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Theorem List for Intuitionistic Logic Explorer - 1901-2000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsb8euh 1901 Variable substitution in uniqueness quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Andrew Salmon, 9-Jul-2011.)
(φyφ)       (∃!xφ∃!y[y / x]φ)

Theoremcbveu 1902 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
yφ    &   xψ    &   (x = y → (φψ))       (∃!xφ∃!yψ)

Theoremeu1 1903* An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.)
(φyφ)       (∃!xφx(φ y([y / x]φx = y)))

Theoremeuor 1904 Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.)
(φxφ)       ((¬ φ ∃!xψ) → ∃!x(φ ψ))

Theoremeuorv 1905* Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)
((¬ φ ∃!xψ) → ∃!x(φ ψ))

Theoremmo2n 1906* There is at most one of something which does not exist. (Contributed by Jim Kingdon, 2-Jul-2018.)
yφ       xφyx(φx = y))

Theoremmon 1907 There is at most one of something which does not exist. (Contributed by Jim Kingdon, 5-Jul-2018.)
xφ∃*xφ)

Theoremeuex 1908 Existential uniqueness implies existence. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(∃!xφxφ)

Theoremeumo0 1909* Existential uniqueness implies "at most one." (Contributed by NM, 8-Jul-1994.)
(φyφ)       (∃!xφyx(φx = y))

Theoremeumo 1910 Existential uniqueness implies "at most one." (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.)
(∃!xφ∃*xφ)

Theoremeumoi 1911 "At most one" inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.)
∃!xφ       ∃*xφ

Theoremmobidh 1912 Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by NM, 8-Mar-1995.)
(φxφ)    &   (φ → (ψχ))       (φ → (∃*xψ∃*xχ))

Theoremmobid 1913 Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by NM, 8-Mar-1995.)
xφ    &   (φ → (ψχ))       (φ → (∃*xψ∃*xχ))

Theoremmobidv 1914* Formula-building rule for "at most one" quantifier (deduction rule). (Contributed by Mario Carneiro, 7-Oct-2016.)
(φ → (ψχ))       (φ → (∃*xψ∃*xχ))

Theoremmobii 1915 Formula-building rule for "at most one" quantifier (inference rule). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.)
(ψχ)       (∃*xψ∃*xχ)

Theoremhbmo1 1916 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 8-Mar-1995.)
(∃*xφx∃*xφ)

Theoremhbmo 1917 Bound-variable hypothesis builder for "at most one." (Contributed by NM, 9-Mar-1995.)
(φxφ)       (∃*yφx∃*yφ)

Theoremcbvmo 1918 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.)
yφ    &   xψ    &   (x = y → (φψ))       (∃*xφ∃*yψ)

Theoremmo23 1919* An implication between two definitions of "there exists at most one." (Contributed by Jim Kingdon, 25-Jun-2018.)
yφ       (yx(φx = y) → xy((φ [y / x]φ) → x = y))

Theoremmor 1920* Converse of mo23 1919 with an additional xφ condition. (Contributed by Jim Kingdon, 25-Jun-2018.)
yφ       (xφ → (xy((φ [y / x]φ) → x = y) → yx(φx = y)))

Theoremmodc 1921* Equivalent definitions of "there exists at most one," given decidable existence. (Contributed by Jim Kingdon, 1-Jul-2018.)
yφ       (DECID xφ → (yx(φx = y) ↔ xy((φ [y / x]φ) → x = y)))

Theoremeu2 1922* An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.)
yφ       (∃!xφ ↔ (xφ xy((φ [y / x]φ) → x = y)))

Theoremeu3h 1923* An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) (New usage is discouraged.)
(φyφ)       (∃!xφ ↔ (xφ yx(φx = y)))

Theoremeu3 1924* An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.)
yφ       (∃!xφ ↔ (xφ yx(φx = y)))

Theoremeu5 1925 Uniqueness in terms of "at most one." (Contributed by NM, 23-Mar-1995.) (Proof rewritten by Jim Kingdon, 27-May-2018.)
(∃!xφ ↔ (xφ ∃*xφ))

Theoremexmoeu2 1926 Existence implies "at most one" is equivalent to uniqueness. (Contributed by NM, 5-Apr-2004.)
(xφ → (∃*xφ∃!xφ))

Theoremmoabs 1927 Absorption of existence condition by "at most one." (Contributed by NM, 4-Nov-2002.)
(∃*xφ ↔ (xφ∃*xφ))

Theoremexmodc 1928 If existence is decidable, something exists or at most one exists. (Contributed by Jim Kingdon, 30-Jun-2018.)
(DECID xφ → (xφ ∃*xφ))

Theoremexmonim 1929 There is at most one of something which does not exist. Unlike exmodc 1928 there is no decidability condition. (Contributed by Jim Kingdon, 22-Sep-2018.)
xφ∃*xφ)

Theoremmo2r 1930* A condition which implies "at most one." (Contributed by Jim Kingdon, 2-Jul-2018.)
yφ       (yx(φx = y) → ∃*xφ)

Theoremmo3h 1931* Alternate definition of "at most one." Definition of [BellMachover] p. 460, except that definition has the side condition that y not occur in φ in place of our hypothesis. (Contributed by NM, 8-Mar-1995.) (New usage is discouraged.)
(φyφ)       (∃*xφxy((φ [y / x]φ) → x = y))

Theoremmo3 1932* Alternate definition of "at most one." Definition of [BellMachover] p. 460, except that definition has the side condition that y not occur in φ in place of our hypothesis. (Contributed by NM, 8-Mar-1995.)
yφ       (∃*xφxy((φ [y / x]φ) → x = y))

Theoremmo2dc 1933* Alternate definition of "at most one" where existence is decidable. (Contributed by Jim Kingdon, 2-Jul-2018.)
yφ       (DECID xφ → (∃*xφyx(φx = y)))

Theoremeuan 1934 Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(φxφ)       (∃!x(φ ψ) ↔ (φ ∃!xψ))

Theoremeuanv 1935* Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)
(∃!x(φ ψ) ↔ (φ ∃!xψ))

Theoremeuor2 1936 Introduce or eliminate a disjunct in a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
xφ → (∃!x(φ ψ) ↔ ∃!xψ))

Theoremsbmo 1937* Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.)
([y / x]∃*zφ∃*z[y / x]φ)

Theoremmo4f 1938* "At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.)
xψ    &   (x = y → (φψ))       (∃*xφxy((φ ψ) → x = y))

Theoremmo4 1939* "At most one" expressed using implicit substitution. (Contributed by NM, 26-Jul-1995.)
(x = y → (φψ))       (∃*xφxy((φ ψ) → x = y))

Theoremeu4 1940* Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.)
(x = y → (φψ))       (∃!xφ ↔ (xφ xy((φ ψ) → x = y)))

Theoremexmoeudc 1941 Existence in terms of "at most one" and uniqueness. (Contributed by Jim Kingdon, 3-Jul-2018.)
(DECID xφ → (xφ ↔ (∃*xφ∃!xφ)))

Theoremmoim 1942 "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 22-Apr-1995.)
(x(φψ) → (∃*xψ∃*xφ))

Theoremmoimi 1943 "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 15-Feb-2006.)
(φψ)       (∃*xψ∃*xφ)

Theoremmoimv 1944* Move antecedent outside of "at most one." (Contributed by NM, 28-Jul-1995.)
(∃*x(φψ) → (φ∃*xψ))

Theoremeuimmo 1945 Uniqueness implies "at most one" through implication. (Contributed by NM, 22-Apr-1995.)
(x(φψ) → (∃!xψ∃*xφ))

Theoremeuim 1946 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.)
((xφ x(φψ)) → (∃!xψ∃!xφ))

Theoremmoan 1947 "At most one" is still the case when a conjunct is added. (Contributed by NM, 22-Apr-1995.)
(∃*xφ∃*x(ψ φ))

Theoremmoani 1948 "At most one" is still true when a conjunct is added. (Contributed by NM, 9-Mar-1995.)
∃*xφ       ∃*x(ψ φ)

Theoremmoor 1949 "At most one" is still the case when a disjunct is removed. (Contributed by NM, 5-Apr-2004.)
(∃*x(φ ψ) → ∃*xφ)

Theoremmooran1 1950 "At most one" imports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
((∃*xφ ∃*xψ) → ∃*x(φ ψ))

Theoremmooran2 1951 "At most one" exports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(∃*x(φ ψ) → (∃*xφ ∃*xψ))

Theoremmoanim 1952 Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 3-Dec-2001.)
xφ       (∃*x(φ ψ) ↔ (φ∃*xψ))

Theoremmoanimv 1953* Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 23-Mar-1995.)
(∃*x(φ ψ) ↔ (φ∃*xψ))

Theoremmoaneu 1954 Nested "at most one" and uniqueness quantifiers. (Contributed by NM, 25-Jan-2006.)
∃*x(φ ∃!xφ)

Theoremmoanmo 1955 Nested "at most one" quantifiers. (Contributed by NM, 25-Jan-2006.)
∃*x(φ ∃*xφ)

Theoremmopick 1956 "At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.)
((∃*xφ x(φ ψ)) → (φψ))

Theoremeupick 1957 Existential uniqueness "picks" a variable value for which another wff is true. If there is only one thing x such that φ is true, and there is also an x (actually the same one) such that φ and ψ are both true, then φ implies ψ regardless of x. 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.)
((∃!xφ x(φ ψ)) → (φψ))

Theoremeupicka 1958 Version of eupick 1957 with closed formulas. (Contributed by NM, 6-Sep-2008.)
((∃!xφ x(φ ψ)) → x(φψ))

Theoremeupickb 1959 Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.)
((∃!xφ ∃!xψ x(φ ψ)) → (φψ))

Theoremeupickbi 1960 Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.)
(∃!xφ → (x(φ ψ) ↔ x(φψ)))

Theoremmopick2 1961 "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 1500. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
((∃*xφ x(φ ψ) x(φ χ)) → x(φ ψ χ))

Theoremmoexexdc 1962 "At most one" double quantification. (Contributed by Jim Kingdon, 5-Jul-2018.)
yφ       (DECID xφ → ((∃*xφ x∃*yψ) → ∃*yx(φ ψ)))

Theoremeuexex 1963 Existential uniqueness and "at most one" double quantification. (Contributed by Jim Kingdon, 28-Dec-2018.)
yφ       ((∃!xφ x∃*yψ) → ∃*yx(φ ψ))

Theorem2moex 1964 Double quantification with "at most one." (Contributed by NM, 3-Dec-2001.)
(∃*xyφy∃*xφ)

Theorem2euex 1965 Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(∃!xyφy∃!xφ)

Theorem2eumo 1966 Double quantification with existential uniqueness and "at most one." (Contributed by NM, 3-Dec-2001.)
(∃!x∃*yφ∃*x∃!yφ)

Theorem2eu2ex 1967 Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
(∃!x∃!yφxyφ)

Theorem2moswapdc 1968 A condition allowing swap of "at most one" and existential quantifiers. (Contributed by Jim Kingdon, 6-Jul-2018.)
(DECID xyφ → (x∃*yφ → (∃*xyφ∃*yxφ)))

Theorem2euswapdc 1969 A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Jim Kingdon, 7-Jul-2018.)
(DECID xyφ → (x∃*yφ → (∃!xyφ∃!yxφ)))

Theorem2exeu 1970 Double existential uniqueness implies double uniqueness quantification. (Contributed by NM, 3-Dec-2001.)
((∃!xyφ ∃!yxφ) → ∃!x∃!yφ)

Theorem2eu4 1971* 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 ∃!x∃!yφ. See 2exeu 1970 for a one-way implication. (Contributed by NM, 3-Dec-2001.)
((∃!xyφ ∃!yxφ) ↔ (xyφ zwxy(φ → (x = z y = w))))

Theorem2eu7 1972 Two equivalent expressions for double existential uniqueness. (Contributed by NM, 19-Feb-2005.)
((∃!xyφ ∃!yxφ) ↔ ∃!x∃!y(xφ yφ))

Theoremeuequ1 1973* Equality has existential uniqueness. (Contributed by Stefan Allan, 4-Dec-2008.)
∃!x x = y

Theoremexists1 1974* 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.)
(∃!x x = xx x = y)

Theoremexists2 1975 A condition implying that at least two things exist. (Contributed by NM, 10-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
((xφ x ¬ φ) → ¬ ∃!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 mpto1 1295) 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 φ, ψ, and χ 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, http://plato.stanford.edu/entries/quantification/. See Principia Mathematica page 22 and *10 for more information (especially *10.3 and *10.26).

Expressions of the form "no φ is ψ " are consistently translated as x(φ → ¬ ψ). These can also be expressed as ¬ x(φ ψ), per alinexa 1472. We translate "all φ is ψ " to x(φψ), "some φ is ψ " to x(φ ψ), and "some φ is not ψ " to x(φ ¬ ψ). 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 A, or xA. 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 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 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), Internet Encyclopedia of Philosophy (A Peer-Reviewed Academic Resource), http://www.iep.utm.edu/aris-log/. 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 1980, celaront 1981, cesaro 1986, camestros 1987, felapton 1992, darapti 1993, calemos 1997, fesapo 1998, and bamalip 1999.

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.

Theorembarbara 1976 "Barbara", one of the fundamental syllogisms of Aristotelian logic. All φ is ψ, and all χ is φ, therefore all χ is ψ. (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 x(x 𝐻x 𝑀) (all men are mortal) and x(x = 𝑆x 𝐻) (Socrates is a man) therefore x(x = 𝑆x 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 1504. 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.)
x(φψ)    &   x(χφ)       x(χψ)

Theoremcelarent 1977 "Celarent", one of the syllogisms of Aristotelian logic. No φ is ψ, and all χ is φ, therefore no χ is ψ. (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.)
x(φ → ¬ ψ)    &   x(χφ)       x(χ → ¬ ψ)

Theoremdarii 1978 "Darii", one of the syllogisms of Aristotelian logic. All φ is ψ, and some χ is φ, therefore some χ is ψ. (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.)
x(φψ)    &   x(χ φ)       x(χ ψ)

Theoremferio 1979 "Ferio" ("Ferioque"), one of the syllogisms of Aristotelian logic. No φ is ψ, and some χ is φ, therefore some χ is not ψ. (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.)
x(φ → ¬ ψ)    &   x(χ φ)       x(χ ¬ ψ)

Theorembarbari 1980 "Barbari", one of the syllogisms of Aristotelian logic. All φ is ψ, all χ is φ, and some χ exist, therefore some χ is ψ. (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.)
x(φψ)    &   x(χφ)    &   xχ       x(χ ψ)

Theoremcelaront 1981 "Celaront", one of the syllogisms of Aristotelian logic. No φ is ψ, all χ is φ, and some χ exist, therefore some χ is not ψ. (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.)
x(φ → ¬ ψ)    &   x(χφ)    &   xχ       x(χ ¬ ψ)

Theoremcesare 1982 "Cesare", one of the syllogisms of Aristotelian logic. No φ is ψ, and all χ is ψ, therefore no χ is φ. (In Aristotelian notation, EAE-2: PeM and SaM therefore SeP.) Related to celarent 1977. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 13-Nov-2016.)
x(φ → ¬ ψ)    &   x(χψ)       x(χ → ¬ φ)

Theoremcamestres 1983 "Camestres", one of the syllogisms of Aristotelian logic. All φ is ψ, and no χ is ψ, therefore no χ is φ. (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.)
x(φψ)    &   x(χ → ¬ ψ)       x(χ → ¬ φ)

Theoremfestino 1984 "Festino", one of the syllogisms of Aristotelian logic. No φ is ψ, and some χ is ψ, therefore some χ is not φ. (In Aristotelian notation, EIO-2: PeM and SiM therefore SoP.) (Contributed by David A. Wheeler, 25-Nov-2016.)
x(φ → ¬ ψ)    &   x(χ ψ)       x(χ ¬ φ)

Theorembaroco 1985 "Baroco", one of the syllogisms of Aristotelian logic. All φ is ψ, and some χ is not ψ, therefore some χ is not φ. (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.)
x(φψ)    &   x(χ ¬ ψ)       x(χ ¬ φ)

Theoremcesaro 1986 "Cesaro", one of the syllogisms of Aristotelian logic. No φ is ψ, all χ is ψ, and χ exist, therefore some χ is not φ. (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.)
x(φ → ¬ ψ)    &   x(χψ)    &   xχ       x(χ ¬ φ)

Theoremcamestros 1987 "Camestros", one of the syllogisms of Aristotelian logic. All φ is ψ, no χ is ψ, and χ exist, therefore some χ is not φ. (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.)
x(φψ)    &   x(χ → ¬ ψ)    &   xχ       x(χ ¬ φ)

Theoremdatisi 1988 "Datisi", one of the syllogisms of Aristotelian logic. All φ is ψ, and some φ is χ, therefore some χ is ψ. (In Aristotelian notation, AII-3: MaP and MiS therefore SiP.) (Contributed by David A. Wheeler, 28-Aug-2016.)
x(φψ)    &   x(φ χ)       x(χ ψ)

Theoremdisamis 1989 "Disamis", one of the syllogisms of Aristotelian logic. Some φ is ψ, and all φ is χ, therefore some χ is ψ. (In Aristotelian notation, IAI-3: MiP and MaS therefore SiP.) (Contributed by David A. Wheeler, 28-Aug-2016.)
x(φ ψ)    &   x(φχ)       x(χ ψ)

Theoremferison 1990 "Ferison", one of the syllogisms of Aristotelian logic. No φ is ψ, and some φ is χ, therefore some χ is not ψ. (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.)
x(φ → ¬ ψ)    &   x(φ χ)       x(χ ¬ ψ)

Theorembocardo 1991 "Bocardo", one of the syllogisms of Aristotelian logic. Some φ is not ψ, and all φ is χ, therefore some χ is not ψ. (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 1989; prefer using that instead. (Contributed by David A. Wheeler, 28-Aug-2016.) (New usage is discouraged.)
x(φ ¬ ψ)    &   x(φχ)       x(χ ¬ ψ)

Theoremfelapton 1992 "Felapton", one of the syllogisms of Aristotelian logic. No φ is ψ, all φ is χ, and some φ exist, therefore some χ is not ψ. (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.)
x(φ → ¬ ψ)    &   x(φχ)    &   xφ       x(χ ¬ ψ)

Theoremdarapti 1993 "Darapti", one of the syllogisms of Aristotelian logic. All φ is ψ, all φ is χ, and some φ exist, therefore some χ is ψ. (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.)
x(φψ)    &   x(φχ)    &   xφ       x(χ ψ)

Theoremcalemes 1994 "Calemes", one of the syllogisms of Aristotelian logic. All φ is ψ, and no ψ is χ, therefore no χ is φ. (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.)
x(φψ)    &   x(ψ → ¬ χ)       x(χ → ¬ φ)

Theoremdimatis 1995 "Dimatis", one of the syllogisms of Aristotelian logic. Some φ is ψ, and all ψ is χ, therefore some χ is φ. (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 1978 with positions interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.)
x(φ ψ)    &   x(ψχ)       x(χ φ)

Theoremfresison 1996 "Fresison", one of the syllogisms of Aristotelian logic. No φ is ψ (PeM), and some ψ is χ (MiS), therefore some χ is not φ (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.)
x(φ → ¬ ψ)    &   x(ψ χ)       x(χ ¬ φ)

Theoremcalemos 1997 "Calemos", one of the syllogisms of Aristotelian logic. All φ is ψ (PaM), no ψ is χ (MeS), and χ exist, therefore some χ is not φ (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.)
x(φψ)    &   x(ψ → ¬ χ)    &   xχ       x(χ ¬ φ)

Theoremfesapo 1998 "Fesapo", one of the syllogisms of Aristotelian logic. No φ is ψ, all ψ is χ, and ψ exist, therefore some χ is not φ. (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.)
x(φ → ¬ ψ)    &   x(ψχ)    &   xψ       x(χ ¬ φ)

Theorembamalip 1999 "Bamalip", one of the syllogisms of Aristotelian logic. All φ is ψ, all ψ is χ, and φ exist, therefore some χ is φ. (In Aristotelian notation, AAI-4: PaM and MaS therefore SiP.) Like barbari 1980. (Contributed by David A. Wheeler, 28-Aug-2016.)
x(φψ)    &   x(ψχ)    &   xφ       x(χ φ)

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 symbol. Starting with the simplest mathematical object, called the empty set, set theory builds up more and more complex structures whose existence follows from the axioms, eventually resulting in extremely complicated sets that we identify with the real numbers and other familiar mathematical objects.

Here we develop set theory based on the Intuitionistic Zermelo-Fraenkel (IZF) system, mostly following the IZF axioms as laid out in [Crosilla]. Constructive Zermelo-Fraenkel (CZF), also described in Crosilla, is not as easy to formalize in metamath because the Axiom of Restricted Separation would require us to develop the ability to classify formulas as bounded formulas, similar to the machinery we have built up for asserting on whether variables are free in formulas.

2.1.1  Introduce the Axiom of Extensionality

Axiomax-ext 2000* Axiom of Extensionality. It states that two sets are identical if they contain the same elements. Axiom 1 of [Crosilla] p. "Axioms of CZF and IZF" (with unnecessary quantifiers removed).

Set theory can also be formulated with a single primitive predicate on top of traditional predicate calculus without equality. In that case the Axiom of Extensionality becomes (w(w xw y) → (x zy z)), and equality x = y is defined as w(w xw 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 1372 through ax-16 1673 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 2000 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 2000 scheme exactly logically equivalent to each other and in particular to the actual axiom of the textbook version. (Contributed by NM, 5-Aug-1993.)

(z(z xz y) → x = y)

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