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Theorem List for Metamath Proof Explorer - 2501-2600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremeuorv 2501* Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)
((¬ 𝜑 ∧ ∃!𝑥𝜓) → ∃!𝑥(𝜑𝜓))

Theoremeuor2 2502 Introduce or eliminate a disjunct in a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
(¬ ∃𝑥𝜑 → (∃!𝑥(𝜑𝜓) ↔ ∃!𝑥𝜓))

Theoremsbmo 2503* Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.)
([𝑦 / 𝑥]∃*𝑧𝜑 ↔ ∃*𝑧[𝑦 / 𝑥]𝜑)

Theoremmo4f 2504* "At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦))

Theoremmo4 2505* "At most one" expressed using implicit substitution. (Contributed by NM, 26-Jul-1995.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦))

Theoremeu4 2506* Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦)))

Theoremmoim 2507 "At most one" reverses implication. (Contributed by NM, 22-Apr-1995.)
(∀𝑥(𝜑𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑))

Theoremmoimi 2508 "At most one" reverses implication. (Contributed by NM, 15-Feb-2006.)
(𝜑𝜓)       (∃*𝑥𝜓 → ∃*𝑥𝜑)

Theoremmoa1 2509 If an implication holds for at most one value, then its consequent holds for at most one value. See also ala1 1755 and exa1 1756. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Wolf Lammen, 22-Dec-2018.) (Revised by BJ, 29-Mar-2021.)
(∃*𝑥(𝜑𝜓) → ∃*𝑥𝜓)

Theoremeuimmo 2510 Uniqueness implies "at most one" through reverse implication. (Contributed by NM, 22-Apr-1995.)
(∀𝑥(𝜑𝜓) → (∃!𝑥𝜓 → ∃*𝑥𝜑))

Theoremeuim 2511 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 2512 "At most one" is still the case when a conjunct is added. (Contributed by NM, 22-Apr-1995.)
(∃*𝑥𝜑 → ∃*𝑥(𝜓𝜑))

Theoremmoani 2513 "At most one" is still true when a conjunct is added. (Contributed by NM, 9-Mar-1995.)
∃*𝑥𝜑       ∃*𝑥(𝜓𝜑)

Theoremmoor 2514 "At most one" is still the case when a disjunct is removed. (Contributed by NM, 5-Apr-2004.)
(∃*𝑥(𝜑𝜓) → ∃*𝑥𝜑)

Theoremmooran1 2515 "At most one" imports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
((∃*𝑥𝜑 ∨ ∃*𝑥𝜓) → ∃*𝑥(𝜑𝜓))

Theoremmooran2 2516 "At most one" exports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(∃*𝑥(𝜑𝜓) → (∃*𝑥𝜑 ∧ ∃*𝑥𝜓))

Theoremmoanim 2517 Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 24-Dec-2018.)
𝑥𝜑       (∃*𝑥(𝜑𝜓) ↔ (𝜑 → ∃*𝑥𝜓))

Theoremeuan 2518 Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 24-Dec-2018.)
𝑥𝜑       (∃!𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))

Theoremmoanimv 2519* Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 23-Mar-1995.)
(∃*𝑥(𝜑𝜓) ↔ (𝜑 → ∃*𝑥𝜓))

Theoremmoanmo 2520 Nested "at most one" quantifiers. (Contributed by NM, 25-Jan-2006.)
∃*𝑥(𝜑 ∧ ∃*𝑥𝜑)

Theoremmoaneu 2521 Nested "at most one" and uniqueness quantifiers. (Contributed by NM, 25-Jan-2006.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
∃*𝑥(𝜑 ∧ ∃!𝑥𝜑)

Theoremeuanv 2522* Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)
(∃!𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))

Theoremmopick 2523 "At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.) (Proof shortened by Wolf Lammen, 17-Sep-2019.)
((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))

Theoremeupick 2524 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(𝜑𝜓)) → (𝜑𝜓))

Theoremeupicka 2525 Version of eupick 2524 with closed formulas. (Contributed by NM, 6-Sep-2008.)
((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ∀𝑥(𝜑𝜓))

Theoremeupickb 2526 Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))

Theoremeupickbi 2527 Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
(∃!𝑥𝜑 → (∃𝑥(𝜑𝜓) ↔ ∀𝑥(𝜑𝜓)))

Theoremmopick2 2528 "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 1785. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓) ∧ ∃𝑥(𝜑𝜒)) → ∃𝑥(𝜑𝜓𝜒))

Theoremmoexex 2529 "At most one" double quantification. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 28-Dec-2018.)
𝑦𝜑       ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦𝑥(𝜑𝜓))

Theoremmoexexv 2530* "At most one" double quantification. (Contributed by NM, 26-Jan-1997.)
((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦𝑥(𝜑𝜓))

Theorem2moex 2531 Double quantification with "at most one." (Contributed by NM, 3-Dec-2001.)
(∃*𝑥𝑦𝜑 → ∀𝑦∃*𝑥𝜑)

Theorem2euex 2532 Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
(∃!𝑥𝑦𝜑 → ∃𝑦∃!𝑥𝜑)

Theorem2eumo 2533 Double quantification with existential uniqueness and "at most one." (Contributed by NM, 3-Dec-2001.)
(∃!𝑥∃*𝑦𝜑 → ∃*𝑥∃!𝑦𝜑)

Theorem2eu2ex 2534 Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
(∃!𝑥∃!𝑦𝜑 → ∃𝑥𝑦𝜑)

Theorem2moswap 2535 A condition allowing swap of "at most one" and existential quantifiers. (Contributed by NM, 10-Apr-2004.)
(∀𝑥∃*𝑦𝜑 → (∃*𝑥𝑦𝜑 → ∃*𝑦𝑥𝜑))

Theorem2euswap 2536 A condition allowing swap of uniqueness and existential quantifiers. (Contributed by NM, 10-Apr-2004.)
(∀𝑥∃*𝑦𝜑 → (∃!𝑥𝑦𝜑 → ∃!𝑦𝑥𝜑))

Theorem2exeu 2537 Double existential uniqueness implies double uniqueness quantification. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Mario Carneiro, 22-Dec-2016.)
((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)

Theorem2mo2 2538* This theorem extends the idea of "at most one" to expressions in two set variables ("at most one pair 𝑥 and 𝑦". Note: this is not expressed by ∃*𝑥∃*𝑦𝜑). 2eu4 2544 relates this extension to double existential uniqueness, if at least one pair exists. (Contributed by Wolf Lammen, 26-Oct-2019.)
((∃*𝑥𝑦𝜑 ∧ ∃*𝑦𝑥𝜑) ↔ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))

Theorem2mo 2539* Two equivalent expressions for double "at most one." (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 2-Nov-2019.)
(∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑥𝑦𝑧𝑤((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧𝑦 = 𝑤)))

Theorem2mos 2540* Double "exists at most one", using implicit substitution. (Contributed by NM, 10-Feb-2005.)
((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑥𝑦𝑧𝑤((𝜑𝜓) → (𝑥 = 𝑧𝑦 = 𝑤)))

Theorem2eu1 2541 Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 11-Nov-2019.)
(∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))

Theorem2eu2 2542 Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
(∃!𝑦𝑥𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ ∃!𝑥𝑦𝜑))

Theorem2eu3 2543 Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
(∀𝑥𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))

Theorem2eu4 2544* 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 2eu1 2541 for a condition under which the naive definition holds and 2exeu 2537 for a one-way implication. See 2eu5 2545 and 2eu8 2548 for alternate definitions. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 14-Sep-2019.)
((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))

Theorem2eu5 2545* An alternate definition of double existential uniqueness (see 2eu4 2544). A mistake sometimes made in the literature is to use ∃!𝑥∃!𝑦 to mean "exactly one 𝑥 and exactly one 𝑦." (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 𝑥∃*𝑦𝜑 as an additional condition. The correct definition apparently has never been published. (∃* means "exists at most one."). (Contributed by NM, 26-Oct-2003.)
((∃!𝑥∃!𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))

Theorem2eu6 2546* Two equivalent expressions for double existential uniqueness. (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 2-Oct-2019.)
((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ ∃𝑧𝑤𝑥𝑦(𝜑 ↔ (𝑥 = 𝑧𝑦 = 𝑤)))

Theorem2eu7 2547 Two equivalent expressions for double existential uniqueness. (Contributed by NM, 19-Feb-2005.)
((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ ∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑))

Theorem2eu8 2548 Two equivalent expressions for double existential uniqueness. Curiously, we can put ∃! on either of the internal conjuncts but not both. We can also commute ∃!𝑥∃!𝑦 using 2eu7 2547. (Contributed by NM, 20-Feb-2005.)
(∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑥∃!𝑦(∃!𝑥𝜑 ∧ ∃𝑦𝜑))

Theoremexists1 2549* 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 4783. (Contributed by NM, 5-Apr-2004.)
(∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)

Theoremexists2 2550 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)

1.7  Other axiomatizations related to classical predicate calculus

1.7.1  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 1684) 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 approach 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(𝜑𝜓), per alinexa 1759. We translate "all 𝜑 is 𝜓 " to 𝑥(𝜑𝜓), "some 𝜑 is 𝜓 " to 𝑥(𝜑𝜓), 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 more specific constructs such as 𝑥 = 𝐴, 𝑥𝐴, 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 Aristotelian logic are the forerunners of predicate calculus. If we used restricted forms like 𝑥𝐴 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. Using such specific constructs would also be anti-historical; Aristotle and others who directly followed his work focused on relating wholes to their parts, an approach now called part-whole theory. The work of Cantor and Peano (over 2,000 years later) led to a sharper distinction between inclusion () and membership (); this distinction was not directly made in Aristotle's work.

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 2555, celaront 2556, cesaro 2561, camestros 2562, felapton 2567, darapti 2568, calemos 2572, fesapo 2573, and bamalip 2574.

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.

Aristotelian 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 2551 "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(𝑥𝐻𝑥𝑀) (all men are mortal) and 𝑥(𝑥 = 𝑆𝑥𝐻) (Socrates is a man) therefore 𝑥(𝑥 = 𝑆𝑥𝑀) (Socrates is mortal). Russell and Whitehead note that the "syllogism in Barbara is derived..." from syl 17. (quote after Theorem *2.06 of [WhiteheadRussell] p. 101). Most of the proof is in alsyl 1811. 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 2552 "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 2553 "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(𝜒𝜑)       𝑥(𝜒𝜓)

Theoremferio 2554 "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(𝜒𝜑)       𝑥(𝜒 ∧ ¬ 𝜓)

Theorembarbari 2555 "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(𝜒𝜓)

Theoremcelaront 2556 "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 2557 "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 2552. (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 2558 "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 2559 "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(𝜒𝜓)       𝑥(𝜒 ∧ ¬ 𝜑)

Theorembaroco 2560 "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 2561 "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 2562 "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 2563 "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(𝜑𝜒)       𝑥(𝜒𝜓)

Theoremdisamis 2564 "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(𝜑𝜓)    &   𝑥(𝜑𝜒)       𝑥(𝜒𝜓)

Theoremferison 2565 "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(𝜑𝜒)       𝑥(𝜒 ∧ ¬ 𝜓)

Theorembocardo 2566 "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 2564; 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 2567 "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 2568 "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(𝜒𝜓)

Theoremcalemes 2569 "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 2570 "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 2553 with positions interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.)
𝑥(𝜑𝜓)    &   𝑥(𝜓𝜒)       𝑥(𝜒𝜑)

Theoremfresison 2571 "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(𝜓𝜒)       𝑥(𝜒 ∧ ¬ 𝜑)

Theoremcalemos 2572 "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 2573 "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 2574 "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 2555. (Contributed by David A. Wheeler, 28-Aug-2016.)
𝑥(𝜑𝜓)    &   𝑥(𝜓𝜒)    &   𝑥𝜑       𝑥(𝜒𝜑)

1.7.2  Intuitionistic logic

Intuitionistic (constructive) logic is similar to classical logic with the notable omission of ax-3 8 and theorems such as exmid 430 or peirce 192. 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 384 and df-ex 1696 which are not valid in intuitionistic 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.

The following axioms are unchanged between set.mm and iset.mm: ax-1 6, ax-2 7, ax-mp 5, ax-4 1728, ax-11 2021, ax-gen 1713, ax-7 1922, ax-12 2034, ax-8 1979, ax-9 1986, and ax-5 1827.

In this list of axioms, the ones that repeat earlier theorems are marked "(New usage is discouraged.)" so that the earlier theorems will be used consistently in other proofs.

Theoremaxia1 2575 Left 'and' elimination (intuitionistic logic axiom ax-ia1). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
((𝜑𝜓) → 𝜑)

Theoremaxia2 2576 Right 'and' elimination (intuitionistic logic axiom ax-ia2). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
((𝜑𝜓) → 𝜓)

Theoremaxia3 2577 'And' introduction (intuitionistic logic axiom ax-ia3). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜑𝜓)))

Theoremaxin1 2578 'Not' introduction (intuitionistic logic axiom ax-in1). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
((𝜑 → ¬ 𝜑) → ¬ 𝜑)

Theoremaxin2 2579 'Not' elimination (intuitionistic logic axiom ax-in2). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
𝜑 → (𝜑𝜓))

Theoremaxio 2580 Definition of 'or' (intuitionistic logic axiom ax-io). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
(((𝜑𝜒) → 𝜓) ↔ ((𝜑𝜓) ∧ (𝜒𝜓)))

Theoremaxi4 2581 Specialization (intuitionistic logic axiom ax-4). This is just sp 2041 by another name. (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
(∀𝑥𝜑𝜑)

Theoremaxi5r 2582 Converse of ax-c4 (intuitionistic logic axiom ax-i5r). (Contributed by Jim Kingdon, 31-Dec-2017.)
((∀𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∀𝑥𝜑𝜓))

Theoremaxial 2583 The setvar 𝑥 is not free in 𝑥𝜑 (intuitionistic logic axiom ax-ial). (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
(∀𝑥𝜑 → ∀𝑥𝑥𝜑)

Theoremaxie1 2584 The setvar 𝑥 is not free in 𝑥𝜑 (intuitionistic logic axiom ax-ie1). (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
(∃𝑥𝜑 → ∀𝑥𝑥𝜑)

Theoremaxie2 2585 A key property of existential quantification (intuitionistic logic axiom ax-ie2). (Contributed by Jim Kingdon, 31-Dec-2017.)
(∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓)))

Theoremaxi9 2586 Axiom of existence (intuitionistic logic axiom ax-i9). In classical logic, this is equivalent to ax-6 1875 but in intuitionistic logic it needs to be stated using the existential quantifier. (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
𝑥 𝑥 = 𝑦

Theoremaxi10 2587 Axiom of Quantifier Substitution (intuitionistic logic axiom ax-10). This is just axc11n 2295 by another name. (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theoremaxi12 2588 Axiom of Quantifier Introduction (intuitionistic logic axiom ax-i12). In classical logic, this is mostly a restatement of axc9 2290 (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.)
(∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))

Theoremaxbnd 2589 Axiom of Bundling (intuitionistic logic axiom ax-bnd). In classical logic, this and axi12 2588 are fairly straightforward consequences of axc9 2290. But in intuitionistic logic, it is not easy to add the extra 𝑥 to axi12 2588 and so we treat the two as separate axioms. (Contributed by Jim Kingdon, 22-Mar-2018.)
(∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))

PART 2  ZF (ZERMELO-FRAENKEL) 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.

A simplistic concept of sets, sometimes called "naive set theory", is vulnerable to a paradox called "Russell's Paradox" (ru 3401), 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 2591.) 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

Axiomax-ext 2590* 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 single primitive predicate on top of traditional predicate calculus without equality. In that case the Axiom of Extensionality becomes (∀𝑤(𝑤𝑥𝑤𝑦) → (𝑥𝑧𝑦𝑧)), and equality 𝑥 = 𝑦 is defined as 𝑤(𝑤𝑥𝑤𝑦). 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 predicate calculus axioms, we would rewrite all axioms involving equality with equality expanded according to the above definition. Some of those axioms may be provable from ax-ext and would become redundant, but this hasn't been studied carefully.

General remarks: Our set theory axioms are presented using defined connectives (, , etc.) for convenience. However, it is implicitly understood that the actual axioms use only the primitive connectives , ¬, , =, and . It is straightforward to establish the equivalence between the actual axioms and the ones we display, and we will not do so.

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 2590 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 Replacement ax-rep 4699, which involves a wff metavariable). 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 2590 scheme exactly logically equivalent to each other and in particular to the actual axiom of the textbook version. (Contributed by NM, 21-May-1993.)

(∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)

Theoremaxext2 2591* The Axiom of Extensionality (ax-ext 2590) restated so that it postulates the existence of a set 𝑧 given two arbitrary sets 𝑥 and 𝑦. 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.)
𝑧((𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)

Theoremaxext3 2592* 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.) Remove dependencies on ax-10 2006, ax-12 2034, ax-13 2234. (Revised by Wolf Lammen, 9-Dec-2019.)
(∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)

Theoremaxext3ALT 2593* Alternate proof of axext3 2592, shorter but uses more axioms. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)

Theoremaxext4 2594* 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 2590 and df-cleq 2603. (Contributed by NM, 14-Nov-2008.)
(𝑥 = 𝑦 ↔ ∀𝑧(𝑧𝑥𝑧𝑦))

Theorembm1.1 2595* 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.) (Proof shortened by Wolf Lammen, 13-Nov-2019.)
𝑥𝜑       (∃𝑥𝑦(𝑦𝑥𝜑) → ∃!𝑥𝑦(𝑦𝑥𝜑))

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

Syntaxcab 2596 Introduce the class builder or class abstraction notation ("the class of sets 𝑥 such that 𝜑 is true"). Our class variables 𝐴, 𝐵, etc. range over class builders (implicitly in the case of defined class terms such as df-nul 3875). Note that a setvar variable can be expressed as a class builder per theorem cvjust 2605, justifying the assignment of setvar variables to class variables via the use of cv 1474.
class {𝑥𝜑}

Definitiondf-clab 2597 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, 𝜑 will have 𝑦 as a free variable, and "{𝑦𝜑} " is read "the class of all sets 𝑦 such that 𝜑(𝑦) is true." We do not define {𝑦𝜑} 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 1977, which extends or "overloads" the wel 1978 definition connecting setvar variables, requires that both sides of be classes. In df-cleq 2603 and df-clel 2606, we introduce a new kind of variable (class variable) that can be substituted with expressions such as {𝑦𝜑}. In the present definition, the 𝑥 on the left-hand side is a setvar variable. Syntax definition cv 1474 allows us to substitute a setvar variable 𝑥 for a class variable: all sets are classes by cvjust 2605 (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 2719 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 done with theorems such as vtoclg 3239 which is used, for example, to convert elirrv 8387 to elirr 8388.

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 {𝑦𝜑} a "class term".

While the three class definitions df-clab 2597, df-cleq 2603, and df-clel 2606 are eliminable and conservative and thus meet the requirements for sound definitions, they are technically axioms in that they do not satisfy the requirements for the current definition checker. The proofs of conservativity require external justification that is beyond the scope of the definition checker.

For a general discussion of the theory of classes, see mmset.html#class. (Contributed by NM, 26-May-1993.)

(𝑥 ∈ {𝑦𝜑} ↔ [𝑥 / 𝑦]𝜑)

Theoremabid 2598 Simplification of class abstraction notation when the free and bound variables are identical. (Contributed by NM, 26-May-1993.)
(𝑥 ∈ {𝑥𝜑} ↔ 𝜑)

Theoremhbab1 2599* Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 26-May-1993.)
(𝑦 ∈ {𝑥𝜑} → ∀𝑥 𝑦 ∈ {𝑥𝜑})

Theoremnfsab1 2600* Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥 𝑦 ∈ {𝑥𝜑}

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