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

Theoremsseldd 3001 Membership inference from subclass relationship. (Contributed by NM, 14-Dec-2004.)
(𝜑𝐴𝐵)    &   (𝜑𝐶𝐴)       (𝜑𝐶𝐵)

Theoremssneld 3002 If a class is not in another class, it is also not in a subclass of that class. Deduction form. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)       (𝜑 → (¬ 𝐶𝐵 → ¬ 𝐶𝐴))

Theoremssneldd 3003 If an element is not in a class, it is also not in a subclass of that class. Deduction form. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)    &   (𝜑 → ¬ 𝐶𝐵)       (𝜑 → ¬ 𝐶𝐴)

Theoremssriv 3004* Inference rule based on subclass definition. (Contributed by NM, 5-Aug-1993.)
(𝑥𝐴𝑥𝐵)       𝐴𝐵

Theoremssrd 3005 Deduction rule based on subclass definition. (Contributed by Thierry Arnoux, 8-Mar-2017.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐵    &   (𝜑 → (𝑥𝐴𝑥𝐵))       (𝜑𝐴𝐵)

Theoremssrdv 3006* Deduction rule based on subclass definition. (Contributed by NM, 15-Nov-1995.)
(𝜑 → (𝑥𝐴𝑥𝐵))       (𝜑𝐴𝐵)

Theoremsstr2 3007 Transitivity of subclasses. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
(𝐴𝐵 → (𝐵𝐶𝐴𝐶))

Theoremsstr 3008 Transitivity of subclasses. Theorem 6 of [Suppes] p. 23. (Contributed by NM, 5-Sep-2003.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)

Theoremsstri 3009 Subclass transitivity inference. (Contributed by NM, 5-May-2000.)
𝐴𝐵    &   𝐵𝐶       𝐴𝐶

Theoremsstrd 3010 Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.)
(𝜑𝐴𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)

Theoremsyl5ss 3011 Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.)
𝐴𝐵    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)

Theoremsyl6ss 3012 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝐴𝐵)    &   𝐵𝐶       (𝜑𝐴𝐶)

Theoremsylan9ss 3013 A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
(𝜑𝐴𝐵)    &   (𝜓𝐵𝐶)       ((𝜑𝜓) → 𝐴𝐶)

Theoremsylan9ssr 3014 A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)
(𝜑𝐴𝐵)    &   (𝜓𝐵𝐶)       ((𝜓𝜑) → 𝐴𝐶)

Theoremeqss 3015 The subclass relationship is antisymmetric. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 5-Aug-1993.)
(𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))

Theoremeqssi 3016 Infer equality from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 9-Sep-1993.)
𝐴𝐵    &   𝐵𝐴       𝐴 = 𝐵

Theoremeqssd 3017 Equality deduction from two subclass relationships. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 27-Jun-2004.)
(𝜑𝐴𝐵)    &   (𝜑𝐵𝐴)       (𝜑𝐴 = 𝐵)

Theoremeqrd 3018 Deduce equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 21-Mar-2017.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐵    &   (𝜑 → (𝑥𝐴𝑥𝐵))       (𝜑𝐴 = 𝐵)

Theoremssid 3019 Any class is a subclass of itself. Exercise 10 of [TakeutiZaring] p. 18. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
𝐴𝐴

Theoremssv 3020 Any class is a subclass of the universal class. (Contributed by NM, 31-Oct-1995.)
𝐴 ⊆ V

Theoremsseq1 3021 Equality theorem for subclasses. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
(𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))

Theoremsseq2 3022 Equality theorem for the subclass relationship. (Contributed by NM, 25-Jun-1998.)
(𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))

Theoremsseq12 3023 Equality theorem for the subclass relationship. (Contributed by NM, 31-May-1999.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶𝐵𝐷))

Theoremsseq1i 3024 An equality inference for the subclass relationship. (Contributed by NM, 18-Aug-1993.)
𝐴 = 𝐵       (𝐴𝐶𝐵𝐶)

Theoremsseq2i 3025 An equality inference for the subclass relationship. (Contributed by NM, 30-Aug-1993.)
𝐴 = 𝐵       (𝐶𝐴𝐶𝐵)

Theoremsseq12i 3026 An equality inference for the subclass relationship. (Contributed by NM, 31-May-1999.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐶𝐵𝐷)

Theoremsseq1d 3027 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶𝐵𝐶))

Theoremsseq2d 3028 An equality deduction for the subclass relationship. (Contributed by NM, 14-Aug-1994.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴𝐶𝐵))

Theoremsseq12d 3029 An equality deduction for the subclass relationship. (Contributed by NM, 31-May-1999.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶𝐵𝐷))

Theoremeqsstri 3030 Substitution of equality into a subclass relationship. (Contributed by NM, 16-Jul-1995.)
𝐴 = 𝐵    &   𝐵𝐶       𝐴𝐶

Theoremeqsstr3i 3031 Substitution of equality into a subclass relationship. (Contributed by NM, 19-Oct-1999.)
𝐵 = 𝐴    &   𝐵𝐶       𝐴𝐶

Theoremsseqtri 3032 Substitution of equality into a subclass relationship. (Contributed by NM, 28-Jul-1995.)
𝐴𝐵    &   𝐵 = 𝐶       𝐴𝐶

Theoremsseqtr4i 3033 Substitution of equality into a subclass relationship. (Contributed by NM, 4-Apr-1995.)
𝐴𝐵    &   𝐶 = 𝐵       𝐴𝐶

Theoremeqsstrd 3034 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)

Theoremeqsstr3d 3035 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
(𝜑𝐵 = 𝐴)    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)

Theoremsseqtrd 3036 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
(𝜑𝐴𝐵)    &   (𝜑𝐵 = 𝐶)       (𝜑𝐴𝐶)

Theoremsseqtr4d 3037 Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
(𝜑𝐴𝐵)    &   (𝜑𝐶 = 𝐵)       (𝜑𝐴𝐶)

Theorem3sstr3i 3038 Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
𝐴𝐵    &   𝐴 = 𝐶    &   𝐵 = 𝐷       𝐶𝐷

Theorem3sstr4i 3039 Substitution of equality in both sides of a subclass relationship. (Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
𝐴𝐵    &   𝐶 = 𝐴    &   𝐷 = 𝐵       𝐶𝐷

Theorem3sstr3g 3040 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)
(𝜑𝐴𝐵)    &   𝐴 = 𝐶    &   𝐵 = 𝐷       (𝜑𝐶𝐷)

Theorem3sstr4g 3041 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
(𝜑𝐴𝐵)    &   𝐶 = 𝐴    &   𝐷 = 𝐵       (𝜑𝐶𝐷)

Theorem3sstr3d 3042 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)
(𝜑𝐴𝐵)    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑𝐶𝐷)

Theorem3sstr4d 3043 Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 30-Nov-1995.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
(𝜑𝐴𝐵)    &   (𝜑𝐶 = 𝐴)    &   (𝜑𝐷 = 𝐵)       (𝜑𝐶𝐷)

Theoremsyl5eqss 3044 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
𝐴 = 𝐵    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)

Theoremsyl5eqssr 3045 B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
𝐵 = 𝐴    &   (𝜑𝐵𝐶)       (𝜑𝐴𝐶)

Theoremsyl6sseq 3046 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
(𝜑𝐴𝐵)    &   𝐵 = 𝐶       (𝜑𝐴𝐶)

Theoremsyl6sseqr 3047 A chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)
(𝜑𝐴𝐵)    &   𝐶 = 𝐵       (𝜑𝐴𝐶)

Theoremsyl5sseq 3048 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
𝐵𝐴    &   (𝜑𝐴 = 𝐶)       (𝜑𝐵𝐶)

Theoremsyl5sseqr 3049 Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
𝐵𝐴    &   (𝜑𝐶 = 𝐴)       (𝜑𝐵𝐶)

Theoremsyl6eqss 3050 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝐴 = 𝐵)    &   𝐵𝐶       (𝜑𝐴𝐶)

Theoremsyl6eqssr 3051 A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)
(𝜑𝐵 = 𝐴)    &   𝐵𝐶       (𝜑𝐴𝐶)

Theoremeqimss 3052 Equality implies the subclass relation. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
(𝐴 = 𝐵𝐴𝐵)

Theoremeqimss2 3053 Equality implies the subclass relation. (Contributed by NM, 23-Nov-2003.)
(𝐵 = 𝐴𝐴𝐵)

Theoremeqimssi 3054 Infer subclass relationship from equality. (Contributed by NM, 6-Jan-2007.)
𝐴 = 𝐵       𝐴𝐵

Theoremeqimss2i 3055 Infer subclass relationship from equality. (Contributed by NM, 7-Jan-2007.)
𝐴 = 𝐵       𝐵𝐴

Theoremnssne1 3056 Two classes are different if they don't include the same class. (Contributed by NM, 23-Apr-2015.)
((𝐴𝐵 ∧ ¬ 𝐴𝐶) → 𝐵𝐶)

Theoremnssne2 3057 Two classes are different if they are not subclasses of the same class. (Contributed by NM, 23-Apr-2015.)
((𝐴𝐶 ∧ ¬ 𝐵𝐶) → 𝐴𝐵)

Theoremnssr 3058* 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 3059* Quantification restricted to a subclass. (Contributed by NM, 11-Mar-2006.)
(𝐴𝐵 → (∀𝑥𝐵 𝜑 → ∀𝑥𝐴 𝜑))

Theoremssrexv 3060* Existential quantification restricted to a subclass. (Contributed by NM, 11-Jan-2007.)
(𝐴𝐵 → (∃𝑥𝐴 𝜑 → ∃𝑥𝐵 𝜑))

Theoremralss 3061* Restricted universal quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝐴𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 (𝑥𝐴𝜑)))

Theoremrexss 3062* Restricted existential quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015.)
(𝐴𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 (𝑥𝐴𝜑)))

Theoremss2ab 3063 Class abstractions in a subclass relationship. (Contributed by NM, 3-Jul-1994.)
({𝑥𝜑} ⊆ {𝑥𝜓} ↔ ∀𝑥(𝜑𝜓))

Theoremabss 3064* Class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
({𝑥𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))

Theoremssab 3065* Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.)
(𝐴 ⊆ {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))

Theoremssabral 3066* 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 3067 Inference of abstraction subclass from implication. (Contributed by NM, 31-Mar-1995.)
(𝜑𝜓)       {𝑥𝜑} ⊆ {𝑥𝜓}

Theoremss2abdv 3068* Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011.)
(𝜑 → (𝜓𝜒))       (𝜑 → {𝑥𝜓} ⊆ {𝑥𝜒})

Theoremabssdv 3069* Deduction of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
(𝜑 → (𝜓𝑥𝐴))       (𝜑 → {𝑥𝜓} ⊆ 𝐴)

Theoremabssi 3070* Inference of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
(𝜑𝑥𝐴)       {𝑥𝜑} ⊆ 𝐴

Theoremss2rab 3071 Restricted abstraction classes in a subclass relationship. (Contributed by NM, 30-May-1999.)
({𝑥𝐴𝜑} ⊆ {𝑥𝐴𝜓} ↔ ∀𝑥𝐴 (𝜑𝜓))

Theoremrabss 3072* Restricted class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
({𝑥𝐴𝜑} ⊆ 𝐵 ↔ ∀𝑥𝐴 (𝜑𝑥𝐵))

Theoremssrab 3073* Subclass of a restricted class abstraction. (Contributed by NM, 16-Aug-2006.)
(𝐵 ⊆ {𝑥𝐴𝜑} ↔ (𝐵𝐴 ∧ ∀𝑥𝐵 𝜑))

Theoremssrabdv 3074* Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 31-Aug-2006.)
(𝜑𝐵𝐴)    &   ((𝜑𝑥𝐵) → 𝜓)       (𝜑𝐵 ⊆ {𝑥𝐴𝜓})

Theoremrabssdv 3075* Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 2-Feb-2015.)
((𝜑𝑥𝐴𝜓) → 𝑥𝐵)       (𝜑 → {𝑥𝐴𝜓} ⊆ 𝐵)

Theoremss2rabdv 3076* Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐴𝜒})

Theoremss2rabi 3077 Inference of restricted abstraction subclass from implication. (Contributed by NM, 14-Oct-1999.)
(𝑥𝐴 → (𝜑𝜓))       {𝑥𝐴𝜑} ⊆ {𝑥𝐴𝜓}

Theoremrabss2 3078* Subclass law for restricted abstraction. (Contributed by NM, 18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴𝐵 → {𝑥𝐴𝜑} ⊆ {𝑥𝐵𝜑})

Theoremssab2 3079* Subclass relation for the restriction of a class abstraction. (Contributed by NM, 31-Mar-1995.)
{𝑥 ∣ (𝑥𝐴𝜑)} ⊆ 𝐴

Theoremssrab2 3080* Subclass relation for a restricted class. (Contributed by NM, 19-Mar-1997.)
{𝑥𝐴𝜑} ⊆ 𝐴

Theoremssrabeq 3081* 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 3082 A restricted class is a subclass of the corresponding unrestricted class. (Contributed by Mario Carneiro, 23-Dec-2016.)
{𝑥𝐴𝜑} ⊆ {𝑥𝜑}

Theoremuniiunlem 3083* 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))

2.1.13  The difference, union, and intersection of two classes

2.1.13.1  The difference of two classes

Theoremdifeq1 3084 Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))

Theoremdifeq2 3085 Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))

Theoremdifeq12 3086 Equality theorem for class difference. (Contributed by FL, 31-Aug-2009.)
((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))

Theoremdifeq1i 3087 Inference adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)
𝐴 = 𝐵       (𝐴𝐶) = (𝐵𝐶)

Theoremdifeq2i 3088 Inference adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
𝐴 = 𝐵       (𝐶𝐴) = (𝐶𝐵)

Theoremdifeq12i 3089 Equality inference for class difference. (Contributed by NM, 29-Aug-2004.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (𝐴𝐶) = (𝐵𝐷)

Theoremdifeq1d 3090 Deduction adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐴𝐶) = (𝐵𝐶))

Theoremdifeq2d 3091 Deduction adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐶𝐴) = (𝐶𝐵))

Theoremdifeq12d 3092 Equality deduction for class difference. (Contributed by FL, 29-May-2014.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶) = (𝐵𝐷))

Theoremdifeqri 3093* 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 3094 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 3095 Implication of membership in a class difference. (Contributed by NM, 29-Apr-1994.)
(𝐴 ∈ (𝐵𝐶) → 𝐴𝐵)

Theoremeldifn 3096 Implication of membership in a class difference. (Contributed by NM, 3-May-1994.)
(𝐴 ∈ (𝐵𝐶) → ¬ 𝐴𝐶)

Theoremelndif 3097 A set does not belong to a class excluding it. (Contributed by NM, 27-Jun-1994.)
(𝐴𝐵 → ¬ 𝐴 ∈ (𝐶𝐵))

Theoremdifdif 3098 Double class difference. Exercise 11 of [TakeutiZaring] p. 22. (Contributed by NM, 17-May-1998.)
(𝐴 ∖ (𝐵𝐴)) = 𝐴

Theoremdifss 3099 Subclass relationship for class difference. Exercise 14 of [TakeutiZaring] p. 22. (Contributed by NM, 29-Apr-1994.)
(𝐴𝐵) ⊆ 𝐴

Theoremdifssd 3100 A difference of two classes is contained in the minuend. Deduction form of difss 3099. (Contributed by David Moews, 1-May-2017.)
(𝜑 → (𝐴𝐵) ⊆ 𝐴)

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