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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | exmoeudc 2101 | Existence in terms of "at most one" and uniqueness. (Contributed by Jim Kingdon, 3-Jul-2018.) |
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Theorem | moim 2102 | "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 22-Apr-1995.) |
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Theorem | moimi 2103 | "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 15-Feb-2006.) |
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Theorem | moimv 2104* | Move antecedent outside of "at most one". (Contributed by NM, 28-Jul-1995.) |
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Theorem | euimmo 2105 | Uniqueness implies "at most one" through implication. (Contributed by NM, 22-Apr-1995.) |
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Theorem | euim 2106 | Add existential unique existential quantifiers to an implication. Note the reversed implication in the antecedent. (Contributed by NM, 19-Oct-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
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Theorem | moan 2107 | "At most one" is still the case when a conjunct is added. (Contributed by NM, 22-Apr-1995.) |
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Theorem | moani 2108 | "At most one" is still true when a conjunct is added. (Contributed by NM, 9-Mar-1995.) |
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Theorem | moor 2109 | "At most one" is still the case when a disjunct is removed. (Contributed by NM, 5-Apr-2004.) |
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Theorem | mooran1 2110 | "At most one" imports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
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Theorem | mooran2 2111 | "At most one" exports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
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Theorem | moanim 2112 | Introduction of a conjunct into at-most-one quantifier. (Contributed by NM, 3-Dec-2001.) |
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Theorem | moanimv 2113* | Introduction of a conjunct into at-most-one quantifier. (Contributed by NM, 23-Mar-1995.) |
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Theorem | moaneu 2114 | Nested at-most-one and unique existential quantifiers. (Contributed by NM, 25-Jan-2006.) |
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Theorem | moanmo 2115 | Nested at-most-one quantifiers. (Contributed by NM, 25-Jan-2006.) |
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Theorem | mopick 2116 | "At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.) |
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Theorem | eupick 2117 |
Existential uniqueness "picks" a variable value for which another wff
is
true. If there is only one thing ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | eupicka 2118 | Version of eupick 2117 with closed formulas. (Contributed by NM, 6-Sep-2008.) |
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Theorem | eupickb 2119 | Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.) |
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Theorem | eupickbi 2120 | Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) |
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Theorem | mopick2 2121 | "At most one" can show the existence of a common value. In this case we can infer existence of conjunction from a conjunction of existence, and it is one way to achieve the converse of 19.40 1642. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
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Theorem | moexexdc 2122 | "At most one" double quantification. (Contributed by Jim Kingdon, 5-Jul-2018.) |
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Theorem | euexex 2123 | Existential uniqueness and "at most one" double quantification. (Contributed by Jim Kingdon, 28-Dec-2018.) |
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Theorem | 2moex 2124 | Double quantification with "at most one". (Contributed by NM, 3-Dec-2001.) |
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Theorem | 2euex 2125 | Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
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Theorem | 2eumo 2126 | Double quantification with existential uniqueness and "at most one." (Contributed by NM, 3-Dec-2001.) |
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Theorem | 2eu2ex 2127 | Double existential uniqueness. (Contributed by NM, 3-Dec-2001.) |
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Theorem | 2moswapdc 2128 | A condition allowing swap of "at most one" and existential quantifiers. (Contributed by Jim Kingdon, 6-Jul-2018.) |
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Theorem | 2euswapdc 2129 | A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Jim Kingdon, 7-Jul-2018.) |
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Theorem | 2exeu 2130 | Double existential uniqueness implies double unique existential quantification. (Contributed by NM, 3-Dec-2001.) |
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Theorem | 2eu4 2131* |
This theorem provides us with a definition of double existential
uniqueness ("exactly one ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | 2eu7 2132 | Two equivalent expressions for double existential uniqueness. (Contributed by NM, 19-Feb-2005.) |
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Theorem | euequ1 2133* | Equality has existential uniqueness. (Contributed by Stefan Allan, 4-Dec-2008.) |
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Theorem | exists1 2134* | Two ways to express "only one thing exists". The left-hand side requires only one variable to express this. Both sides are false in set theory. (Contributed by NM, 5-Apr-2004.) |
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Theorem | exists2 2135 | A condition implying that at least two things exist. (Contributed by NM, 10-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
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Model the Aristotelian assertic syllogisms using modern notation. This section shows that the Aristotelian assertic syllogisms can be proven with our axioms of logic, and also provides generally useful theorems. In antiquity Aristotelian logic and Stoic logic (see mptnan 1434) were the leading logical systems. Aristotelian logic became the leading system in medieval Europe; this section models this system (including later refinements to it). Aristotle defined syllogisms very generally ("a discourse in which certain (specific) things having been supposed, something different from the things supposed results of necessity because these things are so") Aristotle, Prior Analytics 24b18-20. However, in Prior Analytics he limits himself to categorical syllogisms that consist of three categorical propositions with specific structures. The syllogisms are the valid subset of the possible combinations of these structures. The medieval schools used vowels to identify the types of terms (a=all, e=none, i=some, and o=some are not), and named the different syllogisms with Latin words that had the vowels in the intended order. "There is a surprising amount of scholarly debate about how best to formalize Aristotle's syllogisms..." according to Aristotle's Modal Proofs: Prior Analytics A8-22 in Predicate Logic, Adriane Rini, Springer, 2011, ISBN 978-94-007-0049-9, page 28. For example, Lukasiewicz believes it is important to note that "Aristotle does not introduce singular terms or premisses into his system". Lukasiewicz also believes that Aristotelian syllogisms are predicates (having a true/false value), not inference rules: "The characteristic sign of an inference is the word 'therefore'... no syllogism is formulated by Aristotle primarily as an inference, but they are all implications." Jan Lukasiewicz, Aristotle's Syllogistic from the Standpoint of Modern Formal Logic, Second edition, Oxford, 1957, page 1-2. Lukasiewicz devised a specialized prefix notation for representing Aristotelian syllogisms instead of using standard predicate logic notation.
We instead translate each Aristotelian syllogism into an inference rule,
and each rule is defined using standard predicate logic notation and
predicates. The predicates are represented by wff variables
that may depend on the quantified variable
Expressions of the form "no
In traditional Aristotelian syllogisms the predicates
have a restricted form ("x is a ..."); those predicates
could be modeled in modern notation by constructs such as
There are some widespread misconceptions about the existential assumptions made by Aristotle (aka "existential import"). Aristotle was not trying to develop something exactly corresponding to modern logic. Aristotle devised "a companion-logic for science. He relegates fictions like fairy godmothers and mermaids and unicorns to the realms of poetry and literature. In his mind, they exist outside the ambit of science. This is why he leaves no room for such nonexistent entities in his logic. This is a thoughtful choice, not an inadvertent omission. Technically, Aristotelian science is a search for definitions, where a definition is "a phrase signifying a thing's essence." (Topics, I.5.102a37, Pickard-Cambridge.)... Because nonexistent entities cannot be anything, they do not, in Aristotle's mind, possess an essence... This is why he leaves no place for fictional entities like goat-stags (or unicorns)." Source: Louis F. Groarke, "Aristotle: Logic", section 7. (Existential Assumptions), Internet Encyclopedia of Philosophy (A Peer-Reviewed Academic Resource), https://iep.utm.edu/aristotle-log/ 1614. Thus, some syllogisms have "extra" existence hypotheses that do not directly appear in Aristotle's original materials (since they were always assumed); they are added where they are needed. This affects barbari 2140, celaront 2141, cesaro 2146, camestros 2147, felapton 2152, darapti 2153, calemos 2157, fesapo 2158, and bamalip 2159. These are only the assertic syllogisms. Aristotle also defined modal syllogisms that deal with modal qualifiers such as "necessarily" and "possibly". Historically Aristotelian modal syllogisms were not as widely used. For more about modal syllogisms in a modern context, see Rini as well as Aristotle's Modal Syllogistic by Marko Malink, Harvard University Press, November 2013. We do not treat them further here. Aristotelean logic is essentially the forerunner of predicate calculus (as well as set theory since it discusses membership in groups), while Stoic logic is essentially the forerunner of propositional calculus. | ||
Theorem | barbara 2136 |
"Barbara", one of the fundamental syllogisms of Aristotelian logic.
All
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Theorem | celarent 2137 |
"Celarent", one of the syllogisms of Aristotelian logic. No ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | darii 2138 |
"Darii", one of the syllogisms of Aristotelian logic. All ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | ferio 2139 |
"Ferio" ("Ferioque"), one of the syllogisms of Aristotelian
logic. No
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Theorem | barbari 2140 |
"Barbari", one of the syllogisms of Aristotelian logic. All ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | celaront 2141 |
"Celaront", one of the syllogisms of Aristotelian logic. No ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | cesare 2142 |
"Cesare", one of the syllogisms of Aristotelian logic. No ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | camestres 2143 |
"Camestres", one of the syllogisms of Aristotelian logic. All ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | festino 2144 |
"Festino", one of the syllogisms of Aristotelian logic. No ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | baroco 2145 |
"Baroco", one of the syllogisms of Aristotelian logic. All ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | cesaro 2146 |
"Cesaro", one of the syllogisms of Aristotelian logic. No ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | camestros 2147 |
"Camestros", one of the syllogisms of Aristotelian logic. All ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | datisi 2148 |
"Datisi", one of the syllogisms of Aristotelian logic. All ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | disamis 2149 |
"Disamis", one of the syllogisms of Aristotelian logic. Some ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | ferison 2150 |
"Ferison", one of the syllogisms of Aristotelian logic. No ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | bocardo 2151 |
"Bocardo", one of the syllogisms of Aristotelian logic. Some ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | felapton 2152 |
"Felapton", one of the syllogisms of Aristotelian logic. No ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | darapti 2153 |
"Darapti", one of the syllogisms of Aristotelian logic. All ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | calemes 2154 |
"Calemes", one of the syllogisms of Aristotelian logic. All ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | dimatis 2155 |
"Dimatis", one of the syllogisms of Aristotelian logic. Some ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | fresison 2156 |
"Fresison", one of the syllogisms of Aristotelian logic. No ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | calemos 2157 |
"Calemos", one of the syllogisms of Aristotelian logic. All ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | fesapo 2158 |
"Fesapo", one of the syllogisms of Aristotelian logic. No ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | bamalip 2159 |
"Bamalip", one of the syllogisms of Aristotelian logic. All ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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This section adds one non-logical binary predicate to the first-order logic developed until here. We call it "the membership predicate" since it will be used in the next part as the membership predicate of set theory, but in this section, it has no other property than being "a binary predicate". "Non-logical" means that it does not belong to the logic. In our logic (and in most treatments), the only logical predicate is the equality predicate (see weq 1514). | ||
Syntax | wcel 2160 |
Extend wff definition to include the membership connective between
classes.
For a general discussion of the theory of classes, see mmset.html#class.
The purpose of introducing |
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Theorem | wel 2161 |
Extend wff definition to include atomic formulas with the membership
predicate. This is read either "![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]()
This syntactical construction introduces a binary non-logical predicate
symbol
Instead of introducing wel 2161 as an axiomatic statement, as was done in an
older version of this database, we introduce it by "proving" a
special
case of set theory's more general wcel 2160. This lets us avoid overloading
the |
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Axiom | ax-13 2162 |
Axiom of left equality for the binary predicate ![]() ![]() |
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Axiom | ax-14 2163 |
Axiom of right equality for the binary predicate ![]() ![]() |
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Theorem | elequ1 2164 | An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.) |
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Theorem | elequ2 2165 | An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.) |
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Theorem | cleljust 2166* | When the class variables of set theory are replaced with setvar variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the setvar variables in wel 2161 with the class variables in wcel 2160. (Contributed by NM, 28-Jan-2004.) |
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Theorem | elsb1 2167* | Substitution for the first argument of the non-logical predicate in an atomic formula. See elsb2 2168 for substitution for the second argument. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
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Theorem | elsb2 2168* | Substitution for the second argument of the non-logical predicate in an atomic formula. See elsb1 2167 for substitution for the first argument. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) |
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Theorem | dveel1 2169* | Quantifier introduction when one pair of variables is disjoint. (Contributed by NM, 2-Jan-2002.) |
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Theorem | dveel2 2170* | Quantifier introduction when one pair of variables is disjoint. (Contributed by NM, 2-Jan-2002.) |
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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
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 statement of some of its axioms uses the notion of "bounded formula". Since Metamath has, purposefully, a very weak metalogic, that notion must be developed in the logic itself. This is similar to our treatment of substitution (df-sb 1774) and our definition of the nonfreeness predicate (df-nf 1472), whereas substitution and bound and free variables are ordinarily defined in the metalogic. The development of CZF has begun in BJ's mathbox, see wbd 15042. | ||
Axiom | ax-ext 2171* |
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
To use the above "equality-free" version of Extensionality with Metamath's logical axioms, we would rewrite ax-8 1515 through ax-16 1825 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 |
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Theorem | axext3 2172* |
A generalization of the Axiom of Extensionality in which ![]() ![]() |
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Theorem | axext4 2173* | 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 2171. (Contributed by NM, 14-Nov-2008.) |
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Theorem | bm1.1 2174* | 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.) |
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Syntax | cab 2175 |
Introduce the class builder or class abstraction notation ("the class of
sets ![]() ![]() ![]() ![]() |
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Definition | df-clab 2176 |
Define class abstraction notation (so-called by Quine), also called a
"class builder" in the literature. ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]()
This is our first use of the 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 For a general discussion of the theory of classes, see https://us.metamath.org/mpeuni/mmset.html#class 2298. (Contributed by NM, 5-Aug-1993.) |
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Theorem | abid 2177 | Simplification of class abstraction notation when the free and bound variables are identical. (Contributed by NM, 5-Aug-1993.) |
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Theorem | hbab1 2178* | Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 5-Aug-1993.) |
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Theorem | nfsab1 2179* | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
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Theorem | hbab 2180* | Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995.) |
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Theorem | nfsab 2181* | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
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Definition | df-cleq 2182* |
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
We could avoid this complication by introducing a new symbol, say
=2,
in place of 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 2176, df-clel 2185, and abeq2 2298. In the form of dfcleq 2183, 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 https://us.metamath.org/mpeuni/mmset.html#class 2183. (Contributed by NM, 15-Sep-1993.) |
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Theorem | dfcleq 2183* | The same as df-cleq 2182 with the hypothesis removed using the Axiom of Extensionality ax-ext 2171. (Contributed by NM, 15-Sep-1993.) |
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Theorem | cvjust 2184* | 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 1363, which allows us to substitute a setvar variable for a class variable. See also cab 2175 and df-clab 2176. Note that this is not a rigorous justification, because cv 1363 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.) |
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Definition | df-clel 2185* |
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 2182 it extends or "overloads" the
use of the existing membership symbol, but unlike df-cleq 2182 it does not
strengthen the set of valid wffs of logic when the class variables are
replaced with setvar variables (see cleljust 2166), so we don't include
any set theory axiom as a hypothesis. See also comments about the
syntax under df-clab 2176.
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 https://us.metamath.org/mpeuni/mmset.html#class 2176. (Contributed by NM, 5-Aug-1993.) |
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Theorem | eqriv 2186* | Infer equality of classes from equivalence of membership. (Contributed by NM, 5-Aug-1993.) |
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Theorem | eqrdv 2187* | Deduce equality of classes from equivalence of membership. (Contributed by NM, 17-Mar-1996.) |
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Theorem | eqrdav 2188* | Deduce equality of classes from an equivalence of membership that depends on the membership variable. (Contributed by NM, 7-Nov-2008.) |
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Theorem | eqid 2189 |
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.) |
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Theorem | eqidd 2190 | Class identity law with antecedent. (Contributed by NM, 21-Aug-2008.) |
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Theorem | eqcom 2191 | Commutative law for class equality. Theorem 6.5 of [Quine] p. 41. (Contributed by NM, 5-Aug-1993.) |
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Theorem | eqcoms 2192 | Inference applying commutative law for class equality to an antecedent. (Contributed by NM, 5-Aug-1993.) |
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Theorem | eqcomi 2193 | Inference from commutative law for class equality. (Contributed by NM, 5-Aug-1993.) |
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Theorem | neqcomd 2194 | Commute an inequality. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
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Theorem | eqcomd 2195 | Deduction from commutative law for class equality. (Contributed by NM, 15-Aug-1994.) |
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Theorem | eqeq1 2196 | Equality implies equivalence of equalities. (Contributed by NM, 5-Aug-1993.) |
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Theorem | eqeq1i 2197 | Inference from equality to equivalence of equalities. (Contributed by NM, 5-Aug-1993.) |
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Theorem | eqeq1d 2198 | Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.) |
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Theorem | eqeq2 2199 | Equality implies equivalence of equalities. (Contributed by NM, 5-Aug-1993.) |
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Theorem | eqeq2i 2200 | Inference from equality to equivalence of equalities. (Contributed by NM, 5-Aug-1993.) |
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