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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | calemos 2001 | "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 2002 | "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 2003 | "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 1984. (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 2004* |
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 1376 through ax-16 1677 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 2004 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 2004 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 2005* | 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 2006* | 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 2004. (Contributed by NM, 14-Nov-2008.) |
Theorem | bm1.1 2007* | 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 2008 | 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 2017, justifying the assignment of setvar variables to class variables via the use of cv 1227. |
Definition | df-clab 2009 |
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 1374, which extends or "overloads" the wel 1375 definition connecting setvar variables, requires that both sides of be a class. In df-cleq 2015 and df-clel 2018, 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 1227 allows us to substitute a setvar variable for a class variable: all sets are classes by cvjust 2017 (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 2128 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 2010 | Simplification of class abstraction notation when the free and bound variables are identical. (Contributed by NM, 5-Aug-1993.) |
Theorem | hbab1 2011* | Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 5-Aug-1993.) |
Theorem | nfsab1 2012* | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | hbab 2013* | Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995.) |
Theorem | nfsab 2014* | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Definition | df-cleq 2015* |
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 2006). 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 2009, df-clel 2018, and abeq2 2128. In the form of dfcleq 2016, 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 2016* | The same as df-cleq 2015 with the hypothesis removed using the Axiom of Extensionality ax-ext 2004. (Contributed by NM, 15-Sep-1993.) |
Theorem | cvjust 2017* | 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 1227, which allows us to substitute a setvar variable for a class variable. See also cab 2008 and df-clab 2009. Note that this is not a rigorous justification, because cv 1227 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 2018* |
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 2015 it extends or "overloads" the
use of the existing membership symbol, but unlike df-cleq 2015 it does not
strengthen the set of valid wffs of logic when the class variables are
replaced with setvar variables (see cleljust 1795), so we don't include
any set theory axiom as a hypothesis. See also comments about the
syntax under df-clab 2009.
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 2019* | Infer equality of classes from equivalence of membership. (Contributed by NM, 5-Aug-1993.) |
Theorem | eqrdv 2020* | Deduce equality of classes from equivalence of membership. (Contributed by NM, 17-Mar-1996.) |
Theorem | eqrdav 2021* | Deduce equality of classes from an equivalence of membership that depends on the membership variable. (Contributed by NM, 7-Nov-2008.) |
Theorem | eqid 2022 |
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 2023 | Class identity law with antecedent. (Contributed by NM, 21-Aug-2008.) |
Theorem | eqcom 2024 | Commutative law for class equality. Theorem 6.5 of [Quine] p. 41. (Contributed by NM, 5-Aug-1993.) |
Theorem | eqcoms 2025 | Inference applying commutative law for class equality to an antecedent. (Contributed by NM, 5-Aug-1993.) |
Theorem | eqcomi 2026 | Inference from commutative law for class equality. (Contributed by NM, 5-Aug-1993.) |
Theorem | eqcomd 2027 | Deduction from commutative law for class equality. (Contributed by NM, 15-Aug-1994.) |
Theorem | eqeq1 2028 | Equality implies equivalence of equalities. (Contributed by NM, 5-Aug-1993.) |
Theorem | eqeq1i 2029 | Inference from equality to equivalence of equalities. (Contributed by NM, 5-Aug-1993.) |
Theorem | eqeq1d 2030 | Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.) |
Theorem | eqeq2 2031 | Equality implies equivalence of equalities. (Contributed by NM, 5-Aug-1993.) |
Theorem | eqeq2i 2032 | Inference from equality to equivalence of equalities. (Contributed by NM, 5-Aug-1993.) |
Theorem | eqeq2d 2033 | Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.) |
Theorem | eqeq12 2034 | Equality relationship among 4 classes. (Contributed by NM, 3-Aug-1994.) |
Theorem | eqeq12i 2035 | 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 2036 | 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 2037 | 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 2038 | A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.) |
Theorem | eqtr 2039 | Transitive law for class equality. Proposition 4.7(3) of [TakeutiZaring] p. 13. (Contributed by NM, 25-Jan-2004.) |
Theorem | eqtr2 2040 | A transitive law for class equality. (Contributed by NM, 20-May-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | eqtr3 2041 | A transitive law for class equality. (Contributed by NM, 20-May-2005.) |
Theorem | eqtri 2042 | An equality transitivity inference. (Contributed by NM, 5-Aug-1993.) |
Theorem | eqtr2i 2043 | An equality transitivity inference. (Contributed by NM, 21-Feb-1995.) |
Theorem | eqtr3i 2044 | An equality transitivity inference. (Contributed by NM, 6-May-1994.) |
Theorem | eqtr4i 2045 | An equality transitivity inference. (Contributed by NM, 5-Aug-1993.) |
Theorem | 3eqtri 2046 | An inference from three chained equalities. (Contributed by NM, 29-Aug-1993.) |
Theorem | 3eqtrri 2047 | An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | 3eqtr2i 2048 | An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.) |
Theorem | 3eqtr2ri 2049 | An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | 3eqtr3i 2050 | An inference from three chained equalities. (Contributed by NM, 6-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | 3eqtr3ri 2051 | An inference from three chained equalities. (Contributed by NM, 15-Aug-2004.) |
Theorem | 3eqtr4i 2052 | An inference from three chained equalities. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | 3eqtr4ri 2053 | An inference from three chained equalities. (Contributed by NM, 2-Sep-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | eqtrd 2054 | An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) |
Theorem | eqtr2d 2055 | An equality transitivity deduction. (Contributed by NM, 18-Oct-1999.) |
Theorem | eqtr3d 2056 | An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.) |
Theorem | eqtr4d 2057 | An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.) |
Theorem | 3eqtrd 2058 | A deduction from three chained equalities. (Contributed by NM, 29-Oct-1995.) |
Theorem | 3eqtrrd 2059 | A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | 3eqtr2d 2060 | A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.) |
Theorem | 3eqtr2rd 2061 | A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.) |
Theorem | 3eqtr3d 2062 | A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | 3eqtr3rd 2063 | A deduction from three chained equalities. (Contributed by NM, 14-Jan-2006.) |
Theorem | 3eqtr4d 2064 | A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | 3eqtr4rd 2065 | A deduction from three chained equalities. (Contributed by NM, 21-Sep-1995.) |
Theorem | syl5eq 2066 | An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) |
Theorem | syl5req 2067 | An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.) |
Theorem | syl5eqr 2068 | An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) |
Theorem | syl5reqr 2069 | An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.) |
Theorem | syl6eq 2070 | An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) |
Theorem | syl6req 2071 | An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.) |
Theorem | syl6eqr 2072 | An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.) |
Theorem | syl6reqr 2073 | An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.) |
Theorem | sylan9eq 2074 | An equality transitivity deduction. (Contributed by NM, 8-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | sylan9req 2075 | An equality transitivity deduction. (Contributed by NM, 23-Jun-2007.) |
Theorem | sylan9eqr 2076 | An equality transitivity deduction. (Contributed by NM, 8-May-1994.) |
Theorem | 3eqtr3g 2077 | A chained equality inference, useful for converting from definitions. (Contributed by NM, 15-Nov-1994.) |
Theorem | 3eqtr3a 2078 | A chained equality inference, useful for converting from definitions. (Contributed by Mario Carneiro, 6-Nov-2015.) |
Theorem | 3eqtr4g 2079 | A chained equality inference, useful for converting to definitions. (Contributed by NM, 5-Aug-1993.) |
Theorem | 3eqtr4a 2080 | 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 2081 | A compound transitive inference for class equality. (Contributed by NM, 22-Jan-2004.) |
Theorem | eleq1 2082 | Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.) |
Theorem | eleq2 2083 | Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.) |
Theorem | eleq12 2084 | Equality implies equivalence of membership. (Contributed by NM, 31-May-1999.) |
Theorem | eleq1i 2085 | Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.) |
Theorem | eleq2i 2086 | Inference from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.) |
Theorem | eleq12i 2087 | Inference from equality to equivalence of membership. (Contributed by NM, 31-May-1994.) |
Theorem | eleq1d 2088 | Deduction from equality to equivalence of membership. (Contributed by NM, 5-Aug-1993.) |
Theorem | eleq2d 2089 | Deduction from equality to equivalence of membership. (Contributed by NM, 27-Dec-1993.) |
Theorem | eleq12d 2090 | Deduction from equality to equivalence of membership. (Contributed by NM, 31-May-1994.) |
Theorem | eleq1a 2091 | A transitive-type law relating membership and equality. (Contributed by NM, 9-Apr-1994.) |
Theorem | eqeltri 2092 | Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.) |
Theorem | eqeltrri 2093 | Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.) |
Theorem | eleqtri 2094 | Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.) |
Theorem | eleqtrri 2095 | Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.) |
Theorem | eqeltrd 2096 | Substitution of equal classes into membership relation, deduction form. (Contributed by Raph Levien, 10-Dec-2002.) |
Theorem | eqeltrrd 2097 | Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.) |
Theorem | eleqtrd 2098 | Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.) |
Theorem | eleqtrrd 2099 | Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.) |
Theorem | 3eltr3i 2100 | Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.) |
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