
((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))
 
Theoremsseqin2 3183 A relationship between subclass and intersection. Similar to Exercise 9 of [TakeutiZaring] p. 18. (Contributed by NM, 17-May-1994.)
(𝐴𝐵 ↔ (𝐵𝐴) = 𝐴)
 
Theoreminss1 3184 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 3185 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 3186 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 3187 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 3188 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 3189 Add right intersection to subclass relation. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
 
Theoremsslin 3190 Add left intersection to subclass relation. (Contributed by NM, 19-Oct-1999.)
(𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
 
Theoremss2in 3191 Intersection of subclasses. (Contributed by NM, 5-May-2000.)
((𝐴𝐵𝐶𝐷) → (𝐴𝐶) ⊆ (𝐵𝐷))
 
Theoremssinss1 3192 Intersection preserves subclass relationship. (Contributed by NM, 14-Sep-1999.)
(𝐴𝐶 → (𝐴𝐵) ⊆ 𝐶)
 
Theoreminss 3193 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 3194 Absorption law for union. (Contributed by NM, 16-Apr-2006.)
(𝐴 ∪ (𝐴𝐵)) = 𝐴
 
Theoreminabs 3195 Absorption law for intersection. (Contributed by NM, 16-Apr-2006.)
(𝐴 ∩ (𝐴𝐵)) = 𝐴
 
Theoremnssinpss 3196 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 3197 Negation of subclass expressed in terms of proper subclass and union. (Contributed by NM, 15-Sep-2004.)
𝐴𝐵𝐵 ⊊ (𝐴𝐵))
 
Theoremssddif 3198 Double complement and subset. Similar to ddifss 3202 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 3199 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 3200 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))
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