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Theorem List for Metamath Proof Explorer - 2701-2800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmoanim 2701 Introduction of a conjunct into "at most one" quantifier. For a version requiring disjoint variables, but fewer axioms, see moanimv 2700. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 24-Dec-2018.)
𝑥𝜑       (∃*𝑥(𝜑𝜓) ↔ (𝜑 → ∃*𝑥𝜓))

Theoremeuan 2702 Introduction of a conjunct into unique existential 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))

Theoremmoanmo 2703 Nested at-most-one quantifiers. (Contributed by NM, 25-Jan-2006.)
∃*𝑥(𝜑 ∧ ∃*𝑥𝜑)

Theoremmoaneu 2704 Nested at-most-one and unique existential quantifiers. (Contributed by NM, 25-Jan-2006.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
∃*𝑥(𝜑 ∧ ∃!𝑥𝜑)

Theoremeuanv 2705* Introduction of a conjunct into unique existential quantifier. (Contributed by NM, 23-Mar-1995.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 14-Jan-2023.)
(∃!𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃!𝑥𝜓))

Theoremmopick 2706 "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))

Theoremmoexexlem 2707 Factor out the proof skeleton of moexex 2719 and moexexvw 2709. (Contributed by Wolf Lammen, 2-Oct-2023.)
𝑦𝜑    &   𝑦∃*𝑥𝜑    &   𝑥∃*𝑦𝑥(𝜑𝜓)       ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦𝑥(𝜑𝜓))

Theorem2moexv 2708* Double quantification with "at most one". (Contributed by NM, 3-Dec-2001.)
(∃*𝑥𝑦𝜑 → ∀𝑦∃*𝑥𝜑)

Theoremmoexexvw 2709* Version of moexexv 2720 with an additional disjoint variable condition, which does not require ax-13 2386. (Contributed by Gino Giotto, 22-Aug-2023.) Factor out common proof lines with moexex 2719. (Revised by Wolf Lammen, 2-Oct-2023.)
((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦𝑥(𝜑𝜓))

Theorem2moswapv 2710* Version of 2moswap 2725 with a disjoint variable condition, which does not require ax-13 2386. (Contributed by Gino Giotto, 22-Aug-2023.) Factor out common proof lines with moexexvw 2709. (Revised by Wolf Lammen, 2-Oct-2023.)
(∀𝑥∃*𝑦𝜑 → (∃*𝑥𝑦𝜑 → ∃*𝑦𝑥𝜑))

Theorem2euswapv 2711* Version of 2euswap 2726 with a disjoint variable condition, which does not require ax-13 2386. (Contributed by Gino Giotto, 22-Aug-2023.)
(∀𝑥∃*𝑦𝜑 → (∃!𝑥𝑦𝜑 → ∃!𝑦𝑥𝜑))

Theorem2euexv 2712* Version of 2euex 2722 with 𝑥 and 𝑦 distinct, but not requiring ax-13 2386. (Contributed by Wolf Lammen, 2-Oct-2023.)
(∃!𝑥𝑦𝜑 → ∃𝑦∃!𝑥𝜑)

Theorem2exeuv 2713* Version of 2exeu 2727 with 𝑥 and 𝑦 distinct, but not requiring ax-13 2386. (Contributed by Wolf Lammen, 2-Oct-2023.)
((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)

Theoremeupick 2714 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 2715 Version of eupick 2714 with closed formulas. (Contributed by NM, 6-Sep-2008.)
((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ∀𝑥(𝜑𝜓))

Theoremeupickb 2716 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 2717 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 2718 "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 1883. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝜓) ∧ ∃𝑥(𝜑𝜒)) → ∃𝑥(𝜑𝜓𝜒))

Theoremmoexex 2719 "At most one" double quantification. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the version moexexvw 2709 when possible. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 28-Dec-2018.) Factor out common proof lines with moexexvw 2709. (Revised by Wolf Lammen, 2-Oct-2023.) (New usage is discouraged.)
𝑦𝜑       ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦𝑥(𝜑𝜓))

Theoremmoexexv 2720* "At most one" double quantification. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker moexexvw 2709 when possible. (Contributed by NM, 26-Jan-1997.) (New usage is discouraged.)
((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦𝑥(𝜑𝜓))

Theorem2moex 2721 Double quantification with "at most one". Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker 2moexv 2708 when possible. (Contributed by NM, 3-Dec-2001.) (New usage is discouraged.)
(∃*𝑥𝑦𝜑 → ∀𝑦∃*𝑥𝜑)

Theorem2euex 2722 Double quantification with existential uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker 2euexv 2712 when possible. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (New usage is discouraged.)
(∃!𝑥𝑦𝜑 → ∃𝑦∃!𝑥𝜑)

Theorem2eumo 2723 Nested unique existential quantifier and at-most-one quantifier. (Contributed by NM, 3-Dec-2001.)
(∃!𝑥∃*𝑦𝜑 → ∃*𝑥∃!𝑦𝜑)

Theorem2eu2ex 2724 Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
(∃!𝑥∃!𝑦𝜑 → ∃𝑥𝑦𝜑)

Theorem2moswap 2725 A condition allowing to swap an existential quantifier and at at-most-one quantifier. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker 2moswapv 2710 when possible. (Contributed by NM, 10-Apr-2004.) (New usage is discouraged.)
(∀𝑥∃*𝑦𝜑 → (∃*𝑥𝑦𝜑 → ∃*𝑦𝑥𝜑))

Theorem2euswap 2726 A condition allowing to swap an existential quantifier and a unique existential quantifier. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker 2euswapv 2711 when possible. (Contributed by NM, 10-Apr-2004.) (New usage is discouraged.)
(∀𝑥∃*𝑦𝜑 → (∃!𝑥𝑦𝜑 → ∃!𝑦𝑥𝜑))

Theorem2exeu 2727 Double existential uniqueness implies double unique existential quantification. The converse does not hold. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker 2exeuv 2713 when possible. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)

Theorem2mo2 2728* Two ways of expressing "there exists at most one ordered pair 𝑥, 𝑦 such that 𝜑(𝑥, 𝑦) holds. Note that this is not equivalent to ∃*𝑥∃*𝑦𝜑. See also 2mo 2729. This is the analogue of 2eu4 2737 for existential uniqueness. (Contributed by Wolf Lammen, 26-Oct-2019.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 3-Jan-2023.)
((∃*𝑥𝑦𝜑 ∧ ∃*𝑦𝑥𝜑) ↔ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))

Theorem2mo 2729* Two ways of expressing "there exists at most one ordered pair 𝑥, 𝑦 such that 𝜑(𝑥, 𝑦) holds. See also 2mo2 2728. (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 2730* 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 2731 Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker 2eu1v 2733 when possible. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 23-Apr-2023.) (New usage is discouraged.)
(∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))

Theorem2eu1OLD 2732 Obsolete version of 2eu1 2731 as of 23-Apr-2023. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 11-Nov-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
(∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))

Theorem2eu1v 2733* Version of 2eu1 2731 with 𝑥 and 𝑦 distinct, but not requiring ax-13 2386. (Contributed by Wolf Lammen, 2-Oct-2023.)
(∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))

Theorem2eu2 2734 Double existential uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 3-Dec-2001.) (New usage is discouraged.)
(∃!𝑦𝑥𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ ∃!𝑥𝑦𝜑))

Theorem2eu3 2735 Double existential uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 23-Apr-2023.) (New usage is discouraged.)
(∀𝑥𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))

Theorem2eu3OLD 2736 Obsolete version of 2eu3 2735 as of 23-Apr-2023. (Contributed by NM, 3-Dec-2001.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))

Theorem2eu4 2737* 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 2731 for a condition under which the naive definition holds and 2exeu 2727 for a one-way implication. See 2eu5 2738 and 2eu8 2742 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 2738* An alternate definition of double existential uniqueness (see 2eu4 2737). 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.) Avoid ax-13 2386. (Revised by Wolf Lammen, 2-Oct-2023.)
((∃!𝑥∃!𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))

Theorem2eu5OLD 2739* Obsolete version of 2eu5 2738 as of 2-Oct-2023. (Contributed by NM, 26-Oct-2003.) (Proof modification is discouraged.) (New usage is discouraged.)
((∃!𝑥∃!𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))

Theorem2eu6 2740* 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 2741 Two equivalent expressions for double existential uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 19-Feb-2005.) (New usage is discouraged.)
((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ ∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑))

Theorem2eu8 2742 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 2741. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 20-Feb-2005.) (New usage is discouraged.)
(∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑥∃!𝑦(∃!𝑥𝜑 ∧ ∃𝑦𝜑))

Theoremeuae 2743* Two ways to express "exactly one thing exists". To paraphrase the statement and explain the label: there Exists a Unique thing if and only if for All 𝑥, 𝑥 Equals some given (and disjoint) 𝑦. Both sides are false in set theory, see theorems neutru 33750 and dtru 5264. (Contributed by NM, 5-Apr-2004.) State the theorem using truth constant . (Revised by BJ, 7-Oct-2022.) Reduce axiom dependencies. (Revised by Wolf Lammen, 2-Mar-2023.)
(∃!𝑥⊤ ↔ ∀𝑥 𝑥 = 𝑦)

Theoremexists1 2744* Two ways to express "exactly one thing exists". The left-hand side requires only one variable to express this. Both sides are false in set theory, see theorem dtru 5264. (Contributed by NM, 5-Apr-2004.) (Proof shortened by BJ, 7-Oct-2022.)
(∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)

Theoremexists2 2745 A condition implying that at least two things exist. (Contributed by NM, 10-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) Reduce axiom usage. (Revised by Wolf Lammen, 4-Mar-2023.)
((∃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 1765) were the leading logical systems. Aristotelian logic became the leading system in medieval Europe. This section models this system (including later refinements). 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 used 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/ 1765. 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 1839. 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 form "is", not the plural form "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 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 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/ 1839. 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 2752, celaront 2754, cesaro 2761, camestros 2762, felapton 2769, darapti 2767, calemos 2773, fesapo 2774, and bamalip 2775.

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.

The following twenty-four syllogisms (from barbara 2746 to bamalip 2775) are all proven from { ax-mp 5, ax-1 6, ax-2 7, ax-3 8, ax-gen 1792, ax-4 1806 }, which corresponds in the usual translation to modal logic (a universal (resp. existential) quantifier maps to necessity (resp. possibility)) to the weakest normal modal logic (K). Some proofs could be shortened by using additionally spi 2178 (inference form of sp 2177, which corresponds to the axiom (T) of modal logic), as demonstrated by dariiALT 2749, barbariALT 2753, festinoALT 2758, barocoALT 2760, daraptiALT 2768.

Theorembarbara 2746 "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 1890. There are a legion of sources for Barbara, including http://www.friesian.com/aristotl.htm 1890, http://plato.stanford.edu/entries/aristotle-logic/ 1890, and https://en.wikipedia.org/wiki/Syllogism 1890. (Contributed by David A. Wheeler, 24-Aug-2016.)
𝑥(𝜑𝜓)    &   𝑥(𝜒𝜑)       𝑥(𝜒𝜓)

Theoremcelarent 2747 "Celarent", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and all 𝜒 is 𝜑, therefore no 𝜒 is 𝜓. Instance of barbara 2746. 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 2746. (Contributed by David A. Wheeler, 24-Aug-2016.)
𝑥(𝜑 → ¬ 𝜓)    &   𝑥(𝜒𝜑)       𝑥(𝜒 → ¬ 𝜓)

Theoremdarii 2748 "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. See dariiALT 2749 for a shorter proof requiring more axioms. (Contributed by David A. Wheeler, 24-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
𝑥(𝜑𝜓)    &   𝑥(𝜒𝜑)       𝑥(𝜒𝜓)

TheoremdariiALT 2749 Alternate proof of darii 2748, shorter but using more axioms. This shows how the use of spi 2178 may shorten some proofs of the Aristotelian syllogisms, even though this adds axiom dependencies. Note that spi 2178 is the inference associated with sp 2177, which corresponds to the axiom (T) of modal logic. (Contributed by David A. Wheeler, 27-Aug-2016.) Added precisions on axiom usage. (Revised by BJ, 27-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥(𝜑𝜓)    &   𝑥(𝜒𝜑)       𝑥(𝜒𝜓)

Theoremferio 2750 "Ferio" ("Ferioque"), one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and some 𝜒 is 𝜑, therefore some 𝜒 is not 𝜓. Instance of darii 2748. 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 2748. (Contributed by David A. Wheeler, 24-Aug-2016.)
𝑥(𝜑 → ¬ 𝜓)    &   𝑥(𝜒𝜑)       𝑥(𝜒 ∧ ¬ 𝜓)

Theorembarbarilem 2751 Lemma for barbari 2752 and the other Aristotelian syllogisms with existential assumption. (Contributed by BJ, 16-Sep-2022.)
𝑥𝜑    &   𝑥(𝜑𝜓)       𝑥(𝜑𝜓)

Theorembarbari 2752 "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.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
𝑥(𝜑𝜓)    &   𝑥(𝜒𝜑)    &   𝑥𝜒       𝑥(𝜒𝜓)

TheorembarbariALT 2753 Alternate proof of barbari 2752, shorter but using more axioms. See comment of dariiALT 2749. (Contributed by David A. Wheeler, 27-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥(𝜑𝜓)    &   𝑥(𝜒𝜑)    &   𝑥𝜒       𝑥(𝜒𝜓)

Theoremcelaront 2754 "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. Instance of barbari 2752. 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 2752. (Contributed by David A. Wheeler, 27-Aug-2016.)
𝑥(𝜑 → ¬ 𝜓)    &   𝑥(𝜒𝜑)    &   𝑥𝜒       𝑥(𝜒 ∧ ¬ 𝜓)

Theoremcesare 2755 "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 2747. (Contributed by David A. Wheeler, 27-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
𝑥(𝜑 → ¬ 𝜓)    &   𝑥(𝜒𝜓)       𝑥(𝜒 → ¬ 𝜑)

Theoremcamestres 2756 "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.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
𝑥(𝜑𝜓)    &   𝑥(𝜒 → ¬ 𝜓)       𝑥(𝜒 → ¬ 𝜑)

Theoremfestino 2757 "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.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
𝑥(𝜑 → ¬ 𝜓)    &   𝑥(𝜒𝜓)       𝑥(𝜒 ∧ ¬ 𝜑)

TheoremfestinoALT 2758 Alternate proof of festino 2757, shorter but using more axioms. See comment of dariiALT 2749. (Contributed by David A. Wheeler, 27-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥(𝜑 → ¬ 𝜓)    &   𝑥(𝜒𝜓)       𝑥(𝜒 ∧ ¬ 𝜑)

Theorembaroco 2759 "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.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
𝑥(𝜑𝜓)    &   𝑥(𝜒 ∧ ¬ 𝜓)       𝑥(𝜒 ∧ ¬ 𝜑)

TheorembarocoALT 2760 Alternate proof of festino 2757, shorter but using more axioms. See comment of dariiALT 2749. (Contributed by David A. Wheeler, 27-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥(𝜑𝜓)    &   𝑥(𝜒 ∧ ¬ 𝜓)       𝑥(𝜒 ∧ ¬ 𝜑)

Theoremcesaro 2761 "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.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
𝑥(𝜑 → ¬ 𝜓)    &   𝑥(𝜒𝜓)    &   𝑥𝜒       𝑥(𝜒 ∧ ¬ 𝜑)

Theoremcamestros 2762 "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.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
𝑥(𝜑𝜓)    &   𝑥(𝜒 → ¬ 𝜓)    &   𝑥𝜒       𝑥(𝜒 ∧ ¬ 𝜑)

Theoremdatisi 2763 "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.) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
𝑥(𝜑𝜓)    &   𝑥(𝜑𝜒)       𝑥(𝜒𝜓)

Theoremdisamis 2764 "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.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
𝑥(𝜑𝜓)    &   𝑥(𝜑𝜒)       𝑥(𝜒𝜓)

Theoremferison 2765 "Ferison", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and some 𝜑 is 𝜒, therefore some 𝜒 is not 𝜓. Instance of datisi 2763. In Aristotelian notation, EIO-3: MeP and MiS therefore SoP. (Contributed by David A. Wheeler, 28-Aug-2016.)
𝑥(𝜑 → ¬ 𝜓)    &   𝑥(𝜑𝜒)       𝑥(𝜒 ∧ ¬ 𝜓)

Theorembocardo 2766 "Bocardo", one of the syllogisms of Aristotelian logic. Some 𝜑 is not 𝜓, and all 𝜑 is 𝜒, therefore some 𝜒 is not 𝜓. Instance of disamis 2764. 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". (Contributed by David A. Wheeler, 28-Aug-2016.)
𝑥(𝜑 ∧ ¬ 𝜓)    &   𝑥(𝜑𝜒)       𝑥(𝜒 ∧ ¬ 𝜓)

Theoremdarapti 2767 "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.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
𝑥(𝜑𝜓)    &   𝑥(𝜑𝜒)    &   𝑥𝜑       𝑥(𝜒𝜓)

TheoremdaraptiALT 2768 Alternate proof of darapti 2767, shorter but using more axioms. See comment of dariiALT 2749. (Contributed by David A. Wheeler, 27-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥(𝜑𝜓)    &   𝑥(𝜑𝜒)    &   𝑥𝜑       𝑥(𝜒𝜓)

Theoremfelapton 2769 "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. Instance of darapti 2767. 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.)
𝑥(𝜑 → ¬ 𝜓)    &   𝑥(𝜑𝜒)    &   𝑥𝜑       𝑥(𝜒 ∧ ¬ 𝜓)

Theoremcalemes 2770 "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.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
𝑥(𝜑𝜓)    &   𝑥(𝜓 → ¬ 𝜒)       𝑥(𝜒 → ¬ 𝜑)

Theoremdimatis 2771 "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 2748 with positions interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
𝑥(𝜑𝜓)    &   𝑥(𝜓𝜒)       𝑥(𝜒𝜑)

Theoremfresison 2772 "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.) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
𝑥(𝜑 → ¬ 𝜓)    &   𝑥(𝜓𝜒)       𝑥(𝜒 ∧ ¬ 𝜑)

Theoremcalemos 2773 "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.) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
𝑥(𝜑𝜓)    &   𝑥(𝜓 → ¬ 𝜒)    &   𝑥𝜒       𝑥(𝜒 ∧ ¬ 𝜑)

Theoremfesapo 2774 "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.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
𝑥(𝜑 → ¬ 𝜓)    &   𝑥(𝜓𝜒)    &   𝑥𝜓       𝑥(𝜒 ∧ ¬ 𝜑)

Theorembamalip 2775 "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. Very similar to barbari 2752. (Contributed by David A. Wheeler, 28-Aug-2016.) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)
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 891 or peirce 204. 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 844 and df-ex 1777 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 1806, ax-11 2156, ax-gen 1792, ax-7 2011, ax-12 2172, ax-8 2112, ax-9 2120, and ax-5 1907.

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 2776 Left 'and' elimination (intuitionistic logic axiom ax-ia1). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
((𝜑𝜓) → 𝜑)

Theoremaxia2 2777 Right 'and' elimination (intuitionistic logic axiom ax-ia2). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
((𝜑𝜓) → 𝜓)

Theoremaxia3 2778 'And' introduction (intuitionistic logic axiom ax-ia3). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜑𝜓)))

Theoremaxin1 2779 'Not' introduction (intuitionistic logic axiom ax-in1). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
((𝜑 → ¬ 𝜑) → ¬ 𝜑)

Theoremaxin2 2780 'Not' elimination (intuitionistic logic axiom ax-in2). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
𝜑 → (𝜑𝜓))

Theoremaxio 2781 Definition of 'or' (intuitionistic logic axiom ax-io). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.)
(((𝜑𝜒) → 𝜓) ↔ ((𝜑𝜓) ∧ (𝜒𝜓)))

Theoremaxi4 2782 Specialization (intuitionistic logic axiom ax-4). This is just sp 2177 by another name. (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
(∀𝑥𝜑𝜑)

Theoremaxi5r 2783 Converse of axc4 2336 (intuitionistic logic axiom ax-i5r). (Contributed by Jim Kingdon, 31-Dec-2017.)
((∀𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∀𝑥𝜑𝜓))

Theoremaxial 2784 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 2785 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 2786 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 2787 Axiom of existence (intuitionistic logic axiom ax-i9). In classical logic, this is equivalent to ax-6 1966 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 2788 Axiom of Quantifier Substitution (intuitionistic logic axiom ax-10). This is just axc11n 2444 by another name. (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theoremaxi12 2789 Axiom of Quantifier Introduction (intuitionistic logic axiom ax-i12). In classical logic, this is mostly a restatement of axc9 2396 (with one additional quantifier). But in intuitionistic logic, changing the negations and implications to disjunctions makes it stronger. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by Jim Kingdon, 31-Dec-2017.) Avoid ax-11 2156. (Revised by Wolf Lammen, 24-Apr-2023.) (New usage is discouraged.)
(∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))

Theoremaxi12OLD 2790 Obsolete version of axi12 2789 as of 24-Apr-2023. (Contributed by Jim Kingdon, 31-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))

Theoremaxbnd 2791 Axiom of Bundling (intuitionistic logic axiom ax-bnd). In classical logic, this and axi12 2789 are fairly straightforward consequences of axc9 2396. But in intuitionistic logic, it is not easy to add the extra 𝑥 to axi12 2789 and so we treat the two as separate axioms. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by Jim Kingdon, 22-Mar-2018.) (Proof shortened by Wolf Lammen, 24-Apr-2023.) (New usage is discouraged.)
(∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑥𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))

TheoremaxbndOLD 2792 Obsolete version of axbnd 2791 as of 24-Apr-2023. (Contributed by Jim Kingdon, 22-Mar-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀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 3771), 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 axexte 2794.) 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.

2.1.1  Introduce the Axiom of Extensionality

Axiomax-ext 2793* 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. Its converse is a theorem of predicate logic, elequ2g 2127.

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 2793 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 5183, 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 2793 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)

Theoremaxexte 2794* The axiom of extensionality (ax-ext 2793) 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)

Theoremaxextg 2795* 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 2141, ax-12 2172, ax-13 2386. (Revised by BJ, 12-Jul-2019.) (Revised by Wolf Lammen, 9-Dec-2019.)
(∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)

Theoremaxextb 2796* A bidirectional version of the axiom of extensionality. Although this theorem looks like a definition of equality, it requires the axiom of extensionality for its proof under our axiomatization. See the comments for ax-ext 2793 and df-cleq 2814. (Contributed by NM, 14-Nov-2008.)
(𝑥 = 𝑦 ↔ ∀𝑧(𝑧𝑥𝑧𝑦))

Theoremaxextmo 2797* There exists at most one set with prescribed elements. Theorem 1.1 of [BellMachover] p. 462. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Wolf Lammen, 13-Nov-2019.) Use the at-most-one quantifier. (Revised by BJ, 17-Sep-2022.)
𝑥𝜑       ∃*𝑥𝑦(𝑦𝑥𝜑)

Theoremnulmo 2798* There exists at most one empty set. With either axnul 5202 or axnulALT 5201 or ax-nul 5203, this proves that there exists a unique empty set. In practice, once the language of classes is available, we use the stronger characterization among classes eq0 4308. (Contributed by NM, 22-Dec-2007.) Use the at-most-one quantifier. (Revised by BJ, 17-Sep-2022.) (Proof shortened by Wolf Lammen, 26-Apr-2023.)
∃*𝑥𝑦 ¬ 𝑦𝑥

2.1.2  Classes

2.1.2.1  Class abstractions

Syntaxcab 2799 Introduce the class builder or class abstraction notation ("the class of sets 𝑥 such that 𝜑"). Our class variables 𝐴, 𝐵, etc. range over class builders (implicitly in the case of defined class terms such as df-nul 4292). Note that a setvar variable can be expressed as a class builder per theorem cvjust 2816, justifying the assignment of setvar variables to class variables via the use of cv 1532.
class {𝑥𝜑}

Definitiondf-clab 2800 Define class abstractions, that is, classes of the form {𝑦𝜑}, which is read "the class of sets 𝑦 such that 𝜑(𝑦)".

A few remarks are in order:

1. The axiomatic statement df-clab 2800 does not define the class abstraction {𝑦𝜑} itself, that is, it does not have the form {𝑦𝜑} = ... that a standard definition should have (for a good reason: equality itself has not yet been defined or axiomatized for class abstractions; it is defined later in df-cleq 2814). Instead, df-clab 2800 has the form (𝑥 ∈ {𝑦𝜑} ↔ ...), meaning that it only defines what it means for a setvar to be a member of a class abstraction. As a consequence, one can say that df-clab 2800 defines class abstractions if and only if a class abstraction is completely determined by which elements belong to it, which is the content of the axiom of extensionality ax-ext 2793. Therefore, df-clab 2800 can be considered a definition only in systems that can prove ax-ext 2793 (and the necessary first-order logic).

2. As in all definitions, the LHS (definiendum) has no disjoint variable conditions. In particular, the setvar variables 𝑥 and 𝑦 need not be distinct, and the formula 𝜑 may depend on both 𝑥 and 𝑦. This is necessary, as with all definitions, since if there was for instance a disjoint variable condition on 𝑥, 𝑦, then one could not do anything with expressions like 𝑥 ∈ {𝑥𝜑} which are sometimes useful to shorten proofs (because of abid 2803). Most often, however, 𝑥 does not occur in {𝑦𝜑} and 𝑦 is free in 𝜑.

3. Remark 1 stresses that df-clab 2800 does not have the standard form of a definition for a class, but one could be led to think it has the standard form of a definition for a formula. However, it also fails that test since the membership precidate has already appeared earlier (e.g., in the non-syntactic statement ax-8 2112). Indeed, the LHS extends, or "overloads", the membership predicate from formulas of the form "setvar setvar" to formulas of the form "setvar class abstraction". This is possible because of wcel 2110 and cab 2799, and it can be called an "extension" of the membership predicate because of wel 2111, whose proof uses cv 1532. An a posteriori justification for cv 1532 is given by cvjust 2816, stating that every setvar can be written as a class abstraction (though conversely not every class abstraction is a set, as illustrated by Russell's paradox ru 3771).

4. Proof techniques. 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 3568 which is used, for example, to convert elirrv 9054 to elirr 9055.

5. Definition or axiom? The question arises with the three axiomatic statements introducing classes, df-clab 2800, df-cleq 2814, and df-clel 2893, to decide if they qualify as definitions or if they should be called axioms. Under the strict definition of "definition" (see conventions 28173), they are not definitions (see Remarks 1 and 3 above, and similarly for df-cleq 2814 and df-clel 2893). One could be less strict and decide to call "definition" every axiomatic statement which provides an eliminable and conservative extension of the considered axiom system. But the notion of conservativity may be given two different meanings in set.mm, due to the difference between the "scheme level" of set.mm and the "object level" of classical treatments. For a proof that these three axiomatic statements yield an eliminable and weakly (that is, object-level) conservative extension of FOL= plus ax-ext 2793, see Appendix of [Levy] p. 357.

6. References and history. The concept of class abstraction dates back to at least Frege, and is used by Whitehead and Russell. This definition is Definition 2.1 of [Quine] p. 16 and Axiom 4.3.1 of [Levy] p. 12. It is called the "axiom of class comprehension" by [Levy] p. 358, who treats the theory of classes as an extralogical extension to predicate logic and set theory axioms. He calls the construction {𝑦𝜑} a "class term". For a full description of how classes are introduced and how to recover the primitive language, see the books of Quine and Levy (and the comment of abeq2 2945 for a quick overview). For a general discussion of the theory of classes, see mmset.html#class 2945. (Contributed by NM, 26-May-1993.) (Revised by BJ, 19-Aug-2023.)

(𝑥 ∈ {𝑦𝜑} ↔ [𝑥 / 𝑦]𝜑)

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