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Type | Label | Description |
---|---|---|

Statement | ||

Theorem | euimmo 2301 | Uniqueness implies "at most one" through implication. (Contributed by NM, 22-Apr-1995.) |

Theorem | euim 2302 | Add existential uniqueness 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.) |

Theorem | moan 2303 | "At most one" is still the case when a conjunct is added. (Contributed by NM, 22-Apr-1995.) |

Theorem | moani 2304 | "At most one" is still true when a conjunct is added. (Contributed by NM, 9-Mar-1995.) |

Theorem | moor 2305 | "At most one" is still the case when a disjunct is removed. (Contributed by NM, 5-Apr-2004.) |

Theorem | mooran1 2306 | "At most one" imports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |

Theorem | mooran2 2307 | "At most one" exports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |

Theorem | moanim 2308 | Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 3-Dec-2001.) |

Theorem | euan 2309 | Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |

Theorem | moanimv 2310* | Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 23-Mar-1995.) |

Theorem | moaneu 2311 | Nested "at most one" and uniqueness quantifiers. (Contributed by NM, 25-Jan-2006.) |

Theorem | moanmo 2312 | Nested "at most one" quantifiers. (Contributed by NM, 25-Jan-2006.) |

Theorem | euanv 2313* | Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 23-Mar-1995.) |

Theorem | mopick 2314 | "At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.) |

Theorem | eupick 2315 | Existential uniqueness "picks" a variable value for which another wff is true. If there is only one thing such that is true, and there is also an (actually the same one) such that and are both true, then implies regardless of . This theorem can be useful for eliminating existential quantifiers in a hypothesis. Compare Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by NM, 10-Jul-1994.) |

Theorem | eupicka 2316 | Version of eupick 2315 with closed formulas. (Contributed by NM, 6-Sep-2008.) |

Theorem | eupickb 2317 | Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.) |

Theorem | eupickbi 2318 | Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) |

Theorem | mopick2 2319 | "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 1616. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |

Theorem | euor2 2320 | Introduce or eliminate a disjunct in a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |

Theorem | moexex 2321 | "At most one" double quantification. (Contributed by NM, 3-Dec-2001.) |

Theorem | moexexv 2322* | "At most one" double quantification. (Contributed by NM, 26-Jan-1997.) |

Theorem | 2moex 2323 | Double quantification with "at most one." (Contributed by NM, 3-Dec-2001.) |

Theorem | 2euex 2324 | Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |

Theorem | 2eumo 2325 | Double quantification with existential uniqueness and "at most one." (Contributed by NM, 3-Dec-2001.) |

Theorem | 2eu2ex 2326 | Double existential uniqueness. (Contributed by NM, 3-Dec-2001.) |

Theorem | 2moswap 2327 | A condition allowing swap of "at most one" and existential quantifiers. (Contributed by NM, 10-Apr-2004.) |

Theorem | 2euswap 2328 | A condition allowing swap of uniqueness and existential quantifiers. (Contributed by NM, 10-Apr-2004.) |

Theorem | 2exeu 2329 | Double existential uniqueness implies double uniqueness quantification. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) |

Theorem | 2mo 2330* | Two equivalent expressions for double "at most one." (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) |

Theorem | 2mos 2331* | Double "exists at most one", using implicit substitution. (Contributed by NM, 10-Feb-2005.) |

Theorem | 2eu1 2332 | Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one. (Contributed by NM, 3-Dec-2001.) |

Theorem | 2eu2 2333 | Double existential uniqueness. (Contributed by NM, 3-Dec-2001.) |

Theorem | 2eu3 2334 | Double existential uniqueness. (Contributed by NM, 3-Dec-2001.) |

Theorem | 2eu4 2335* | This theorem provides us with a definition of double existential uniqueness ("exactly one and exactly one "). Naively one might think (incorrectly) that it could be defined by . See 2eu1 2332 for a condition under which the naive definition holds and 2exeu 2329 for a one-way implication. See 2eu5 2336 and 2eu8 2339 for alternate definitions. (Contributed by NM, 3-Dec-2001.) |

Theorem | 2eu5 2336* | An alternate definition of double existential uniqueness (see 2eu4 2335). A mistake sometimes made in the literature is to use to mean "exactly one and exactly one ." (For example, see Proposition 7.53 of [TakeutiZaring] p. 53.) It turns out that this is actually a weaker assertion, as can be seen by expanding out the formal definitions. This theorem shows that the erroneous definition can be repaired by conjoining as an additional condition. The correct definition apparently has never been published. ( means "exists at most one.") (Contributed by NM, 26-Oct-2003.) |

Theorem | 2eu6 2337* | Two equivalent expressions for double existential uniqueness. (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) |

Theorem | 2eu7 2338 | Two equivalent expressions for double existential uniqueness. (Contributed by NM, 19-Feb-2005.) |

Theorem | 2eu8 2339 | Two equivalent expressions for double existential uniqueness. Curiously, we can put on either of the internal conjuncts but not both. We can also commute using 2eu7 2338. (Contributed by NM, 20-Feb-2005.) |

Theorem | euequ1 2340* | Equality has existential uniqueness. Special case of eueq1 3065 proved using only predicate calculus. (Contributed by Stefan Allan, 4-Dec-2008.) |

Theorem | exists1 2341* | 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; see theorem dtru 4348. (Contributed by NM, 5-Apr-2004.) |

Theorem | exists2 2342 | A condition implying that at least two things exist. (Contributed by NM, 10-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |

1.8 Other axiomatizations related to classical
predicate calculus | ||

1.8.1 Predicate calculus with all distinct
variables | ||

Axiom | ax-7d 2343* | Distinct variable version of ax-7 1745. (Contributed by Mario Carneiro, 14-Aug-2015.) |

Axiom | ax-8d 2344* | Distinct variable version of ax-8 1683. (Contributed by Mario Carneiro, 14-Aug-2015.) |

Axiom | ax-9d1 2345 | Distinct variable version of ax9 1949, equal variables case. (Contributed by Mario Carneiro, 14-Aug-2015.) |

Axiom | ax-9d2 2346* | Distinct variable version of ax9 1949, distinct variables case. (Contributed by Mario Carneiro, 14-Aug-2015.) |

Axiom | ax-10d 2347* | Distinct variable version of ax10 1991. (Contributed by Mario Carneiro, 14-Aug-2015.) |

Axiom | ax-11d 2348* | Distinct variable version of ax-11 1757. (Contributed by Mario Carneiro, 14-Aug-2015.) |

1.8.2 Aristotelian logic: Assertic
syllogismsModel 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 mpto1 1539) 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,
"There is a surprising amount of scholarly debate
about how best to formalize Aristotle's syllogisms..." according to
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 .
Our translation is essentially identical to the one
use in Rini page 18, Table 2 "Non-Modal Syllogisms in
Lower Predicate Calculus (LPC)", which uses
standard predicate logic with predicates. Rini states,
"the crucial point is that we capture the meaning Aristotle intends,
and the method by which we represent that meaning is less important."
There are two differences: we make the existence criteria explicit, and
we use , , and in the order they
appear
(a common Metamath convention).
Patzig also uses standard predicate logic notation and predicates
(though he interprets them as conditional propositions, not as
inference rules); see
Gunther Patzig, Expressions of the form "no is " are consistently translated as . These can also be expressed as , per alinexa 1585. We translate "all is " to , "some is " to , and "some is not " to . It is traditional to use the singular verb "is", not the plural verb "are", in the generic expressions. By convention the major premise is listed first. 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 , , or . Here we use wff variables instead of specialized restricted forms. This generalization makes the syllogisms more useful in more circumstances. In addition, these expressions make it clearer that the syllogisms of Aristolean logic are the forerunners of predicate calculus. If we used restricted forms like instead, we would not only unnecessarily limit their use, but we would also need to use set and class axioms, making their relationship to predicate calculus less clear.
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 non-existent
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 non-existent 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),
These are only the 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 2349 | "Barbara", one of the fundamental syllogisms of Aristotelian logic. All is , and all is , therefore all is . (In Aristotelian notation, AAA-1: MaP and SaM therefore SaP.) For example, given "All men are mortal" and "Socrates is a man", we can prove "Socrates is mortal". If H is the set of men, M is the set of mortal beings, and S is Socrates, these word phrases can be represented as (all men are mortal) and (Socrates is a man) therefore (Socrates is mortal). Russell and Whitehead note that the "syllogism in Barbara is derived..." from syl 16. (quote after Theorem *2.06 of [WhiteheadRussell] p. 101). Most of the proof is in alsyl 1622. There are a legion of sources for Barbara, including http://www.friesian.com/aristotl.htm, http://plato.stanford.edu/entries/aristotle-logic/, and https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.) |

Theorem | celarent 2350 | "Celarent", one of the syllogisms of Aristotelian logic. No is , and all is , therefore no is . (In Aristotelian notation, EAE-1: MeP and SaM therefore SeP.) For example, given the "No reptiles have fur" and "All snakes are reptiles", therefore "No snakes have fur". Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) |

Theorem | darii 2351 | "Darii", one of the syllogisms of Aristotelian logic. All is , and some is , therefore some is . (In Aristotelian notation, AII-1: MaP and SiM therefore SiP.) For example, given "All rabbits have fur" and "Some pets are rabbits", therefore "Some pets have fur". Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.) |

Theorem | ferio 2352 | "Ferio" ("Ferioque"), one of the syllogisms of Aristotelian logic. No is , and some is , therefore some is not . (In Aristotelian notation, EIO-1: MeP and SiM therefore SoP.) For example, given "No homework is fun" and "Some reading is homework", therefore "Some reading is not fun". This is essentially a logical axiom in Aristotelian logic. Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) |

Theorem | barbari 2353 | "Barbari", one of the syllogisms of Aristotelian logic. All is , all is , and some exist, therefore some is . (In Aristotelian notation, AAI-1: MaP and SaM therefore SiP.) For example, given "All men are mortal", "All Greeks are men", and "Greeks exist", therefore "Some Greeks are mortal". Note the existence hypothesis (to prove the "some" in the conclusion). Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 30-Aug-2016.) |

Theorem | celaront 2354 | "Celaront", one of the syllogisms of Aristotelian logic. No is , all is , and some exist, therefore some is not . (In Aristotelian notation, EAO-1: MeP and SaM therefore SoP.) For example, given "No reptiles have fur", "All snakes are reptiles.", and "Snakes exist.", prove "Some snakes have no fur". Note the existence hypothesis. Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) |

Theorem | cesare 2355 | "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 2350. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 13-Nov-2016.) |

Theorem | camestres 2356 | "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 2357 | "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 2358 | "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 2359 | "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 2360 | "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 2361 | "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 2362 | "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 2363 | "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 2364 | "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 2362; prefer using that instead. (Contributed by David A. Wheeler, 28-Aug-2016.) (New usage is discouraged.) |

Theorem | felapton 2365 | "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 2366 | "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 2367 | "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 2368 | "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 2351 with positions interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.) |

Theorem | fresison 2369 | "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 2370 | "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 2371 | "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 2372 | "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 2353. (Contributed by David A. Wheeler, 28-Aug-2016.) |

1.8.3 Intuitionistic logicIntuitionistic (constructive) logic is similar to classical logic with the notable omission of ax-3 7 and theorems such as exmid 405 or peirce 174. We mostly treat intuitionistic logic in a separate file, iset.mm, which is known as the Intuitionistic Logic Explorer on the web site. However, iset.mm has a number of additional axioms (mainly to replace definitions like df-or 360 and df-ex 1548 which are not valid in intitionistic logic) and we want to prove those axioms here to demonstrate that adding those axioms in iset.mm does not make iset.mm any less consistent than set.mm. | ||

Theorem | axi4 2373 | Specialization (intuitionistic logic axiom ax-4). This is just sp 1759 by another name. (Contributed by Jim Kingdon, 31-Dec-2017.) |

Theorem | axi5r 2374 | Converse of ax-5o (intuitionistic logic axiom ax-i5r). (Contributed by Jim Kingdon, 31-Dec-2017.) |

Theorem | axial 2375 | is not free in (intuitionistic logic axiom ax-ial). (Contributed by Jim Kingdon, 31-Dec-2017.) |

Theorem | axie1 2376 | is bound in (intuitionistic logic axiom ax-ie1). (Contributed by Jim Kingdon, 31-Dec-2017.) |

Theorem | axie2 2377 | A key property of existential quantification (intuitionistic logic axiom ax-ie2). (Contributed by Jim Kingdon, 31-Dec-2017.) |

Theorem | axi9 2378 | Axiom of existence (intuitionistic logic axiom ax-i9). In classical logic, this is equivalent to ax-9 1662 but in intuitionistic logic it needs to be stated using the existential quantifier. (Contributed by Jim Kingdon, 31-Dec-2017.) |

Theorem | axi10 2379 | Axiom of Quantifier Substitution (intuitionistic logic axiom ax-10). This is just ax10 1991 by another name. (Contributed by Jim Kingdon, 31-Dec-2017.) |

Theorem | axi11e 2380 | Axiom of Variable Substitution for Existence (intuitionistic logic axiom ax-i11e). This can be derived from ax-11 1757 in a classical context but a separate axiom is needed for intuitionistic predicate calculus. (Contributed by Jim Kingdon, 31-Dec-2017.) |

Theorem | axi12 2381 |
Axiom of Quantifier Introduction (intuitionistic logic axiom ax-i12).
In classical logic, this is mostly a restatement of ax12o 1976 (with one additional quantifier). But in intuitionistic logic, changing the negations and implications to disjunctions makes it stronger. (Contributed by Jim Kingdon, 31-Dec-2017.) |

Theorem | axbnd 2382 |
Axiom of Bundling (intuitionistic logic axiom ax-bnd).
In classical logic, this and axi12 2381 are fairly straightforward consequences of ax12o 1976. But in intuitionistic logic, it is not easy to add the extra to axi12 2381 and so we treat the two as separate axioms. (Contributed by Jim Kingdon, 22-Mar-2018.) |

PART 2 ZF (ZERMELO-FRAENKEL) SET
THEORYSet theory uses the formalism of propositional and predicate calculus to assert properties of arbitrary mathematical objects called "sets." A set can be contained in 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. A simplistic concept of sets, sometimes called "naive set theory", is vulnerable to a paradox called "Russell's paradox" (ru 3118), a discovery that revolutionized the foundations of mathematics and logic. Russell's Paradox spawned the development of set theories that countered the paradox, including the ZF set theory that is most widely used and is defined here. Except for Extensionality, the axioms basically say, "given an arbitrary set x (and, in the cases of Replacement and Regularity, provided that an antecedent is satisfied), there exists another set y based on or constructed from it, with the stated properties." (The axiom of Extensionality can also be restated this way as shown by axext2 2384.) The individual axiom links provide more detailed descriptions. We derive the redundant ZF axioms of Separation, Null Set, and Pairing from the others as theorems. | ||

2.1 ZF Set Theory - start with the Axiom of
Extensionality | ||

2.1.1 Introduce the Axiom of
Extensionality | ||

Axiom | ax-ext 2383* |
Axiom of Extensionality. An axiom of Zermelo-Fraenkel set theory. It
states that two sets are identical if they contain the same elements.
Axiom Ext of [BellMachover] p.
461.
Set theory can also be formulated with a To use the above "equality-free" version of Extensionality with Metamath's logical axioms, we would rewrite ax-8 1683 through ax-16 2192 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 2383 can represent any actual
variable |

Theorem | axext2 2384* | The Axiom of Extensionality (ax-ext 2383) restated so that it postulates the existence of a set given two arbitrary sets and . This way to express it follows the general idea of the other ZFC axioms, which is to postulate the existence of sets given other sets. (Contributed by NM, 28-Sep-2003.) |

Theorem | axext3 2385* | 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 2386* | 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 2383 and df-cleq 2395. (Contributed by NM, 14-Nov-2008.) |

Theorem | bm1.1 2387* | 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.) |

2.1.2 Class abstractions (a.k.a. class
builders) | ||

Syntax | cab 2388 | Introduce the class builder or class abstraction notation ("the class of sets such that is true"). Our class variables , , etc. range over class builders (implicitly in the case of defined class terms such as df-nul 3587). Note that a set variable can be expressed as a class builder per theorem cvjust 2397, justifying the assignment of set variables to class variables via the use of cv 1648. |

Definition | df-clab 2389 |
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 1721, which extends or "overloads" the wel 1722 definition connecting set variables, requires that both sides of be a class. In df-cleq 2395 and df-clel 2398, 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 set variable. Syntax definition cv 1648 allows us to substitute a set variable for a class variable: all sets are classes by cvjust 2397 (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 2507 for a quick overview). Because class variables can be substituted with compound expressions and set variables cannot, it is often useful to convert a theorem containing a free set variable to a more general version with a class variable. This is done with theorems such as vtoclg 2969 which is used, for example, to convert elirrv 7519 to elirr 7520. 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 2390 | Simplification of class abstraction notation when the free and bound variables are identical. (Contributed by NM, 5-Aug-1993.) |

Theorem | hbab1 2391* | Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 5-Aug-1993.) |

Theorem | nfsab1 2392* | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |

Theorem | hbab 2393* | Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995.) |

Theorem | nfsab 2394* | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |

Definition | df-cleq 2395* |
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 2386). 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
= 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 2389, df-clel 2398, and abeq2 2507. In the form of dfcleq 2396, 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 2396* | The same as df-cleq 2395 with the hypothesis removed using the Axiom of Extensionality ax-ext 2383. (Contributed by NM, 15-Sep-1993.) |

Theorem | cvjust 2397* | Every set is a class. Proposition 4.9 of [TakeutiZaring] p. 13. This theorem shows that a set variable can be expressed as a class abstraction. This provides a motivation for the class syntax construction cv 1648, which allows us to substitute a set variable for a class variable. See also cab 2388 and df-clab 2389. Note that this is not a rigorous justification, because cv 1648 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 2398* |
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 2395 it extends or "overloads" the
use of the existing membership symbol, but unlike df-cleq 2395 it does not
strengthen the set of valid wffs of logic when the class variables are
replaced with set variables (see cleljust 2062), so we don't include any
set theory axiom as a hypothesis. See also comments about the syntax
under df-clab 2389. Alternate definitions of
(but that require
either or to be a set) are shown by
clel2 3030, clel3 3032, and
clel4 3033.
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 2399* | Infer equality of classes from equivalence of membership. (Contributed by NM, 5-Aug-1993.) |

Theorem | eqrdv 2400* | Deduce equality of classes from equivalence of membership. (Contributed by NM, 17-Mar-1996.) |

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