Theorem List for Intuitionistic Logic Explorer - 3001-3100 *Has distinct variable
group(s)
Type | Label | Description |
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Theorem | ssneld 3001 |
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.)
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Theorem | ssneldd 3002 |
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.)
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Theorem | ssriv 3003* |
Inference rule based on subclass definition. (Contributed by NM,
5-Aug-1993.)
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Theorem | ssrd 3004 |
Deduction rule based on subclass definition. (Contributed by Thierry
Arnoux, 8-Mar-2017.)
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Theorem | ssrdv 3005* |
Deduction rule based on subclass definition. (Contributed by NM,
15-Nov-1995.)
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Theorem | sstr2 3006 |
Transitivity of subclasses. Exercise 5 of [TakeutiZaring] p. 17.
(Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon,
14-Jun-2011.)
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Theorem | sstr 3007 |
Transitivity of subclasses. Theorem 6 of [Suppes] p. 23. (Contributed by
NM, 5-Sep-2003.)
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Theorem | sstri 3008 |
Subclass transitivity inference. (Contributed by NM, 5-May-2000.)
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Theorem | sstrd 3009 |
Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.)
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Theorem | syl5ss 3010 |
Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.)
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Theorem | syl6ss 3011 |
Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim,
3-Jun-2011.)
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Theorem | sylan9ss 3012 |
A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)
(Proof shortened by Andrew Salmon, 14-Jun-2011.)
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Theorem | sylan9ssr 3013 |
A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.)
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Theorem | eqss 3014 |
The subclass relationship is antisymmetric. Compare Theorem 4 of
[Suppes] p. 22. (Contributed by NM,
5-Aug-1993.)
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Theorem | eqssi 3015 |
Infer equality from two subclass relationships. Compare Theorem 4 of
[Suppes] p. 22. (Contributed by NM,
9-Sep-1993.)
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Theorem | eqssd 3016 |
Equality deduction from two subclass relationships. Compare Theorem 4
of [Suppes] p. 22. (Contributed by NM,
27-Jun-2004.)
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Theorem | eqrd 3017 |
Deduce equality of classes from equivalence of membership. (Contributed
by Thierry Arnoux, 21-Mar-2017.)
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Theorem | ssid 3018 |
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.)
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Theorem | ssv 3019 |
Any class is a subclass of the universal class. (Contributed by NM,
31-Oct-1995.)
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Theorem | sseq1 3020 |
Equality theorem for subclasses. (Contributed by NM, 5-Aug-1993.) (Proof
shortened by Andrew Salmon, 21-Jun-2011.)
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Theorem | sseq2 3021 |
Equality theorem for the subclass relationship. (Contributed by NM,
25-Jun-1998.)
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Theorem | sseq12 3022 |
Equality theorem for the subclass relationship. (Contributed by NM,
31-May-1999.)
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Theorem | sseq1i 3023 |
An equality inference for the subclass relationship. (Contributed by
NM, 18-Aug-1993.)
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Theorem | sseq2i 3024 |
An equality inference for the subclass relationship. (Contributed by
NM, 30-Aug-1993.)
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Theorem | sseq12i 3025 |
An equality inference for the subclass relationship. (Contributed by
NM, 31-May-1999.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
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Theorem | sseq1d 3026 |
An equality deduction for the subclass relationship. (Contributed by
NM, 14-Aug-1994.)
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Theorem | sseq2d 3027 |
An equality deduction for the subclass relationship. (Contributed by
NM, 14-Aug-1994.)
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Theorem | sseq12d 3028 |
An equality deduction for the subclass relationship. (Contributed by
NM, 31-May-1999.)
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Theorem | eqsstri 3029 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 16-Jul-1995.)
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Theorem | eqsstr3i 3030 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 19-Oct-1999.)
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Theorem | sseqtri 3031 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 28-Jul-1995.)
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Theorem | sseqtr4i 3032 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 4-Apr-1995.)
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Theorem | eqsstrd 3033 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 25-Apr-2004.)
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Theorem | eqsstr3d 3034 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 25-Apr-2004.)
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Theorem | sseqtrd 3035 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 25-Apr-2004.)
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Theorem | sseqtr4d 3036 |
Substitution of equality into a subclass relationship. (Contributed by
NM, 25-Apr-2004.)
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Theorem | 3sstr3i 3037 |
Substitution of equality in both sides of a subclass relationship.
(Contributed by NM, 13-Jan-1996.) (Proof shortened by Eric Schmidt,
26-Jan-2007.)
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Theorem | 3sstr4i 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.)
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Theorem | 3sstr3g 3039 |
Substitution of equality into both sides of a subclass relationship.
(Contributed by NM, 1-Oct-2000.)
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Theorem | 3sstr4g 3040 |
Substitution of equality into both sides of a subclass relationship.
(Contributed by NM, 16-Aug-1994.) (Proof shortened by Eric Schmidt,
26-Jan-2007.)
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Theorem | 3sstr3d 3041 |
Substitution of equality into both sides of a subclass relationship.
(Contributed by NM, 1-Oct-2000.)
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Theorem | 3sstr4d 3042 |
Substitution of equality into both sides of a subclass relationship.
(Contributed by NM, 30-Nov-1995.) (Proof shortened by Eric Schmidt,
26-Jan-2007.)
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Theorem | syl5eqss 3043 |
B chained subclass and equality deduction. (Contributed by NM,
25-Apr-2004.)
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Theorem | syl5eqssr 3044 |
B chained subclass and equality deduction. (Contributed by NM,
25-Apr-2004.)
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Theorem | syl6sseq 3045 |
A chained subclass and equality deduction. (Contributed by NM,
25-Apr-2004.)
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Theorem | syl6sseqr 3046 |
A chained subclass and equality deduction. (Contributed by NM,
25-Apr-2004.)
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Theorem | syl5sseq 3047 |
Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim,
3-Jun-2011.)
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Theorem | syl5sseqr 3048 |
Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim,
3-Jun-2011.)
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Theorem | syl6eqss 3049 |
A chained subclass and equality deduction. (Contributed by Mario
Carneiro, 2-Jan-2017.)
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Theorem | syl6eqssr 3050 |
A chained subclass and equality deduction. (Contributed by Mario
Carneiro, 2-Jan-2017.)
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Theorem | eqimss 3051 |
Equality implies the subclass relation. (Contributed by NM, 5-Aug-1993.)
(Proof shortened by Andrew Salmon, 21-Jun-2011.)
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Theorem | eqimss2 3052 |
Equality implies the subclass relation. (Contributed by NM,
23-Nov-2003.)
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Theorem | eqimssi 3053 |
Infer subclass relationship from equality. (Contributed by NM,
6-Jan-2007.)
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Theorem | eqimss2i 3054 |
Infer subclass relationship from equality. (Contributed by NM,
7-Jan-2007.)
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Theorem | nssne1 3055 |
Two classes are different if they don't include the same class.
(Contributed by NM, 23-Apr-2015.)
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Theorem | nssne2 3056 |
Two classes are different if they are not subclasses of the same class.
(Contributed by NM, 23-Apr-2015.)
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Theorem | nssr 3057* |
Negation of subclass relationship. One direction of Exercise 13 of
[TakeutiZaring] p. 18.
(Contributed by Jim Kingdon, 15-Jul-2018.)
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Theorem | ssralv 3058* |
Quantification restricted to a subclass. (Contributed by NM,
11-Mar-2006.)
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Theorem | ssrexv 3059* |
Existential quantification restricted to a subclass. (Contributed by
NM, 11-Jan-2007.)
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Theorem | ralss 3060* |
Restricted universal quantification on a subset in terms of superset.
(Contributed by Stefan O'Rear, 3-Apr-2015.)
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Theorem | rexss 3061* |
Restricted existential quantification on a subset in terms of superset.
(Contributed by Stefan O'Rear, 3-Apr-2015.)
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Theorem | ss2ab 3062 |
Class abstractions in a subclass relationship. (Contributed by NM,
3-Jul-1994.)
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Theorem | abss 3063* |
Class abstraction in a subclass relationship. (Contributed by NM,
16-Aug-2006.)
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Theorem | ssab 3064* |
Subclass of a class abstraction. (Contributed by NM, 16-Aug-2006.)
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Theorem | ssabral 3065* |
The relation for a subclass of a class abstraction is equivalent to
restricted quantification. (Contributed by NM, 6-Sep-2006.)
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Theorem | ss2abi 3066 |
Inference of abstraction subclass from implication. (Contributed by NM,
31-Mar-1995.)
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Theorem | ss2abdv 3067* |
Deduction of abstraction subclass from implication. (Contributed by NM,
29-Jul-2011.)
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Theorem | abssdv 3068* |
Deduction of abstraction subclass from implication. (Contributed by NM,
20-Jan-2006.)
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Theorem | abssi 3069* |
Inference of abstraction subclass from implication. (Contributed by NM,
20-Jan-2006.)
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Theorem | ss2rab 3070 |
Restricted abstraction classes in a subclass relationship. (Contributed
by NM, 30-May-1999.)
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Theorem | rabss 3071* |
Restricted class abstraction in a subclass relationship. (Contributed
by NM, 16-Aug-2006.)
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Theorem | ssrab 3072* |
Subclass of a restricted class abstraction. (Contributed by NM,
16-Aug-2006.)
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Theorem | ssrabdv 3073* |
Subclass of a restricted class abstraction (deduction rule).
(Contributed by NM, 31-Aug-2006.)
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Theorem | rabssdv 3074* |
Subclass of a restricted class abstraction (deduction rule).
(Contributed by NM, 2-Feb-2015.)
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Theorem | ss2rabdv 3075* |
Deduction of restricted abstraction subclass from implication.
(Contributed by NM, 30-May-2006.)
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Theorem | ss2rabi 3076 |
Inference of restricted abstraction subclass from implication.
(Contributed by NM, 14-Oct-1999.)
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Theorem | rabss2 3077* |
Subclass law for restricted abstraction. (Contributed by NM,
18-Dec-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
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Theorem | ssab2 3078* |
Subclass relation for the restriction of a class abstraction.
(Contributed by NM, 31-Mar-1995.)
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Theorem | ssrab2 3079* |
Subclass relation for a restricted class. (Contributed by NM,
19-Mar-1997.)
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Theorem | ssrabeq 3080* |
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.)
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Theorem | rabssab 3081 |
A restricted class is a subclass of the corresponding unrestricted class.
(Contributed by Mario Carneiro, 23-Dec-2016.)
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Theorem | uniiunlem 3082* |
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.)
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2.1.13 The difference, union, and intersection
of two classes
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2.1.13.1 The difference of two
classes
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Theorem | difeq1 3083 |
Equality theorem for class difference. (Contributed by NM,
10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
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Theorem | difeq2 3084 |
Equality theorem for class difference. (Contributed by NM,
10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
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Theorem | difeq12 3085 |
Equality theorem for class difference. (Contributed by FL,
31-Aug-2009.)
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Theorem | difeq1i 3086 |
Inference adding difference to the right in a class equality.
(Contributed by NM, 15-Nov-2002.)
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Theorem | difeq2i 3087 |
Inference adding difference to the left in a class equality.
(Contributed by NM, 15-Nov-2002.)
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Theorem | difeq12i 3088 |
Equality inference for class difference. (Contributed by NM,
29-Aug-2004.)
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Theorem | difeq1d 3089 |
Deduction adding difference to the right in a class equality.
(Contributed by NM, 15-Nov-2002.)
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Theorem | difeq2d 3090 |
Deduction adding difference to the left in a class equality.
(Contributed by NM, 15-Nov-2002.)
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Theorem | difeq12d 3091 |
Equality deduction for class difference. (Contributed by FL,
29-May-2014.)
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Theorem | difeqri 3092* |
Inference from membership to difference. (Contributed by NM,
17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
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Theorem | nfdif 3093 |
Bound-variable hypothesis builder for class difference. (Contributed by
NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
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Theorem | eldifi 3094 |
Implication of membership in a class difference. (Contributed by NM,
29-Apr-1994.)
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Theorem | eldifn 3095 |
Implication of membership in a class difference. (Contributed by NM,
3-May-1994.)
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Theorem | elndif 3096 |
A set does not belong to a class excluding it. (Contributed by NM,
27-Jun-1994.)
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Theorem | difdif 3097 |
Double class difference. Exercise 11 of [TakeutiZaring] p. 22.
(Contributed by NM, 17-May-1998.)
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Theorem | difss 3098 |
Subclass relationship for class difference. Exercise 14 of
[TakeutiZaring] p. 22.
(Contributed by NM, 29-Apr-1994.)
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Theorem | difssd 3099 |
A difference of two classes is contained in the minuend. Deduction form
of difss 3098. (Contributed by David Moews, 1-May-2017.)
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Theorem | difss2 3100 |
If a class is contained in a difference, it is contained in the minuend.
(Contributed by David Moews, 1-May-2017.)
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