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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cesare 2001 | "Cesare", one of the syllogisms of Aristotelian logic. No is , and all is , therefore no is . (In Aristotelian notation, EAE-2: PeM and SaM therefore SeP.) Related to celarent 1996. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 13-Nov-2016.) |
Theorem | camestres 2002 | "Camestres", one of the syllogisms of Aristotelian logic. All is , and no is , therefore no is . (In Aristotelian notation, AEE-2: PaM and SeM therefore SeP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) |
Theorem | festino 2003 | "Festino", one of the syllogisms of Aristotelian logic. No is , and some is , therefore some is not . (In Aristotelian notation, EIO-2: PeM and SiM therefore SoP.) (Contributed by David A. Wheeler, 25-Nov-2016.) |
Theorem | baroco 2004 | "Baroco", one of the syllogisms of Aristotelian logic. All is , and some is not , therefore some is not . (In Aristotelian notation, AOO-2: PaM and SoM therefore SoP.) For example, "All informative things are useful", "Some websites are not useful", therefore "Some websites are not informative." (Contributed by David A. Wheeler, 28-Aug-2016.) |
Theorem | cesaro 2005 | "Cesaro", one of the syllogisms of Aristotelian logic. No is , all is , and exist, therefore some is not . (In Aristotelian notation, EAO-2: PeM and SaM therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) |
Theorem | camestros 2006 | "Camestros", one of the syllogisms of Aristotelian logic. All is , no is , and exist, therefore some is not . (In Aristotelian notation, AEO-2: PaM and SeM therefore SoP.) For example, "All horses have hooves", "No humans have hooves", and humans exist, therefore "Some humans are not horses". (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) |
Theorem | datisi 2007 | "Datisi", one of the syllogisms of Aristotelian logic. All is , and some is , therefore some is . (In Aristotelian notation, AII-3: MaP and MiS therefore SiP.) (Contributed by David A. Wheeler, 28-Aug-2016.) |
Theorem | disamis 2008 | "Disamis", one of the syllogisms of Aristotelian logic. Some is , and all is , therefore some is . (In Aristotelian notation, IAI-3: MiP and MaS therefore SiP.) (Contributed by David A. Wheeler, 28-Aug-2016.) |
Theorem | ferison 2009 | "Ferison", one of the syllogisms of Aristotelian logic. No is , and some is , therefore some is not . (In Aristotelian notation, EIO-3: MeP and MiS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) |
Theorem | bocardo 2010 | "Bocardo", one of the syllogisms of Aristotelian logic. Some is not , and all is , therefore some is not . (In Aristotelian notation, OAO-3: MoP and MaS therefore SoP.) For example, "Some cats have no tails", "All cats are mammals", therefore "Some mammals have no tails". A reorder of disamis 2008; prefer using that instead. (Contributed by David A. Wheeler, 28-Aug-2016.) (New usage is discouraged.) |
Theorem | felapton 2011 | "Felapton", one of the syllogisms of Aristotelian logic. No is , all is , and some exist, therefore some is not . (In Aristotelian notation, EAO-3: MeP and MaS therefore SoP.) For example, "No flowers are animals" and "All flowers are plants", therefore "Some plants are not animals". (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) |
Theorem | darapti 2012 | "Darapti", one of the syllogisms of Aristotelian logic. All is , all is , and some exist, therefore some is . (In Aristotelian notation, AAI-3: MaP and MaS therefore SiP.) For example, "All squares are rectangles" and "All squares are rhombuses", therefore "Some rhombuses are rectangles". (Contributed by David A. Wheeler, 28-Aug-2016.) |
Theorem | calemes 2013 | "Calemes", one of the syllogisms of Aristotelian logic. All is , and no is , therefore no is . (In Aristotelian notation, AEE-4: PaM and MeS therefore SeP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) |
Theorem | dimatis 2014 | "Dimatis", one of the syllogisms of Aristotelian logic. Some is , and all is , therefore some is . (In Aristotelian notation, IAI-4: PiM and MaS therefore SiP.) For example, "Some pets are rabbits.", "All rabbits have fur", therefore "Some fur bearing animals are pets". Like darii 1997 with positions interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.) |
Theorem | fresison 2015 | "Fresison", one of the syllogisms of Aristotelian logic. No is (PeM), and some is (MiS), therefore some is not (SoP). (In Aristotelian notation, EIO-4: PeM and MiS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) |
Theorem | calemos 2016 | "Calemos", one of the syllogisms of Aristotelian logic. All is (PaM), no is (MeS), and exist, therefore some is not (SoP). (In Aristotelian notation, AEO-4: PaM and MeS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) |
Theorem | fesapo 2017 | "Fesapo", one of the syllogisms of Aristotelian logic. No is , all is , and exist, therefore some is not . (In Aristotelian notation, EAO-4: PeM and MaS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) |
Theorem | bamalip 2018 | "Bamalip", one of the syllogisms of Aristotelian logic. All is , all is , and exist, therefore some is . (In Aristotelian notation, AAI-4: PaM and MaS therefore SiP.) Like barbari 1999. (Contributed by David A. Wheeler, 28-Aug-2016.) |
Set theory uses the formalism of propositional and predicate calculus to assert properties of arbitrary mathematical objects called "sets." A set can be an element of another set, and this relationship is indicated by the symbol. Starting with the simplest mathematical object, called the empty set, set theory builds up more and more complex structures whose existence follows from the axioms, eventually resulting in extremely complicated sets that we identify with the real numbers and other familiar mathematical objects. Here we develop set theory based on the Intuitionistic Zermelo-Fraenkel (IZF) system, mostly following the IZF axioms as laid out in [Crosilla]. Constructive Zermelo-Fraenkel (CZF), also described in Crosilla, is not as easy to formalize in metamath because the Axiom of Restricted Separation would require us to develop the ability to classify formulas as bounded formulas, similar to the machinery we have built up for asserting on whether variables are free in formulas. | ||
Axiom | ax-ext 2019* |
Axiom of Extensionality. It states that two sets are identical if they
contain the same elements. Axiom 1 of [Crosilla] p. "Axioms of CZF and
IZF" (with unnecessary quantifiers removed).
Set theory can also be formulated with a single primitive predicate on top of traditional predicate calculus without equality. In that case the Axiom of Extensionality becomes , and equality is defined as . All of the usual axioms of equality then become theorems of set theory. See, for example, Axiom 1 of [TakeutiZaring] p. 8. To use the above "equality-free" version of Extensionality with Metamath's logical axioms, we would rewrite ax-8 1392 through ax-16 1692 with equality expanded according to the above definition. Some of those axioms could be proved from set theory and would be redundant. Not all of them are redundant, since our axioms of predicate calculus make essential use of equality for the proper substitution that is a primitive notion in traditional predicate calculus. A study of such an axiomatization would be an interesting project for someone exploring the foundations of logic. It is important to understand that strictly speaking, all of our set theory axioms are really schemes that represent an infinite number of actual axioms. This is inherent in the design of Metamath ("metavariable math"), which manipulates only metavariables. For example, the metavariable in ax-ext 2019 can represent any actual variable v1, v2, v3,... . Distinct variable restrictions ($d) prevent us from substituting say v1 for both and . This is in contrast to typical textbook presentations that present actual axioms (except for axioms which involve wff metavariables). In practice, though, the theorems and proofs are essentially the same. The $d restrictions make each of the infinite axioms generated by the ax-ext 2019 scheme exactly logically equivalent to each other and in particular to the actual axiom of the textbook version. (Contributed by NM, 5-Aug-1993.) |
Theorem | axext3 2020* | A generalization of the Axiom of Extensionality in which and need not be distinct. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
Theorem | axext4 2021* | A bidirectional version of Extensionality. Although this theorem "looks" like it is just a definition of equality, it requires the Axiom of Extensionality for its proof under our axiomatization. See the comments for ax-ext 2019. (Contributed by NM, 14-Nov-2008.) |
Theorem | bm1.1 2022* | Any set defined by a property is the only set defined by that property. Theorem 1.1 of [BellMachover] p. 462. (Contributed by NM, 30-Jun-1994.) |
Syntax | cab 2023 | Introduce the class builder or class abstraction notation ("the class of sets such that is true"). Our class variables , , etc. range over class builders (sometimes implicitly). Note that a setvar variable can be expressed as a class builder per theorem cvjust 2032, justifying the assignment of setvar variables to class variables via the use of cv 1241. |
Definition | df-clab 2024 |
Define class abstraction notation (so-called by Quine), also called a
"class builder" in the literature. and need not be distinct.
Definition 2.1 of [Quine] p. 16. Typically,
will have as a
free variable, and " " is read "the class of all sets
such that is true." We do not define in
isolation but only as part of an expression that extends or
"overloads"
the
relationship.
This is our first use of the symbol to connect classes instead of sets. The syntax definition wcel 1390, which extends or "overloads" the wel 1391 definition connecting setvar variables, requires that both sides of be a class. In df-cleq 2030 and df-clel 2033, we introduce a new kind of variable (class variable) that can substituted with expressions such as . In the present definition, the on the left-hand side is a setvar variable. Syntax definition cv 1241 allows us to substitute a setvar variable for a class variable: all sets are classes by cvjust 2032 (but not necessarily vice-versa). For a full description of how classes are introduced and how to recover the primitive language, see the discussion in Quine (and under abeq2 2143 for a quick overview). Because class variables can be substituted with compound expressions and setvar variables cannot, it is often useful to convert a theorem containing a free setvar variable to a more general version with a class variable. This is called the "axiom of class comprehension" by [Levy] p. 338, who treats the theory of classes as an extralogical extension to our logic and set theory axioms. He calls the construction a "class term". For a general discussion of the theory of classes, see http://us.metamath.org/mpeuni/mmset.html#class. (Contributed by NM, 5-Aug-1993.) |
Theorem | abid 2025 | Simplification of class abstraction notation when the free and bound variables are identical. (Contributed by NM, 5-Aug-1993.) |
Theorem | hbab1 2026* | Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 5-Aug-1993.) |
Theorem | nfsab1 2027* | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | hbab 2028* | Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995.) |
Theorem | nfsab 2029* | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Definition | df-cleq 2030* |
Define the equality connective between classes. Definition 2.7 of
[Quine] p. 18. Also Definition 4.5 of [TakeutiZaring] p. 13; Chapter 4
provides its justification and methods for eliminating it. Note that
its elimination will not necessarily result in a single wff in the
original language but possibly a "scheme" of wffs.
This is an example of a somewhat "risky" definition, meaning that it has a more complex than usual soundness justification (outside of Metamath), because it "overloads" or reuses the existing equality symbol rather than introducing a new symbol. This allows us to make statements that may not hold for the original symbol. For example, it permits us to deduce , which is not a theorem of logic but rather presupposes the Axiom of Extensionality (see theorem axext4 2021). We therefore include this axiom as a hypothesis, so that the use of Extensionality is properly indicated. We could avoid this complication by introducing a new symbol, say =_{2}, in place of . This would also have the advantage of making elimination of the definition straightforward, so that we could eliminate Extensionality as a hypothesis. We would then also have the advantage of being able to identify in various proofs exactly where Extensionality truly comes into play rather than just being an artifact of a definition. One of our theorems would then be =_{2} by invoking Extensionality. However, to conform to literature usage, we retain this overloaded definition. This also makes some proofs shorter and probably easier to read, without the constant switching between two kinds of equality. See also comments under df-clab 2024, df-clel 2033, and abeq2 2143. In the form of dfcleq 2031, this is called the "axiom of extensionality" by [Levy] p. 338, who treats the theory of classes as an extralogical extension to our logic and set theory axioms. For a general discussion of the theory of classes, see http://us.metamath.org/mpeuni/mmset.html#class. (Contributed by NM, 15-Sep-1993.) |
Theorem | dfcleq 2031* | The same as df-cleq 2030 with the hypothesis removed using the Axiom of Extensionality ax-ext 2019. (Contributed by NM, 15-Sep-1993.) |
Theorem | cvjust 2032* | Every set is a class. Proposition 4.9 of [TakeutiZaring] p. 13. This theorem shows that a setvar variable can be expressed as a class abstraction. This provides a motivation for the class syntax construction cv 1241, which allows us to substitute a setvar variable for a class variable. See also cab 2023 and df-clab 2024. Note that this is not a rigorous justification, because cv 1241 is used as part of the proof of this theorem, but a careful argument can be made outside of the formalism of Metamath, for example as is done in Chapter 4 of Takeuti and Zaring. See also the discussion under the definition of class in [Jech] p. 4 showing that "Every set can be considered to be a class." (Contributed by NM, 7-Nov-2006.) |
Definition | df-clel 2033* |
Define the membership connective between classes. Theorem 6.3 of
[Quine] p. 41, or Proposition 4.6 of [TakeutiZaring] p. 13, which we
adopt as a definition. See these references for its metalogical
justification. Note that like df-cleq 2030 it extends or "overloads" the
use of the existing membership symbol, but unlike df-cleq 2030 it does not
strengthen the set of valid wffs of logic when the class variables are
replaced with setvar variables (see cleljust 1810), so we don't include
any set theory axiom as a hypothesis. See also comments about the
syntax under df-clab 2024.
This is called the "axiom of membership" by [Levy] p. 338, who treats the theory of classes as an extralogical extension to our logic and set theory axioms. For a general discussion of the theory of classes, see http://us.metamath.org/mpeuni/mmset.html#class. (Contributed by NM, 5-Aug-1993.) |
Theorem | eqriv 2034* | Infer equality of classes from equivalence of membership. (Contributed by NM, 5-Aug-1993.) |
Theorem | eqrdv 2035* | Deduce equality of classes from equivalence of membership. (Contributed by NM, 17-Mar-1996.) |
Theorem | eqrdav 2036* | Deduce equality of classes from an equivalence of membership that depends on the membership variable. (Contributed by NM, 7-Nov-2008.) |
Theorem | eqid 2037 |
Law of identity (reflexivity of class equality). Theorem 6.4 of [Quine]
p. 41.
This law is thought to have originated with Aristotle (Metaphysics, Zeta, 17, 1041 a, 10-20). (Thanks to Stefan Allan and BJ for this information.) (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 14-Oct-2017.) |
Theorem | eqidd 2038 | Class identity law with antecedent. (Contributed by NM, 21-Aug-2008.) |
Theorem | eqcom 2039 | Commutative law for class equality. Theorem 6.5 of [Quine] p. 41. (Contributed by NM, 5-Aug-1993.) |
Theorem | eqcoms 2040 | Inference applying commutative law for class equality to an antecedent. (Contributed by NM, 5-Aug-1993.) |
Theorem | eqcomi 2041 | Inference from commutative law for class equality. (Contributed by NM, 5-Aug-1993.) |
Theorem | eqcomd 2042 | Deduction from commutative law for class equality. (Contributed by NM, 15-Aug-1994.) |
Theorem | eqeq1 2043 | Equality implies equivalence of equalities. (Contributed by NM, 5-Aug-1993.) |
Theorem | eqeq1i 2044 | Inference from equality to equivalence of equalities. (Contributed by NM, 5-Aug-1993.) |
Theorem | eqeq1d 2045 | Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.) |
Theorem | eqeq2 2046 | Equality implies equivalence of equalities. (Contributed by NM, 5-Aug-1993.) |
Theorem | eqeq2i 2047 | Inference from equality to equivalence of equalities. (Contributed by NM, 5-Aug-1993.) |
Theorem | eqeq2d 2048 | Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.) |
Theorem | eqeq12 2049 | Equality relationship among 4 classes. (Contributed by NM, 3-Aug-1994.) |
Theorem | eqeq12i 2050 | A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | eqeq12d 2051 | A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | eqeqan12d 2052 | A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | eqeqan12rd 2053 | A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.) |
Theorem | eqtr 2054 | Transitive law for class equality. Proposition 4.7(3) of [TakeutiZaring] p. 13. (Contributed by NM, 25-Jan-2004.) |
Theorem | eqtr2 2055 | A transitive law for class equality. (Contributed by NM, 20-May-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | eqtr3 2056 | A transitive law for class equality. (Contributed by NM, 20-May-2005.) |
Theorem | eqtri 2057 | An equality transitivity inference. (Contributed by NM, 5-Aug-1993.) |
Theorem | eqtr2i 2058 | An equality transitivity inference. (Contributed by NM, 21-Feb-1995.) |
Theorem | eqtr3i 2059 | An equality transitivity inference. (Contributed by NM, 6-May-1994.) |
Theorem | eqtr4i 2060 | An equality transitivity inference. (Contributed by NM, 5-Aug-1993.) |
Theorem | 3eqtri 2061 | An inference from three chained equalities. (Contributed by NM, 29-Aug-1993.) |
Theorem | 3eqtrri 2062 | An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | 3eqtr2i 2063 | An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.) |
Theorem | 3eqtr2ri 2064 | An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | 3eqtr3i 2065 | An inference from three chained equalities. (Contributed by NM, 6-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | 3eqtr3ri 2066 | An inference from three chained equalities. (Contributed by NM, 15-Aug-2004.) |
Theorem | 3eqtr4i 2067 | An inference from three chained equalities. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | 3eqtr4ri 2068 | An inference from three chained equalities. (Contributed by NM, 2-Sep-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | eqtrd 2069 | An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) |
Theorem | eqtr2d 2070 | An equality transitivity deduction. (Contributed by NM, 18-Oct-1999.) |
Theorem | eqtr3d 2071 | An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.) |
Theorem | eqtr4d 2072 | An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.) |
Theorem | 3eqtrd 2073 | A deduction from three chained equalities. (Contributed by NM, 29-Oct-1995.) |
Theorem | 3eqtrrd 2074 | A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | 3eqtr2d 2075 | A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.) |
Theorem | 3eqtr2rd 2076 | A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.) |
Theorem | 3eqtr3d 2077 | A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | 3eqtr3rd 2078 | A deduction from three chained equalities. (Contributed by NM, 14-Jan-2006.) |
Theorem | 3eqtr4d 2079 | A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | 3eqtr4rd 2080 | A deduction from three chained equalities. (Contributed by NM, 21-Sep-1995.) |
Theorem | syl5eq 2081 | An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) |
Theorem | syl5req 2082 | An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.) |
Theorem | syl5eqr 2083 | An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) |
Theorem | syl5reqr 2084 | An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.) |
Theorem | syl6eq 2085 | An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) |
Theorem | syl6req 2086 | An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.) |
Theorem | syl6eqr 2087 | An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) |
Theorem | syl6reqr 2088 | An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.) |
Theorem | sylan9eq 2089 | An equality transitivity deduction. (Contributed by NM, 8-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | sylan9req 2090 | An equality transitivity deduction. (Contributed by NM, 23-Jun-2007.) |
Theorem | sylan9eqr 2091 | An equality transitivity deduction. (Contributed by NM, 8-May-1994.) |
Theorem | 3eqtr3g 2092 | A chained equality inference, useful for converting from definitions. (Contributed by NM, 15-Nov-1994.) |
Theorem | 3eqtr3a 2093 | A chained equality inference, useful for converting from definitions. (Contributed by Mario Carneiro, 6-Nov-2015.) |
Theorem | 3eqtr4g 2094 | A chained equality inference, useful for converting to definitions. (Contributed by NM, 5-Aug-1993.) |
Theorem | 3eqtr4a 2095 | A chained equality inference, useful for converting to definitions. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | eq2tri 2096 | A compound transitive inference for class equality. (Contributed by NM, 22-Jan-2004.) |
Theorem | eleq1 2097 | Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.) |
Theorem | eleq2 2098 | Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.) |
Theorem | eleq12 2099 | Equality implies equivalence of membership. (Contributed by NM, 31-May-1999.) |
Theorem | eleq1i 2100 | Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.) |
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