P. 22 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 2101-2200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	eupickbi 2101	Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.)
|- ( E! x ph -> ( E. x ( ph /\ ps ) <-> A. x ( ph -> ps ) ) )
Theorem	mopick2 2102	"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 1624. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- ( ( E* x ph /\ E. x ( ph /\ ps ) /\ E. x ( ph /\ ch ) ) -> E. x ( ph /\ ps /\ ch ) )
Theorem	moexexdc 2103	"At most one" double quantification. (Contributed by Jim Kingdon, 5-Jul-2018.)
|- F/ y ph   =>    |- (DECID E. x ph -> ( ( E* x ph /\ A. x E* y ps ) -> E* y E. x ( ph /\ ps ) ) )
Theorem	euexex 2104	Existential uniqueness and "at most one" double quantification. (Contributed by Jim Kingdon, 28-Dec-2018.)
|- F/ y ph   =>    |- ( ( E! x ph /\ A. x E* y ps ) -> E* y E. x ( ph /\ ps ) )
Theorem	2moex 2105	Double quantification with "at most one". (Contributed by NM, 3-Dec-2001.)
|- ( E* x E. y ph -> A. y E* x ph )
Theorem	2euex 2106	Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- ( E! x E. y ph -> E. y E! x ph )
Theorem	2eumo 2107	Double quantification with existential uniqueness and "at most one." (Contributed by NM, 3-Dec-2001.)
|- ( E! x E* y ph -> E* x E! y ph )
Theorem	2eu2ex 2108	Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
|- ( E! x E! y ph -> E. x E. y ph )
Theorem	2moswapdc 2109	A condition allowing swap of "at most one" and existential quantifiers. (Contributed by Jim Kingdon, 6-Jul-2018.)
|- (DECID E. x E. y ph -> ( A. x E* y ph -> ( E* x E. y ph -> E* y E. x ph ) ) )
Theorem	2euswapdc 2110	A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Jim Kingdon, 7-Jul-2018.)
|- (DECID E. x E. y ph -> ( A. x E* y ph -> ( E! x E. y ph -> E! y E. x ph ) ) )
Theorem	2exeu 2111	Double existential uniqueness implies double unique existential quantification. (Contributed by NM, 3-Dec-2001.)
|- ( ( E! x E. y ph /\ E! y E. x ph ) -> E! x E! y ph )
Theorem	2eu4 2112*	This theorem provides us with a definition of double existential uniqueness ("exactly one x and exactly one y"). Naively one might think (incorrectly) that it could be defined by E! x E! y ph. See 2exeu 2111 for a one-way implication. (Contributed by NM, 3-Dec-2001.)
|- ( ( E! x E. y ph /\ E! y E. x ph ) <-> ( E. x E. y ph /\ E. z E. w A. x A. y ( ph -> ( x = z /\ y = w ) ) ) )
Theorem	2eu7 2113	Two equivalent expressions for double existential uniqueness. (Contributed by NM, 19-Feb-2005.)
|- ( ( E! x E. y ph /\ E! y E. x ph ) <-> E! x E! y ( E. x ph /\ E. y ph ) )
Theorem	euequ1 2114*	Equality has existential uniqueness. (Contributed by Stefan Allan, 4-Dec-2008.)
|- E! x x = y
Theorem	exists1 2115*	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.)
|- ( E! x x = x <-> A. x x = y )
Theorem	exists2 2116	A condition implying that at least two things exist. (Contributed by NM, 10-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
|- ( ( E. x ph /\ E. x -. ph ) -> -. E! x x = x )
1.4.7  Aristotelian logic: Assertic syllogisms 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 1418) 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 x. 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 ph, ps, and ch 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, Aristotle's Theory of the Syllogism second edition, 1963, English translation by Jonathan Barnes, 1968, page 38. Terms such as "all" and "some" are translated into predicate logic using the aproach devised by Frege and Russell. "Frege (and Russell) devised an ingenious procedure for regimenting binary quantifiers like "every" and "some" in terms of unary quantifiers like "everything" and "something": they formalized sentences of the form "Some A is B" and "Every A is B" as exists x (Ax and Bx) and all x (Ax implies Bx), respectively." "Quantifiers and Quantification", Stanford Encyclopedia of Philosophy, https://plato.stanford.edu/entries/quantification/ 1418. See Principia Mathematica page 22 and *10 for more information (especially *10.3 and *10.26). Expressions of the form "no ph is ps " are consistently translated as A. x ( ph -> -. ps ). These can also be expressed as -. E. x ( ph /\ ps ), per alinexa 1596. We translate "all ph is ps " to A. x ( ph -> ps ), "some ph is ps " to E. x ( ph /\ ps ), and "some ph is not ps " to E. x ( ph /\ -. ps ). 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 x = A, x e. A, or x C_ A. 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 x e. A 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 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/ 1596. 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 2121, celaront 2122, cesaro 2127, camestros 2128, felapton 2133, darapti 2134, calemos 2138, fesapo 2139, and bamalip 2140. 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 2117	"Barbara", one of the fundamental syllogisms of Aristotelian logic. All ph is ps, and all ch is ph, therefore all ch is ps. (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 A. x ( x e. H -> x e. M ) (all men are mortal) and A. x ( x = S -> x e. H ) (Socrates is a man) therefore A. x ( x = S -> x e. M ) (Socrates is mortal). Russell and Whitehead note that the "syllogism in Barbara is derived..." from syl 14. (quote after Theorem *2.06 of [WhiteheadRussell] p. 101). Most of the proof is in alsyl 1628. There are a legion of sources for Barbara, including https://www.friesian.com/aristotl.htm 1628, https://plato.stanford.edu/entries/aristotle-logic/ 1628, and https://en.wikipedia.org/wiki/Syllogism 1628. (Contributed by David A. Wheeler, 24-Aug-2016.)
|- A. x ( ph -> ps )   &    |- A. x ( ch -> ph )   =>    |- A. x ( ch -> ps )
Theorem	celarent 2118	"Celarent", one of the syllogisms of Aristotelian logic. No ph is ps, and all ch is ph, therefore no ch is ps. (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.)
|- A. x ( ph -> -. ps )   &    |- A. x ( ch -> ph )   =>    |- A. x ( ch -> -. ps )
Theorem	darii 2119	"Darii", one of the syllogisms of Aristotelian logic. All ph is ps, and some ch is ph, therefore some ch is ps. (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.)
|- A. x ( ph -> ps )   &    |- E. x ( ch /\ ph )   =>    |- E. x ( ch /\ ps )
Theorem	ferio 2120	"Ferio" ("Ferioque"), one of the syllogisms of Aristotelian logic. No ph is ps, and some ch is ph, therefore some ch is not ps. (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.)
|- A. x ( ph -> -. ps )   &    |- E. x ( ch /\ ph )   =>    |- E. x ( ch /\ -. ps )
Theorem	barbari 2121	"Barbari", one of the syllogisms of Aristotelian logic. All ph is ps, all ch is ph, and some ch exist, therefore some ch is ps. (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.)
|- A. x ( ph -> ps )   &    |- A. x ( ch -> ph )   &    |- E. x ch   =>    |- E. x ( ch /\ ps )
Theorem	celaront 2122	"Celaront", one of the syllogisms of Aristotelian logic. No ph is ps, all ch is ph, and some ch exist, therefore some ch is not ps. (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.)
|- A. x ( ph -> -. ps )   &    |- A. x ( ch -> ph )   &    |- E. x ch   =>    |- E. x ( ch /\ -. ps )
Theorem	cesare 2123	"Cesare", one of the syllogisms of Aristotelian logic. No ph is ps, and all ch is ps, therefore no ch is ph. (In Aristotelian notation, EAE-2: PeM and SaM therefore SeP.) Related to celarent 2118. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 13-Nov-2016.)
|- A. x ( ph -> -. ps )   &    |- A. x ( ch -> ps )   =>    |- A. x ( ch -> -. ph )
Theorem	camestres 2124	"Camestres", one of the syllogisms of Aristotelian logic. All ph is ps, and no ch is ps, therefore no ch is ph. (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.)
|- A. x ( ph -> ps )   &    |- A. x ( ch -> -. ps )   =>    |- A. x ( ch -> -. ph )
Theorem	festino 2125	"Festino", one of the syllogisms of Aristotelian logic. No ph is ps, and some ch is ps, therefore some ch is not ph. (In Aristotelian notation, EIO-2: PeM and SiM therefore SoP.) (Contributed by David A. Wheeler, 25-Nov-2016.)
|- A. x ( ph -> -. ps )   &    |- E. x ( ch /\ ps )   =>    |- E. x ( ch /\ -. ph )
Theorem	baroco 2126	"Baroco", one of the syllogisms of Aristotelian logic. All ph is ps, and some ch is not ps, therefore some ch is not ph. (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.)
|- A. x ( ph -> ps )   &    |- E. x ( ch /\ -. ps )   =>    |- E. x ( ch /\ -. ph )
Theorem	cesaro 2127	"Cesaro", one of the syllogisms of Aristotelian logic. No ph is ps, all ch is ps, and ch exist, therefore some ch is not ph. (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.)
|- A. x ( ph -> -. ps )   &    |- A. x ( ch -> ps )   &    |- E. x ch   =>    |- E. x ( ch /\ -. ph )
Theorem	camestros 2128	"Camestros", one of the syllogisms of Aristotelian logic. All ph is ps, no ch is ps, and ch exist, therefore some ch is not ph. (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.)
|- A. x ( ph -> ps )   &    |- A. x ( ch -> -. ps )   &    |- E. x ch   =>    |- E. x ( ch /\ -. ph )
Theorem	datisi 2129	"Datisi", one of the syllogisms of Aristotelian logic. All ph is ps, and some ph is ch, therefore some ch is ps. (In Aristotelian notation, AII-3: MaP and MiS therefore SiP.) (Contributed by David A. Wheeler, 28-Aug-2016.)
|- A. x ( ph -> ps )   &    |- E. x ( ph /\ ch )   =>    |- E. x ( ch /\ ps )
Theorem	disamis 2130	"Disamis", one of the syllogisms of Aristotelian logic. Some ph is ps, and all ph is ch, therefore some ch is ps. (In Aristotelian notation, IAI-3: MiP and MaS therefore SiP.) (Contributed by David A. Wheeler, 28-Aug-2016.)
|- E. x ( ph /\ ps )   &    |- A. x ( ph -> ch )   =>    |- E. x ( ch /\ ps )
Theorem	ferison 2131	"Ferison", one of the syllogisms of Aristotelian logic. No ph is ps, and some ph is ch, therefore some ch is not ps. (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.)
|- A. x ( ph -> -. ps )   &    |- E. x ( ph /\ ch )   =>    |- E. x ( ch /\ -. ps )
Theorem	bocardo 2132	"Bocardo", one of the syllogisms of Aristotelian logic. Some ph is not ps, and all ph is ch, therefore some ch is not ps. (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 2130; prefer using that instead. (Contributed by David A. Wheeler, 28-Aug-2016.) (New usage is discouraged.)
|- E. x ( ph /\ -. ps )   &    |- A. x ( ph -> ch )   =>    |- E. x ( ch /\ -. ps )
Theorem	felapton 2133	"Felapton", one of the syllogisms of Aristotelian logic. No ph is ps, all ph is ch, and some ph exist, therefore some ch is not ps. (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.)
|- A. x ( ph -> -. ps )   &    |- A. x ( ph -> ch )   &    |- E. x ph   =>    |- E. x ( ch /\ -. ps )
Theorem	darapti 2134	"Darapti", one of the syllogisms of Aristotelian logic. All ph is ps, all ph is ch, and some ph exist, therefore some ch is ps. (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.)
|- A. x ( ph -> ps )   &    |- A. x ( ph -> ch )   &    |- E. x ph   =>    |- E. x ( ch /\ ps )
Theorem	calemes 2135	"Calemes", one of the syllogisms of Aristotelian logic. All ph is ps, and no ps is ch, therefore no ch is ph. (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.)
|- A. x ( ph -> ps )   &    |- A. x ( ps -> -. ch )   =>    |- A. x ( ch -> -. ph )
Theorem	dimatis 2136	"Dimatis", one of the syllogisms of Aristotelian logic. Some ph is ps, and all ps is ch, therefore some ch is ph. (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 2119 with positions interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.)
|- E. x ( ph /\ ps )   &    |- A. x ( ps -> ch )   =>    |- E. x ( ch /\ ph )
Theorem	fresison 2137	"Fresison", one of the syllogisms of Aristotelian logic. No ph is ps (PeM), and some ps is ch (MiS), therefore some ch is not ph (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.)
|- A. x ( ph -> -. ps )   &    |- E. x ( ps /\ ch )   =>    |- E. x ( ch /\ -. ph )
Theorem	calemos 2138	"Calemos", one of the syllogisms of Aristotelian logic. All ph is ps (PaM), no ps is ch (MeS), and ch exist, therefore some ch is not ph (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.)
|- A. x ( ph -> ps )   &    |- A. x ( ps -> -. ch )   &    |- E. x ch   =>    |- E. x ( ch /\ -. ph )
Theorem	fesapo 2139	"Fesapo", one of the syllogisms of Aristotelian logic. No ph is ps, all ps is ch, and ps exist, therefore some ch is not ph. (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.)
|- A. x ( ph -> -. ps )   &    |- A. x ( ps -> ch )   &    |- E. x ps   =>    |- E. x ( ch /\ -. ph )
Theorem	bamalip 2140	"Bamalip", one of the syllogisms of Aristotelian logic. All ph is ps, all ps is ch, and ph exist, therefore some ch is ph. (In Aristotelian notation, AAI-4: PaM and MaS therefore SiP.) Like barbari 2121. (Contributed by David A. Wheeler, 28-Aug-2016.)
|- A. x ( ph -> ps )   &    |- A. x ( ps -> ch )   &    |- E. x ph   =>    |- E. x ( ch /\ ph )
1.5  First-order logic with one non-logical binary predicate 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 1496).
Syntax	wcel 2141	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 wff A e. B here is to allow us to prove the wel 2142 of predicate calculus in terms of the wceq 1348 of set theory, so that we do not overload the e. connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. The class variables A and B are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-clab 2157 for more information on the set theory usage of wcel 2141.
wff A e. B
Theorem	wel 2142	Extend wff definition to include atomic formulas with the membership predicate. This is read either " x is an element of y", or " x is a member of y", or " x belongs to y", or " y contains x". Note: The phrase " y includes x " means " x is a subset of y"; to use it also for x e. y, as some authors occasionally do, is poor form and causes confusion, according to George Boolos (1992 lecture at MIT). This syntactical construction introduces a binary non-logical predicate symbol e. into our predicate calculus. We will eventually use it for the membership predicate of set theory, but that is irrelevant at this point: the predicate calculus axioms for e. apply to any arbitrary binary predicate symbol. "Non-logical" means that the predicate is presumed to have additional properties beyond the realm of predicate calculus, although these additional properties are not specified by predicate calculus itself but rather by the axioms of a theory (in our case set theory) added to predicate calculus. "Binary" means that the predicate has two arguments. Instead of introducing wel 2142 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 2141. This lets us avoid overloading the e. connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically wel 2142 is considered to be a primitive syntax, even though here it is artificially "derived" from wcel 2141. Note: To see the proof steps of this syntax proof, type "MM> SHOW PROOF wel / ALL" in the Metamath program. (Contributed by NM, 24-Jan-2006.)
wff x e. y
Axiom	ax-13 2143	Axiom of left equality for the binary predicate e.. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the left-hand side of the e. binary predicate. Axiom scheme C12' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Axiom B8 (p. 75) of system S2 of [Tarski] p. 77. "Non-logical" means that the predicate is not a primitive of predicate calculus proper but instead is an extension to it. "Binary" means that the predicate has two arguments. In a system of predicate calculus with equality, like ours, equality is not usually considered to be a non-logical predicate. In systems of predicate calculus without equality, it typically would be. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( x e. z -> y e. z ) )
Axiom	ax-14 2144	Axiom of right equality for the binary predicate e.. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the right-hand side of the e. binary predicate. Axiom scheme C13' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Axiom B8 (p. 75) of system S2 of [Tarski] p. 77. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( z e. x -> z e. y ) )
Theorem	elequ1 2145	An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( x e. z <-> y e. z ) )
Theorem	elequ2 2146	An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)
|- ( x = y -> ( z e. x <-> z e. y ) )
Theorem	cleljust 2147*	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 2142 with the class variables in wcel 2141. (Contributed by NM, 28-Jan-2004.)
|- ( x e. y <-> E. z ( z = x /\ z e. y ) )
Theorem	elsb1 2148*	Substitution for the first argument of the non-logical predicate in an atomic formula. See elsb2 2149 for substitution for the second argument. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
|- ( [ y / x ] x e. z <-> y e. z )
Theorem	elsb2 2149*	Substitution for the second argument of the non-logical predicate in an atomic formula. See elsb1 2148 for substitution for the first argument. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
|- ( [ y / x ] z e. x <-> z e. y )
Theorem	dveel1 2150*	Quantifier introduction when one pair of variables is disjoint. (Contributed by NM, 2-Jan-2002.)
|- ( -. A. x x = y -> ( y e. z -> A. x y e. z ) )
Theorem	dveel2 2151*	Quantifier introduction when one pair of variables is disjoint. (Contributed by NM, 2-Jan-2002.)
|- ( -. A. x x = y -> ( z e. y -> A. x z e. y ) )
PART 2  SET THEORY 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 e. 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 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 1756) and our definition of the nonfreeness predicate (df-nf 1454), 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 13847.
2.1  IZF Set Theory - start with the Axiom of Extensionality
2.1.1  Introduce the Axiom of Extensionality
Axiom	ax-ext 2152*	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 e. on top of traditional predicate calculus without equality. In that case the Axiom of Extensionality becomes ( A. w ( w e. x <-> w e. y ) -> ( x e. z -> y e. z ) ), and equality x = y is defined as A. w ( w e. x <-> w e. y ). 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 1497 through ax-16 1807 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 x in ax-ext 2152 can represent any actual variable v1, v2, v3,... . Distinct variable restrictions ($d) prevent us from substituting say v1 for both x and z. 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 2152 scheme exactly logically equivalent to each other and in particular to the actual axiom of the textbook version. (Contributed by NM, 5-Aug-1993.)
|- ( A. z ( z e. x <-> z e. y ) -> x = y )
Theorem	axext3 2153*	A generalization of the Axiom of Extensionality in which x and y need not be distinct. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
|- ( A. z ( z e. x <-> z e. y ) -> x = y )
Theorem	axext4 2154*	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 2152. (Contributed by NM, 14-Nov-2008.)
|- ( x = y <-> A. z ( z e. x <-> z e. y ) )
Theorem	bm1.1 2155*	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.)
|- F/ x ph   =>    |- ( E. x A. y ( y e. x <-> ph ) -> E! x A. y ( y e. x <-> ph ) )
2.1.2  Class abstractions (a.k.a. class builders)
Syntax	cab 2156	Introduce the class builder or class abstraction notation ("the class of sets x such that ph is true"). Our class variables A, B, etc. range over class builders (sometimes implicitly). Note that a setvar variable can be expressed as a class builder per Theorem cvjust 2165, justifying the assignment of setvar variables to class variables via the use of cv 1347.
class { x | ph }
Definition	df-clab 2157	Define class abstraction notation (so-called by Quine), also called a "class builder" in the literature. x and y need not be distinct. Definition 2.1 of [Quine] p. 16. Typically, ph will have y as a free variable, and " { y | ph } " is read "the class of all sets y such that ph ( y ) is true". We do not define { y | ph } in isolation but only as part of an expression that extends or "overloads" the e. relationship. This is our first use of the e. symbol to connect classes instead of sets. The syntax definition wcel 2141, which extends or "overloads" the wel 2142 definition connecting setvar variables, requires that both sides of e. be a class. In df-cleq 2163 and df-clel 2166, we introduce a new kind of variable (class variable) that can substituted with expressions such as { y | ph }. In the present definition, the x on the left-hand side is a setvar variable. Syntax definition cv 1347 allows us to substitute a setvar variable x for a class variable: all sets are classes by cvjust 2165 (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 2279 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 { y | ph } a "class term". For a general discussion of the theory of classes, see https://us.metamath.org/mpeuni/mmset.html#class 2279. (Contributed by NM, 5-Aug-1993.)
|- ( x e. { y | ph } <-> [ x / y ] ph )
Theorem	abid 2158	Simplification of class abstraction notation when the free and bound variables are identical. (Contributed by NM, 5-Aug-1993.)
|- ( x e. { x | ph } <-> ph )
Theorem	hbab1 2159*	Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 5-Aug-1993.)
|- ( y e. { x | ph } -> A. x y e. { x | ph } )
Theorem	nfsab1 2160*	Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ x y e. { x | ph }
Theorem	hbab 2161*	Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995.)
|- ( ph -> A. x ph )   =>    |- ( z e. { y | ph } -> A. x z e. { y | ph } )
Theorem	nfsab 2162*	Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
|- F/ x ph   =>    |- F/ x z e. { y | ph }
Definition	df-cleq 2163*	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 y = z <-> A. x ( x e. y <-> x e. z ), which is not a theorem of logic but rather presupposes the Axiom of Extensionality (see Theorem axext4 2154). 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 x =2 y <-> x = y 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 2157, df-clel 2166, and abeq2 2279. In the form of dfcleq 2164, 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 2164. (Contributed by NM, 15-Sep-1993.)
|- ( A. x ( x e. y <-> x e. z ) -> y = z )   =>    |- ( A = B <-> A. x ( x e. A <-> x e. B ) )
Theorem	dfcleq 2164*	The same as df-cleq 2163 with the hypothesis removed using the Axiom of Extensionality ax-ext 2152. (Contributed by NM, 15-Sep-1993.)
|- ( A = B <-> A. x ( x e. A <-> x e. B ) )
Theorem	cvjust 2165*	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 1347, which allows us to substitute a setvar variable for a class variable. See also cab 2156 and df-clab 2157. Note that this is not a rigorous justification, because cv 1347 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.)
|- x = { y | y e. x }
Definition	df-clel 2166*	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 2163 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2163 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 2147), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2157. 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 2157. (Contributed by NM, 5-Aug-1993.)
|- ( A e. B <-> E. x ( x = A /\ x e. B ) )
Theorem	eqriv 2167*	Infer equality of classes from equivalence of membership. (Contributed by NM, 5-Aug-1993.)
|- ( x e. A <-> x e. B )   =>    |- A = B
Theorem	eqrdv 2168*	Deduce equality of classes from equivalence of membership. (Contributed by NM, 17-Mar-1996.)
|- ( ph -> ( x e. A <-> x e. B ) )   =>    |- ( ph -> A = B )
Theorem	eqrdav 2169*	Deduce equality of classes from an equivalence of membership that depends on the membership variable. (Contributed by NM, 7-Nov-2008.)
|- ( ( ph /\ x e. A ) -> x e. C )   &    |- ( ( ph /\ x e. B ) -> x e. C )   &    |- ( ( ph /\ x e. C ) -> ( x e. A <-> x e. B ) )   =>    |- ( ph -> A = B )
Theorem	eqid 2170	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.)
|- A = A
Theorem	eqidd 2171	Class identity law with antecedent. (Contributed by NM, 21-Aug-2008.)
|- ( ph -> A = A )
Theorem	eqcom 2172	Commutative law for class equality. Theorem 6.5 of [Quine] p. 41. (Contributed by NM, 5-Aug-1993.)
|- ( A = B <-> B = A )
Theorem	eqcoms 2173	Inference applying commutative law for class equality to an antecedent. (Contributed by NM, 5-Aug-1993.)
|- ( A = B -> ph )   =>    |- ( B = A -> ph )
Theorem	eqcomi 2174	Inference from commutative law for class equality. (Contributed by NM, 5-Aug-1993.)
|- A = B   =>    |- B = A
Theorem	neqcomd 2175	Commute an inequality. (Contributed by Rohan Ridenour, 3-Aug-2023.)
|- ( ph -> -. A = B )   =>    |- ( ph -> -. B = A )
Theorem	eqcomd 2176	Deduction from commutative law for class equality. (Contributed by NM, 15-Aug-1994.)
|- ( ph -> A = B )   =>    |- ( ph -> B = A )
Theorem	eqeq1 2177	Equality implies equivalence of equalities. (Contributed by NM, 5-Aug-1993.)
|- ( A = B -> ( A = C <-> B = C ) )
Theorem	eqeq1i 2178	Inference from equality to equivalence of equalities. (Contributed by NM, 5-Aug-1993.)
|- A = B   =>    |- ( A = C <-> B = C )
Theorem	eqeq1d 2179	Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.)
|- ( ph -> A = B )   =>    |- ( ph -> ( A = C <-> B = C ) )
Theorem	eqeq2 2180	Equality implies equivalence of equalities. (Contributed by NM, 5-Aug-1993.)
|- ( A = B -> ( C = A <-> C = B ) )
Theorem	eqeq2i 2181	Inference from equality to equivalence of equalities. (Contributed by NM, 5-Aug-1993.)
|- A = B   =>    |- ( C = A <-> C = B )
Theorem	eqeq2d 2182	Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.)
|- ( ph -> A = B )   =>    |- ( ph -> ( C = A <-> C = B ) )
Theorem	eqeq12 2183	Equality relationship among 4 classes. (Contributed by NM, 3-Aug-1994.)
|- ( ( A = B /\ C = D ) -> ( A = C <-> B = D ) )
Theorem	eqeq12i 2184	A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- A = B   &    |- C = D   =>    |- ( A = C <-> B = D )
Theorem	eqeq12d 2185	A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( ph -> A = B )   &    |- ( ph -> C = D )   =>    |- ( ph -> ( A = C <-> B = D ) )
Theorem	eqeqan12d 2186	A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( ph -> A = B )   &    |- ( ps -> C = D )   =>    |- ( ( ph /\ ps ) -> ( A = C <-> B = D ) )
Theorem	eqeqan12rd 2187	A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.)
|- ( ph -> A = B )   &    |- ( ps -> C = D )   =>    |- ( ( ps /\ ph ) -> ( A = C <-> B = D ) )
Theorem	eqtr 2188	Transitive law for class equality. Proposition 4.7(3) of [TakeutiZaring] p. 13. (Contributed by NM, 25-Jan-2004.)
|- ( ( A = B /\ B = C ) -> A = C )
Theorem	eqtr2 2189	A transitive law for class equality. (Contributed by NM, 20-May-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- ( ( A = B /\ A = C ) -> B = C )
Theorem	eqtr3 2190	A transitive law for class equality. (Contributed by NM, 20-May-2005.)
|- ( ( A = C /\ B = C ) -> A = B )
Theorem	eqtri 2191	An equality transitivity inference. (Contributed by NM, 5-Aug-1993.)
|- A = B   &    |- B = C   =>    |- A = C
Theorem	eqtr2i 2192	An equality transitivity inference. (Contributed by NM, 21-Feb-1995.)
|- A = B   &    |- B = C   =>    |- C = A
Theorem	eqtr3i 2193	An equality transitivity inference. (Contributed by NM, 6-May-1994.)
|- A = B   &    |- A = C   =>    |- B = C
Theorem	eqtr4i 2194	An equality transitivity inference. (Contributed by NM, 5-Aug-1993.)
|- A = B   &    |- C = B   =>    |- A = C
Theorem	3eqtri 2195	An inference from three chained equalities. (Contributed by NM, 29-Aug-1993.)
|- A = B   &    |- B = C   &    |- C = D   =>    |- A = D
Theorem	3eqtrri 2196	An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- A = B   &    |- B = C   &    |- C = D   =>    |- D = A
Theorem	3eqtr2i 2197	An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.)
|- A = B   &    |- C = B   &    |- C = D   =>    |- A = D
Theorem	3eqtr2ri 2198	An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- A = B   &    |- C = B   &    |- C = D   =>    |- D = A
Theorem	3eqtr3i 2199	An inference from three chained equalities. (Contributed by NM, 6-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)
|- A = B   &    |- A = C   &    |- B = D   =>    |- C = D
Theorem	3eqtr3ri 2200	An inference from three chained equalities. (Contributed by NM, 15-Aug-2004.)
|- A = B   &    |- A = C   &    |- B = D   =>    |- D = C