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Theorem List for Intuitionistic Logic Explorer - 3001-3100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnssne1 3001 Two classes are different if they don't include the same class. (Contributed by NM, 23-Apr-2015.)
((𝐴𝐵 ∧ ¬ 𝐴𝐶) → 𝐵𝐶)

Theoremnssne2 3002 Two classes are different if they are not subclasses of the same class. (Contributed by NM, 23-Apr-2015.)
((𝐴𝐶 ∧ ¬ 𝐵𝐶) → 𝐴𝐵)

Theoremnssr 3003* Negation of subclass relationship. One direction of Exercise 13 of [TakeutiZaring] p. 18. (Contributed by Jim Kingdon, 15-Jul-2018.)
(∃𝑥(𝑥𝐴 ∧ ¬ 𝑥𝐵) → ¬ 𝐴𝐵)

Theoremssralv 3004* Quantification restricted to a subclass. (Contributed by NM, 11-Mar-2006.)
(𝐴𝐵 → (∀𝑥𝐵 𝜑 → ∀𝑥𝐴 𝜑))

Theoremssrexv 3005* Existential quantification restricted to a subclass. (Contributed by NM, 11-Jan-2007.)
(𝐴𝐵 → (∃𝑥𝐴 𝜑 → ∃𝑥𝐵 𝜑))

Theoremralss 3006* Restricted universal quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝐴𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 (𝑥𝐴𝜑)))

Theoremrexss 3007* Restricted existential quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝐴𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 (𝑥𝐴𝜑)))

Theoremss2ab 3008 Class abstractions in a subclass relationship. (Contributed by NM, 3-Jul-1994.)
({𝑥𝜑} ⊆ {𝑥𝜓} ↔ ∀𝑥(𝜑𝜓))

Theoremabss 3009* Class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
({𝑥𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))

Theoremssab 3010* Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.)
(𝐴 ⊆ {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))

Theoremssabral 3011* The relation for a subclass of a class abstraction is equivalent to restricted quantification. (Contributed by NM, 6-Sep-2006.)
(𝐴 ⊆ {𝑥𝜑} ↔ ∀𝑥𝐴 𝜑)

Theoremss2abi 3012 Inference of abstraction subclass from implication. (Contributed by NM, 31-Mar-1995.)
(𝜑𝜓)       {𝑥𝜑} ⊆ {𝑥𝜓}

Theoremss2abdv 3013* Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011.)
(𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝜓} ⊆ {𝑥𝜒})

Theoremabssdv 3014* Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
(𝜑 → (𝜓𝑥𝐴))       (𝜑 → {𝑥𝜓} ⊆ 𝐴)

Theoremabssi 3015* Inference of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
(𝜑𝑥𝐴)       {𝑥𝜑} ⊆ 𝐴

Theoremss2rab 3016 Restricted abstraction classes in a subclass relationship. (Contributed by NM, 30-May-1999.)
({𝑥𝐴𝜑} ⊆ {𝑥𝐴𝜓} ↔ ∀𝑥𝐴 (𝜑𝜓))

Theoremrabss 3017* Restricted class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
({𝑥𝐴𝜑} ⊆ 𝐵 ↔ ∀𝑥𝐴 (𝜑𝑥𝐵))

Theoremssrab 3018* Subclass of a restricted class abstraction. (Contributed by NM, 16-Aug-2006.)
(𝐵 ⊆ {𝑥𝐴𝜑} ↔ (𝐵𝐴 ∧ ∀𝑥𝐵 𝜑))

Theoremssrabdv 3019* Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 31-Aug-2006.)
(𝜑𝐵𝐴)    &   ((𝜑𝑥𝐵) → 𝜓)       (𝜑𝐵 ⊆ {𝑥𝐴𝜓})

Theoremrabssdv 3020* Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 2-Feb-2015.)
((𝜑𝑥𝐴𝜓) → 𝑥𝐵)       (𝜑 → {𝑥𝐴𝜓} ⊆ 𝐵)

Theoremss2rabdv 3021* Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐴𝜒})

Theoremss2rabi 3022 Inference of restricted abstraction subclass from implication. (Contributed by NM, 14-Oct-1999.)
(𝑥𝐴 → (𝜑𝜓))       {𝑥𝐴𝜑} ⊆ {𝑥𝐴𝜓}

Theoremrabss2 3023* Subclass law for restricted abstraction. (Contributed by NM, 18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴𝐵 → {𝑥𝐴𝜑} ⊆ {𝑥𝐵𝜑})

Theoremssab2 3024* Subclass relation for the restriction of a class abstraction. (Contributed by NM, 31-Mar-1995.)
{𝑥 ∣ (𝑥𝐴𝜑)} ⊆ 𝐴

Theoremssrab2 3025* Subclass relation for a restricted class. (Contributed by NM, 19-Mar-1997.)
{𝑥𝐴𝜑} ⊆ 𝐴

Theoremssrabeq 3026* If the restricting class of a restricted class abstraction is a subset of this restricted class abstraction, it is equal to this restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)
(𝑉 ⊆ {𝑥𝑉𝜑} ↔ 𝑉 = {𝑥𝑉𝜑})

Theoremrabssab 3027 A restricted class is a subclass of the corresponding unrestricted class. (Contributed by Mario Carneiro, 23-Dec-2016.)
{𝑥𝐴𝜑} ⊆ {𝑥𝜑}

Theoremuniiunlem 3028* A subset relationship useful for converting union to indexed union using dfiun2 or dfiun2g and intersection to indexed intersection using dfiin2 . (Contributed by NM, 5-Oct-2006.) (Proof shortened by Mario Carneiro, 26-Sep-2015.)
(∀𝑥𝐴 𝐵𝐷 → (∀𝑥𝐴 𝐵𝐶 ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ⊆ 𝐶))

Theoremdfpss2 3029 Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.)
(𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴 = 𝐵))

Theoremdfpss3 3030 Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐵𝐴))

Theorempsseq1 3031 Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)
(𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))

Theorempsseq2 3032 Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996.)
(𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))

Theorempsseq1i 3033 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
𝐴 = 𝐵       (𝐴𝐶𝐵𝐶)

Theorempsseq2i 3034 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
𝐴 = 𝐵       (𝐶𝐴𝐶𝐵)

Theorempsseq12i 3035 An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐶𝐵𝐷)

Theorempsseq1d 3036 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶𝐵𝐶))

Theorempsseq2d 3037 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴𝐶𝐵))

Theorempsseq12d 3038 An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶𝐵𝐷))

Theorempssss 3039 A proper subclass is a subclass. Theorem 10 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)
(𝐴𝐵𝐴𝐵)

Theorempssne 3040 Two classes in a proper subclass relationship are not equal. (Contributed by NM, 16-Feb-2015.)
(𝐴𝐵𝐴𝐵)

Theorempssssd 3041 Deduce subclass from proper subclass. (Contributed by NM, 29-Feb-1996.)
(𝜑𝐴𝐵)       (𝜑𝐴𝐵)

Theorempssned 3042 Proper subclasses are unequal. Deduction form of pssne 3040. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)       (𝜑𝐴𝐵)

Theoremsspssr 3043 Subclass in terms of proper subclass. (Contributed by Jim Kingdon, 16-Jul-2018.)
((𝐴𝐵𝐴 = 𝐵) → 𝐴𝐵)

Theorempssirr 3044 Proper subclass is irreflexive. Theorem 7 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)
¬ 𝐴𝐴

Theorempssn2lp 3045 Proper subclass has no 2-cycle loops. Compare Theorem 8 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
¬ (𝐴𝐵𝐵𝐴)

Theoremsspsstrir 3046 Two ways of stating trichotomy with respect to inclusion. (Contributed by Jim Kingdon, 16-Jul-2018.)
((𝐴𝐵𝐴 = 𝐵𝐵𝐴) → (𝐴𝐵𝐵𝐴))

Theoremssnpss 3047 Partial trichotomy law for subclasses. (Contributed by NM, 16-May-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴𝐵 → ¬ 𝐵𝐴)

Theoremsspssn 3048 Like pssn2lp 3045 but for subset and proper subset. (Contributed by Jim Kingdon, 17-Jul-2018.)
¬ (𝐴𝐵𝐵𝐴)

Theorempsstr 3049 Transitive law for proper subclass. Theorem 9 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)

Theoremsspsstr 3050 Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)

Theorempsssstr 3051 Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)

Theorempsstrd 3052 Proper subclass inclusion is transitive. Deduction form of psstr 3049. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)

Theoremsspsstrd 3053 Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 3050. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)

Theorempsssstrd 3054 Transitivity involving subclass and proper subclass inclusion. Deduction form of psssstr 3051. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)

2.1.13  The difference, union, and intersection of two classes

2.1.13.1  The difference of two classes

Theoremdifeq1 3055 Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))

Theoremdifeq2 3056 Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))

Theoremdifeq12 3057 Equality theorem for class difference. (Contributed by FL, 31-Aug-2009.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))

Theoremdifeq1i 3058 Inference adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)
𝐴 = 𝐵       (𝐴𝐶) = (𝐵𝐶)

Theoremdifeq2i 3059 Inference adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
𝐴 = 𝐵       (𝐶𝐴) = (𝐶𝐵)

Theoremdifeq12i 3060 Equality inference for class difference. (Contributed by NM, 29-Aug-2004.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐶) = (𝐵𝐷)

Theoremdifeq1d 3061 Deduction adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶) = (𝐵𝐶))

Theoremdifeq2d 3062 Deduction adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴) = (𝐶𝐵))

Theoremdifeq12d 3063 Equality deduction for class difference. (Contributed by FL, 29-May-2014.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶) = (𝐵𝐷))

Theoremdifeqri 3064* Inference from membership to difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝑥𝐴 ∧ ¬ 𝑥𝐵) ↔ 𝑥𝐶)       (𝐴𝐵) = 𝐶

Theoremnfdif 3065 Bound-variable hypothesis builder for class difference. (Contributed by NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
𝑥𝐴    &   𝑥𝐵       𝑥(𝐴𝐵)

Theoremeldifi 3066 Implication of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
(𝐴 ∈ (𝐵𝐶) → 𝐴𝐵)

Theoremeldifn 3067 Implication of membership in a class difference. (Contributed by NM, 3-May-1994.)
(𝐴 ∈ (𝐵𝐶) → ¬ 𝐴𝐶)

Theoremelndif 3068 A set does not belong to a class excluding it. (Contributed by NM, 27-Jun-1994.)
(𝐴𝐵 → ¬ 𝐴 ∈ (𝐶𝐵))

Theoremdifdif 3069 Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.)
(𝐴 ∖ (𝐵𝐴)) = 𝐴

Theoremdifss 3070 Subclass relationship for class difference. Exercise 14 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)
(𝐴𝐵) ⊆ 𝐴

Theoremdifssd 3071 A difference of two classes is contained in the minuend. Deduction form of difss 3070. (Contributed by David Moews, 1-May-2017.)
(𝜑 → (𝐴𝐵) ⊆ 𝐴)

Theoremdifss2 3072 If a class is contained in a difference, it is contained in the minuend. (Contributed by David Moews, 1-May-2017.)
(𝐴 ⊆ (𝐵𝐶) → 𝐴𝐵)

Theoremdifss2d 3073 If a class is contained in a difference, it is contained in the minuend. Deduction form of difss2 3072. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ⊆ (𝐵𝐶))       (𝜑𝐴𝐵)

Theoremssdifss 3074 Preservation of a subclass relationship by class difference. (Contributed by NM, 15-Feb-2007.)
(𝐴𝐵 → (𝐴𝐶) ⊆ 𝐵)

Theoremddifnel 3075* 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 3175. (Contributed by Jim Kingdon, 21-Jul-2018.)
𝑥 ∈ (V ∖ 𝐴) → 𝑥𝐴)       (V ∖ (V ∖ 𝐴)) = 𝐴

Theoremssconb 3076 Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)
((𝐴𝐶𝐵𝐶) → (𝐴 ⊆ (𝐶𝐵) ↔ 𝐵 ⊆ (𝐶𝐴)))

Theoremsscon 3077 Contraposition law for subsets. Exercise 15 of [TakeutiZaring] p. 22. (Contributed by NM, 22-Mar-1998.)
(𝐴𝐵 → (𝐶𝐵) ⊆ (𝐶𝐴))

Theoremssdif 3078 Difference law for subsets. (Contributed by NM, 28-May-1998.)
(𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))

Theoremssdifd 3079 If 𝐴 is contained in 𝐵, then (𝐴𝐶) is contained in (𝐵𝐶). Deduction form of ssdif 3078. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)       (𝜑 → (𝐴𝐶) ⊆ (𝐵𝐶))

Theoremsscond 3080 If 𝐴 is contained in 𝐵, then (𝐶𝐵) is contained in (𝐶𝐴). Deduction form of sscon 3077. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)       (𝜑 → (𝐶𝐵) ⊆ (𝐶𝐴))

Theoremssdifssd 3081 If 𝐴 is contained in 𝐵, then (𝐴𝐶) is also contained in 𝐵. Deduction form of ssdifss 3074. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)       (𝜑 → (𝐴𝐶) ⊆ 𝐵)

Theoremssdif2d 3082 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 3083 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 3084 Expansion of membership in class union. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 7-Aug-1994.)
(𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))

Theoremuneqri 3085* Inference from membership to union. (Contributed by NM, 5-Aug-1993.)
((𝑥𝐴𝑥𝐵) ↔ 𝑥𝐶)       (𝐴𝐵) = 𝐶

Theoremunidm 3086 Idempotent law for union of classes. Theorem 23 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
(𝐴𝐴) = 𝐴

Theoremuncom 3087 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 3088 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 3089 Inference form of equncom 3088. (Contributed by Alan Sare, 18-Feb-2012.)
𝐴 = (𝐵𝐶)       𝐴 = (𝐶𝐵)

Theoremuneq1 3090 Equality theorem for union of two classes. (Contributed by NM, 5-Aug-1993.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))

Theoremuneq2 3091 Equality theorem for the union of two classes. (Contributed by NM, 5-Aug-1993.)
(𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))

Theoremuneq12 3092 Equality theorem for union of two classes. (Contributed by NM, 29-Mar-1998.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))

Theoremuneq1i 3093 Inference adding union to the right in a class equality. (Contributed by NM, 30-Aug-1993.)
𝐴 = 𝐵       (𝐴𝐶) = (𝐵𝐶)

Theoremuneq2i 3094 Inference adding union to the left in a class equality. (Contributed by NM, 30-Aug-1993.)
𝐴 = 𝐵       (𝐶𝐴) = (𝐶𝐵)

Theoremuneq12i 3095 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 3096 Deduction adding union to the right in a class equality. (Contributed by NM, 29-Mar-1998.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶) = (𝐵𝐶))

Theoremuneq2d 3097 Deduction adding union to the left in a class equality. (Contributed by NM, 29-Mar-1998.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴) = (𝐶𝐵))

Theoremuneq12d 3098 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 3099 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 3100 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))

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