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Type | Label | Description |
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Statement | ||
Theorem | moexexv 2701* | "At most one" double quantification. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker moexexvw 2690 when possible. (Contributed by NM, 26-Jan-1997.) (New usage is discouraged.) |
⊢ ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦∃𝑥(𝜑 ∧ 𝜓)) | ||
Theorem | 2moex 2702 | Double quantification with "at most one". Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker 2moexv 2689 when possible. (Contributed by NM, 3-Dec-2001.) (New usage is discouraged.) |
⊢ (∃*𝑥∃𝑦𝜑 → ∀𝑦∃*𝑥𝜑) | ||
Theorem | 2euex 2703 | Double quantification with existential uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker 2euexv 2693 when possible. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (New usage is discouraged.) |
⊢ (∃!𝑥∃𝑦𝜑 → ∃𝑦∃!𝑥𝜑) | ||
Theorem | 2eumo 2704 | Nested unique existential quantifier and at-most-one quantifier. (Contributed by NM, 3-Dec-2001.) |
⊢ (∃!𝑥∃*𝑦𝜑 → ∃*𝑥∃!𝑦𝜑) | ||
Theorem | 2eu2ex 2705 | Double existential uniqueness. (Contributed by NM, 3-Dec-2001.) |
⊢ (∃!𝑥∃!𝑦𝜑 → ∃𝑥∃𝑦𝜑) | ||
Theorem | 2moswap 2706 | 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 2379. Use the weaker 2moswapv 2691 when possible. (Contributed by NM, 10-Apr-2004.) (New usage is discouraged.) |
⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑)) | ||
Theorem | 2euswap 2707 | 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 2379. Use the weaker 2euswapv 2692 when possible. (Contributed by NM, 10-Apr-2004.) (New usage is discouraged.) |
⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃𝑦𝜑 → ∃!𝑦∃𝑥𝜑)) | ||
Theorem | 2exeu 2708 | 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 2379. Use the weaker 2exeuv 2694 when possible. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.) |
⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → ∃!𝑥∃!𝑦𝜑) | ||
Theorem | 2mo2 2709* | 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 2710. This is the analogue of 2eu4 2716 for existential uniqueness. (Contributed by Wolf Lammen, 26-Oct-2019.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 3-Jan-2023.) |
⊢ ((∃*𝑥∃𝑦𝜑 ∧ ∃*𝑦∃𝑥𝜑) ↔ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) | ||
Theorem | 2mo 2710* | Two ways of expressing "there exists at most one ordered pair 〈𝑥, 𝑦〉 such that 𝜑(𝑥, 𝑦) holds. See also 2mo2 2709. (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 2-Nov-2019.) |
⊢ (∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∀𝑥∀𝑦∀𝑧∀𝑤((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) | ||
Theorem | 2mos 2711* | Double "there exists at most one", using implicit substitution. (Contributed by NM, 10-Feb-2005.) |
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∀𝑥∀𝑦∀𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) | ||
Theorem | 2eu1 2712 | 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 2379. Use the weaker 2eu1v 2713 when possible. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 23-Apr-2023.) (New usage is discouraged.) |
⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑))) | ||
Theorem | 2eu1v 2713* | Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one. Version of 2eu1 2712 with 𝑥 and 𝑦 distinct, but not requiring ax-13 2379. (Contributed by NM, 3-Dec-2001.) (Revised by Wolf Lammen, 2-Oct-2023.) |
⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑))) | ||
Theorem | 2eu2 2714 | Double existential uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 3-Dec-2001.) (New usage is discouraged.) |
⊢ (∃!𝑦∃𝑥𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ ∃!𝑥∃𝑦𝜑)) | ||
Theorem | 2eu3 2715 | Double existential uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 23-Apr-2023.) (New usage is discouraged.) |
⊢ (∀𝑥∀𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑))) | ||
Theorem | 2eu4 2716* | 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 2712 for a condition under which the naive definition holds and 2exeu 2708 for a one-way implication. See 2eu5 2717 and 2eu8 2721 for alternate definitions. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 14-Sep-2019.) |
⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))) | ||
Theorem | 2eu5 2717* | An alternate definition of double existential uniqueness (see 2eu4 2716). 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 "there exists at most one"). (Contributed by NM, 26-Oct-2003.) Avoid ax-13 2379. (Revised by Wolf Lammen, 2-Oct-2023.) |
⊢ ((∃!𝑥∃!𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))) | ||
Theorem | 2eu5OLD 2718* | Obsolete version of 2eu5 2717 as of 2-Oct-2023. (Contributed by NM, 26-Oct-2003.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((∃!𝑥∃!𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))) | ||
Theorem | 2eu6 2719* | 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.) |
⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ ∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) | ||
Theorem | 2eu7 2720 | Two equivalent expressions for double existential uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 19-Feb-2005.) (New usage is discouraged.) |
⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) ↔ ∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑)) | ||
Theorem | 2eu8 2721 | 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 2720. Usage of this theorem is discouraged because it depends on ax-13 2379. (Contributed by NM, 20-Feb-2005.) (New usage is discouraged.) |
⊢ (∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑥∃!𝑦(∃!𝑥𝜑 ∧ ∃𝑦𝜑)) | ||
Theorem | euae 2722* | 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 33868 and dtru 5236. (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) | ||
Theorem | exists1 2723* | 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 5236. (Contributed by NM, 5-Apr-2004.) (Proof shortened by BJ, 7-Oct-2022.) |
⊢ (∃!𝑥 𝑥 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦) | ||
Theorem | exists2 2724 | 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) | ||
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 1770) 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/ 1770. 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 1844. We translate "all 𝜑 is 𝜓 " to ∀𝑥(𝜑 → 𝜓), "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 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, 𝑥 ∈ 𝐴, or 𝑥 ⊆ 𝐴. Here we use wff variables instead of specialized restricted forms. This generalization makes the syllogisms more useful in more circumstances. In addition, these expressions make it clearer that the syllogisms of Aristotelian 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. 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/ 1844. 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 2731, celaront 2733, cesaro 2740, camestros 2741, felapton 2748, darapti 2746, calemos 2752, fesapo 2753, and bamalip 2754. 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 2725 to bamalip 2754) are all proven from { ax-mp 5, ax-1 6, ax-2 7, ax-3 8, ax-gen 1797, ax-4 1811 }, 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 2181 (inference form of sp 2180, which corresponds to the axiom (T) of modal logic), as demonstrated by dariiALT 2728, barbariALT 2732, festinoALT 2737, barocoALT 2739, daraptiALT 2747. | ||
Theorem | barbara 2725 | "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 ∈ 𝐻 → 𝑥 ∈ 𝑀) (all men are mortal) and ∀𝑥(𝑥 = 𝑆 → 𝑥 ∈ 𝐻) (Socrates is a man) therefore ∀𝑥(𝑥 = 𝑆 → 𝑥 ∈ 𝑀) (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 1894. There are a legion of sources for Barbara, including http://www.friesian.com/aristotl.htm 1894, http://plato.stanford.edu/entries/aristotle-logic/ 1894, and https://en.wikipedia.org/wiki/Syllogism 1894. (Contributed by David A. Wheeler, 24-Aug-2016.) |
⊢ ∀𝑥(𝜑 → 𝜓) & ⊢ ∀𝑥(𝜒 → 𝜑) ⇒ ⊢ ∀𝑥(𝜒 → 𝜓) | ||
Theorem | celarent 2726 | "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 2725. 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 2725. (Contributed by David A. Wheeler, 24-Aug-2016.) |
⊢ ∀𝑥(𝜑 → ¬ 𝜓) & ⊢ ∀𝑥(𝜒 → 𝜑) ⇒ ⊢ ∀𝑥(𝜒 → ¬ 𝜓) | ||
Theorem | darii 2727 | "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 2728 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(𝜑 → 𝜓) & ⊢ ∃𝑥(𝜒 ∧ 𝜑) ⇒ ⊢ ∃𝑥(𝜒 ∧ 𝜓) | ||
Theorem | dariiALT 2728 | Alternate proof of darii 2727, shorter but using more axioms. This shows how the use of spi 2181 may shorten some proofs of the Aristotelian syllogisms, even though this adds axiom dependencies. Note that spi 2181 is the inference associated with sp 2180, 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(𝜑 → 𝜓) & ⊢ ∃𝑥(𝜒 ∧ 𝜑) ⇒ ⊢ ∃𝑥(𝜒 ∧ 𝜓) | ||
Theorem | ferio 2729 | "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 2727. 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 2727. (Contributed by David A. Wheeler, 24-Aug-2016.) |
⊢ ∀𝑥(𝜑 → ¬ 𝜓) & ⊢ ∃𝑥(𝜒 ∧ 𝜑) ⇒ ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓) | ||
Theorem | barbarilem 2730 | Lemma for barbari 2731 and the other Aristotelian syllogisms with existential assumption. (Contributed by BJ, 16-Sep-2022.) |
⊢ ∃𝑥𝜑 & ⊢ ∀𝑥(𝜑 → 𝜓) ⇒ ⊢ ∃𝑥(𝜑 ∧ 𝜓) | ||
Theorem | barbari 2731 | "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(𝜑 → 𝜓) & ⊢ ∀𝑥(𝜒 → 𝜑) & ⊢ ∃𝑥𝜒 ⇒ ⊢ ∃𝑥(𝜒 ∧ 𝜓) | ||
Theorem | barbariALT 2732 | Alternate proof of barbari 2731, shorter but using more axioms. See comment of dariiALT 2728. (Contributed by David A. Wheeler, 27-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ∀𝑥(𝜑 → 𝜓) & ⊢ ∀𝑥(𝜒 → 𝜑) & ⊢ ∃𝑥𝜒 ⇒ ⊢ ∃𝑥(𝜒 ∧ 𝜓) | ||
Theorem | celaront 2733 | "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 2731. 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 2731. (Contributed by David A. Wheeler, 27-Aug-2016.) |
⊢ ∀𝑥(𝜑 → ¬ 𝜓) & ⊢ ∀𝑥(𝜒 → 𝜑) & ⊢ ∃𝑥𝜒 ⇒ ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓) | ||
Theorem | cesare 2734 | "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 2726. (Contributed by David A. Wheeler, 27-Aug-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.) |
⊢ ∀𝑥(𝜑 → ¬ 𝜓) & ⊢ ∀𝑥(𝜒 → 𝜓) ⇒ ⊢ ∀𝑥(𝜒 → ¬ 𝜑) | ||
Theorem | camestres 2735 | "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(𝜑 → 𝜓) & ⊢ ∀𝑥(𝜒 → ¬ 𝜓) ⇒ ⊢ ∀𝑥(𝜒 → ¬ 𝜑) | ||
Theorem | festino 2736 | "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(𝜒 ∧ 𝜓) ⇒ ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑) | ||
Theorem | festinoALT 2737 | Alternate proof of festino 2736, shorter but using more axioms. See comment of dariiALT 2728. (Contributed by David A. Wheeler, 27-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ∀𝑥(𝜑 → ¬ 𝜓) & ⊢ ∃𝑥(𝜒 ∧ 𝜓) ⇒ ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑) | ||
Theorem | baroco 2738 | "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(𝜑 → 𝜓) & ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓) ⇒ ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑) | ||
Theorem | barocoALT 2739 | Alternate proof of festino 2736, shorter but using more axioms. See comment of dariiALT 2728. (Contributed by David A. Wheeler, 27-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ∀𝑥(𝜑 → 𝜓) & ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓) ⇒ ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑) | ||
Theorem | cesaro 2740 | "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(𝜒 → 𝜓) & ⊢ ∃𝑥𝜒 ⇒ ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑) | ||
Theorem | camestros 2741 | "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(𝜑 → 𝜓) & ⊢ ∀𝑥(𝜒 → ¬ 𝜓) & ⊢ ∃𝑥𝜒 ⇒ ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑) | ||
Theorem | datisi 2742 | "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(𝜑 → 𝜓) & ⊢ ∃𝑥(𝜑 ∧ 𝜒) ⇒ ⊢ ∃𝑥(𝜒 ∧ 𝜓) | ||
Theorem | disamis 2743 | "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(𝜑 ∧ 𝜓) & ⊢ ∀𝑥(𝜑 → 𝜒) ⇒ ⊢ ∃𝑥(𝜒 ∧ 𝜓) | ||
Theorem | ferison 2744 | "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 2742. In Aristotelian notation, EIO-3: MeP and MiS therefore SoP. (Contributed by David A. Wheeler, 28-Aug-2016.) |
⊢ ∀𝑥(𝜑 → ¬ 𝜓) & ⊢ ∃𝑥(𝜑 ∧ 𝜒) ⇒ ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓) | ||
Theorem | bocardo 2745 | "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 2743. 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(𝜑 → 𝜒) ⇒ ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓) | ||
Theorem | darapti 2746 | "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(𝜑 → 𝜓) & ⊢ ∀𝑥(𝜑 → 𝜒) & ⊢ ∃𝑥𝜑 ⇒ ⊢ ∃𝑥(𝜒 ∧ 𝜓) | ||
Theorem | daraptiALT 2747 | Alternate proof of darapti 2746, shorter but using more axioms. See comment of dariiALT 2728. (Contributed by David A. Wheeler, 27-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ∀𝑥(𝜑 → 𝜓) & ⊢ ∀𝑥(𝜑 → 𝜒) & ⊢ ∃𝑥𝜑 ⇒ ⊢ ∃𝑥(𝜒 ∧ 𝜓) | ||
Theorem | felapton 2748 | "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 2746. 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(𝜑 → 𝜒) & ⊢ ∃𝑥𝜑 ⇒ ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓) | ||
Theorem | calemes 2749 | "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(𝜑 → 𝜓) & ⊢ ∀𝑥(𝜓 → ¬ 𝜒) ⇒ ⊢ ∀𝑥(𝜒 → ¬ 𝜑) | ||
Theorem | dimatis 2750 | "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 2727 with positions interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.) |
⊢ ∃𝑥(𝜑 ∧ 𝜓) & ⊢ ∀𝑥(𝜓 → 𝜒) ⇒ ⊢ ∃𝑥(𝜒 ∧ 𝜑) | ||
Theorem | fresison 2751 | "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(𝜓 ∧ 𝜒) ⇒ ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑) | ||
Theorem | calemos 2752 | "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(𝜑 → 𝜓) & ⊢ ∀𝑥(𝜓 → ¬ 𝜒) & ⊢ ∃𝑥𝜒 ⇒ ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑) | ||
Theorem | fesapo 2753 | "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(𝜓 → 𝜒) & ⊢ ∃𝑥𝜓 ⇒ ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑) | ||
Theorem | bamalip 2754 | "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 2731. (Contributed by David A. Wheeler, 28-Aug-2016.) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.) |
⊢ ∀𝑥(𝜑 → 𝜓) & ⊢ ∀𝑥(𝜓 → 𝜒) & ⊢ ∃𝑥𝜑 ⇒ ⊢ ∃𝑥(𝜒 ∧ 𝜑) | ||
Intuitionistic (constructive) logic is similar to classical logic with the notable omission of ax-3 8 and theorems such as exmid 892 or peirce 205. 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 845 and df-ex 1782 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 1811, ax-11 2158, ax-gen 1797, ax-7 2015, ax-12 2175, ax-8 2113, ax-9 2121, and ax-5 1911. 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. | ||
Theorem | axia1 2755 | Left 'and' elimination (intuitionistic logic axiom ax-ia1). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜑) | ||
Theorem | axia2 2756 | Right 'and' elimination (intuitionistic logic axiom ax-ia2). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜓) | ||
Theorem | axia3 2757 | 'And' introduction (intuitionistic logic axiom ax-ia3). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓))) | ||
Theorem | axin1 2758 | 'Not' introduction (intuitionistic logic axiom ax-in1). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.) |
⊢ ((𝜑 → ¬ 𝜑) → ¬ 𝜑) | ||
Theorem | axin2 2759 | 'Not' elimination (intuitionistic logic axiom ax-in2). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.) |
⊢ (¬ 𝜑 → (𝜑 → 𝜓)) | ||
Theorem | axio 2760 | Definition of 'or' (intuitionistic logic axiom ax-io). (Contributed by Jim Kingdon, 21-May-2018.) (New usage is discouraged.) |
⊢ (((𝜑 ∨ 𝜒) → 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜒 → 𝜓))) | ||
Theorem | axi4 2761 | Specialization (intuitionistic logic axiom ax-4). This is just sp 2180 by another name. (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.) |
⊢ (∀𝑥𝜑 → 𝜑) | ||
Theorem | axi5r 2762 | Converse of axc4 2329 (intuitionistic logic axiom ax-i5r). (Contributed by Jim Kingdon, 31-Dec-2017.) |
⊢ ((∀𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∀𝑥𝜑 → 𝜓)) | ||
Theorem | axial 2763 | 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 → ∀𝑥∀𝑥𝜑) | ||
Theorem | axie1 2764 | 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 → ∀𝑥∃𝑥𝜑) | ||
Theorem | axie2 2765 | A key property of existential quantification (intuitionistic logic axiom ax-ie2). (Contributed by Jim Kingdon, 31-Dec-2017.) |
⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))) | ||
Theorem | axi9 2766 | Axiom of existence (intuitionistic logic axiom ax-i9). In classical logic, this is equivalent to ax-6 1970 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 | ||
Theorem | axi10 2767 | Axiom of Quantifier Substitution (intuitionistic logic axiom ax-10). This is just axc11n 2437 by another name. (Contributed by Jim Kingdon, 31-Dec-2017.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) | ||
Theorem | axi12 2768 | Axiom of Quantifier Introduction (intuitionistic logic axiom ax-i12). In classical logic, this is mostly a restatement of axc9 2389 (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 2379. (Contributed by Jim Kingdon, 31-Dec-2017.) Avoid ax-11 2158. (Revised by Wolf Lammen, 24-Apr-2023.) (New usage is discouraged.) |
⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) | ||
Theorem | axbnd 2769 | Axiom of Bundling (intuitionistic logic axiom ax-bnd). In classical logic, this and axi12 2768 are fairly straightforward consequences of axc9 2389. But in intuitionistic logic, it is not easy to add the extra ∀𝑥 to axi12 2768 and so we treat the two as separate axioms. Usage of this theorem is discouraged because it depends on ax-13 2379. (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 ∨ ∀𝑥∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) | ||
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 3719), 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 2771.) 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. | ||
Axiom | ax-ext 2770* |
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 2128.
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 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)), and equality 𝑥 = 𝑦 is defined as ∀𝑤(𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 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 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 2770 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 5154, 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 2770 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(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦) | ||
Theorem | axexte 2771* | The axiom of extensionality (ax-ext 2770) 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((𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦) | ||
Theorem | axextg 2772* | 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 2142, ax-12 2175, ax-13 2379. (Revised by BJ, 12-Jul-2019.) (Revised by Wolf Lammen, 9-Dec-2019.) |
⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦) | ||
Theorem | axextb 2773* | 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 2770 and df-cleq 2791. (Contributed by NM, 14-Nov-2008.) |
⊢ (𝑥 = 𝑦 ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)) | ||
Theorem | axextmo 2774* | 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.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ ∃*𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) | ||
Theorem | nulmo 2775* | There exists at most one empty set. With either axnul 5173 or axnulALT 5172 or ax-nul 5174, 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 4258. (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.) |
⊢ ∃*𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 | ||
Syntax | cab 2776 | Introduce the class abstraction (or class builder) notation: {𝑥 ∣ 𝜑} is the class of sets 𝑥 such that 𝜑(𝑥) is true. A setvar variable can be expressed as a class abstraction per theorem cvjust 2793, justifying the substitution of class variables for setvar variables via the use of cv 1537. |
class {𝑥 ∣ 𝜑} | ||
Definition | df-clab 2777 |
Define class abstractions, that is, classes of the form {𝑦 ∣ 𝜑},
which is read "the class of sets 𝑦 such that 𝜑(𝑦)".
A few remarks are in order: 1. The axiomatic statement df-clab 2777 does not define the class abstraction {𝑦 ∣ 𝜑} itself, that is, it does not have the form ⊢ {𝑦 ∣ 𝜑} = ... 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 2791). Instead, df-clab 2777 has the form ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ ...), 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 2777 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 2770. Therefore, df-clab 2777 can be considered a definition only in systems that can prove ax-ext 2770 (and the necessary first-order logic). 2. As in all definitions, the definiendum (the left-hand side of the biconditional) 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 ∈ {𝑥 ∣ 𝜑} which are sometimes useful to shorten proofs (because of abid 2780). Most often, however, 𝑥 does not occur in {𝑦 ∣ 𝜑} and 𝑦 is free in 𝜑. 3. Remark 1 stresses that df-clab 2777 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 predicate ∈ has already appeared earlier (e.g., in the non-syntactic statement ax-8 2113). Indeed, the definiendum 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 2111 and cab 2776, and it can be called an "extension" of the membership predicate because of wel 2112, whose proof uses cv 1537. An a posteriori justification for cv 1537 is given by cvjust 2793, 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 3719). 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 3515 which is used, for example, to convert elirrv 9044 to elirr 9045. 5. Definition or axiom? The question arises with the three axiomatic statements introducing classes, df-clab 2777, df-cleq 2791, and df-clel 2870, to decide if they qualify as definitions or if they should be called axioms. Under the strict definition of "definition" (see conventions 28185), they are not definitions (see Remarks 1 and 3 above, and similarly for df-cleq 2791 and df-clel 2870). 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 2770, 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 {𝑦 ∣ 𝜑} 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 2922 for a quick overview). For a general discussion of the theory of classes, see mmset.html#class 2922. (Contributed by NM, 26-May-1993.) (Revised by BJ, 19-Aug-2023.) |
⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑) | ||
Theorem | eleq1ab 2778 |
Extension (in the sense of Remark 3 of the comment of df-clab 2777) of
elequ1 2118 from formulas of the form "setvar ∈ setvar" to formulas of
the form "setvar ∈ class
abstraction". This extension does not
require ax-8 2113 contrary to elequ1 2118, but recall from Remark 3 of the
comment of df-clab 2777 that it can be considered an extension only
because
of cvjust 2793, which does require ax-8 2113.
This is an instance of eleq1w 2872 where the containing class is a class abstraction, and contrary to it, it can be proved without df-clel 2870. See also eleq1 2877 for general classes. The straightforward yet important fact that this statement can be proved from FOL= plus df-clab 2777 (hence without ax-ext 2770, df-cleq 2791 or df-clel 2870) was stressed by Mario Carneiro. (Contributed by BJ, 17-Aug-2023.) |
⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑧 ∣ 𝜑} ↔ 𝑦 ∈ {𝑧 ∣ 𝜑})) | ||
Theorem | cleljustab 2779* | Extension of cleljust 2120 from formulas of the form "setvar ∈ setvar" to formulas of the form "setvar ∈ class abstraction". This is an instance of dfclel 2871 where the containing class is a class abstraction. The same remarks as for eleq1ab 2778 apply. (Contributed by BJ, 8-Nov-2021.) (Proof shortened by Steven Nguyen, 19-May-2023.) |
⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ {𝑦 ∣ 𝜑})) | ||
Theorem | abid 2780 | Simplification of class abstraction notation when the free and bound variables are identical. (Contributed by NM, 26-May-1993.) |
⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | ||
Theorem | vexwt 2781 | A standard theorem of predicate calculus (stdpc4 2073) expressed using class abstractions. Closed form of vexw 2782. (Contributed by BJ, 14-Jun-2019.) |
⊢ (∀𝑥𝜑 → 𝑦 ∈ {𝑥 ∣ 𝜑}) | ||
Theorem | vexw 2782 |
If 𝜑
is a theorem, then any set belongs to the class
{𝑥
∣ 𝜑}.
Therefore, {𝑥 ∣ 𝜑} is "a" universal class.
This is the closest one can get to defining a universal class, or proving vex 3444, without using ax-ext 2770. Note that this theorem has no disjoint variable condition and does not use df-clel 2870 nor df-cleq 2791 either: only propositional logic and ax-gen 1797 and df-clab 2777. This is sbt 2071 expressed using class abstractions. Without ax-ext 2770, one cannot define "the" universal class, since one could not prove for instance the justification theorem {𝑥 ∣ ⊤} = {𝑦 ∣ ⊤} (see vjust 3442). Indeed, in order to prove any equality of classes, one needs df-cleq 2791, which has ax-ext 2770 as a hypothesis. Therefore, the classes {𝑥 ∣ ⊤}, {𝑦 ∣ (𝜑 → 𝜑)}, {𝑧 ∣ (∀𝑡𝑡 = 𝑡 → ∀𝑡𝑡 = 𝑡)} and countless others are all universal classes whose equality cannot be proved without ax-ext 2770. Once dfcleq 2792 is available, we will define "the" universal class in df-v 3443. Its degenerate instance is also a simple consequence of abid 2780 (using mpbir 234). (Contributed by BJ, 13-Jun-2019.) Reduce axiom dependencies. (Revised by Steven Nguyen, 25-Apr-2023.) |
⊢ 𝜑 ⇒ ⊢ 𝑦 ∈ {𝑥 ∣ 𝜑} | ||
Theorem | vextru 2783 | Every setvar is a member of {𝑥 ∣ ⊤}, which is therefore "a" universal class. Once class extensionality dfcleq 2792 is available, we can say "the" universal class (see df-v 3443). This is sbtru 2072 expressed using class abstractions. (Contributed by BJ, 2-Sep-2023.) |
⊢ 𝑦 ∈ {𝑥 ∣ ⊤} | ||
Theorem | hbab1 2784* | Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 26-May-1993.) |
⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}) | ||
Theorem | nfsab1 2785* | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove use of ax-12 2175. (Revised by SN, 20-Sep-2023.) |
⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} | ||
Theorem | nfsab1OLD 2786* | Obsolete version of nfsab1 2785 as of 20-Sep-2023. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} | ||
Theorem | hbab 2787* | Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995.) Add disjoint variable condition to avoid ax-13 2379. See hbabg 2788 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} → ∀𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}) | ||
Theorem | hbabg 2788* | Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2379. See hbab 2787 for a version with more disjoint variable conditions, but not requiring ax-13 2379. (Contributed by NM, 1-Mar-1995.) (New usage is discouraged.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} → ∀𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}) | ||
Theorem | nfsab 2789* | Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) Add disjoint variable condition to avoid ax-13 2379. See nfsabg 2790 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜑} | ||
Theorem | nfsabg 2790* | Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2379. See nfsab 2789 for a version with more disjoint variable conditions, but not requiring ax-13 2379. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜑} | ||
This section introduces class equality in df-cleq 2791. Note that apart from the local introduction of class variables to state the syntax axioms wceq 1538 and wcel 2111, this section is the first to use class variables. Therefore, the file set.mm contains declarations of class variables at the beginning of this section (not visible on the webpages). | ||
Definition | df-cleq 2791* |
Define the equality connective between classes. Definition 2.7 of
[Quine] p. 18. Also Definition 4.5 of
[TakeutiZaring] p. 13; Chapter 4
provides its justification and methods for eliminating it. Note that
its elimination will not necessarily result in a single wff in the
original language but possibly a "scheme" of wffs.
The hypotheses express that all instances of the conclusion where class variables are replaced with setvar variables hold. Therefore, this definition merely extends to class variables something that is true for setvar variables, hence is conservative. This is only a proof sketch of conservativity; for details see Appendix of [Levy] p. 357. This is the reason why we call this axiomatic statement a "definition", even though it does not have the usual form of a definition. If we required a definition to have the usual form, we would call df-cleq 2791 an axiom. See also comments under df-clab 2777, df-clel 2870, and abeq2 2922. In the form of dfcleq 2792, this is called the "axiom of extensionality" by [Levy] p. 338, who treats the theory of classes as an extralogical extension to our logic and set theory axioms. While the three class definitions df-clab 2777, df-cleq 2791, and df-clel 2870 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 2870. (Contributed by NM, 15-Sep-1993.) (Revised by BJ, 24-Jun-2019.) |
⊢ (𝑦 = 𝑧 ↔ ∀𝑢(𝑢 ∈ 𝑦 ↔ 𝑢 ∈ 𝑧)) & ⊢ (𝑡 = 𝑡 ↔ ∀𝑣(𝑣 ∈ 𝑡 ↔ 𝑣 ∈ 𝑡)) ⇒ ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | ||
Theorem | dfcleq 2792* | The defining characterization of class equality. It is proved, over Tarski's FOL, from the axiom of (set) extensionality (ax-ext 2770) and the definition of class equality (df-cleq 2791). Its forward implication is called "class extensionality". Remark: the proof uses axextb 2773 to prove also the hypothesis of df-cleq 2791 that is a degenerate instance, but it could be proved also from minimal propositional calculus and { ax-gen 1797, equid 2019 }. (Contributed by NM, 15-Sep-1993.) (Revised by BJ, 24-Jun-2019.) |
⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | ||
Theorem | cvjust 2793* | Every set is a class. Proposition 4.9 of [TakeutiZaring] p. 13. This theorem shows that a setvar variable can be expressed as a class abstraction. This provides a motivation for the class syntax construction cv 1537, which allows us to substitute a setvar variable for a class variable. See also cab 2776 and df-clab 2777. Note that this is not a rigorous justification, because cv 1537 is used as part of the proof of this theorem, but a careful argument can be made outside of the formalism of Metamath, for example as is done in Chapter 4 of Takeuti and Zaring. See also the discussion under the definition of class in [Jech] p. 4 showing that "Every set can be considered to be a class." See abid1 2931 for the version of cvjust 2793 extended to classes. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2379. (Revised by Wolf Lammen, 4-May-2023.) |
⊢ 𝑥 = {𝑦 ∣ 𝑦 ∈ 𝑥} | ||
Theorem | ax9ALT 2794 | Proof of ax-9 2121 from Tarski's FOL and dfcleq 2792. For a version not using ax-8 2113 either, see bj-ax9 34340. This shows that dfcleq 2792 is too powerful to be used as a definition instead of df-cleq 2791. Note that ax-ext 2770 is also a direct consequence of dfcleq 2792 (as an instance of its forward implication). (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦)) | ||
Theorem | eqriv 2795* | Infer equality of classes from equivalence of membership. (Contributed by NM, 21-Jun-1993.) |
⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ⇒ ⊢ 𝐴 = 𝐵 | ||
Theorem | eqrdv 2796* | Deduce equality of classes from equivalence of membership. (Contributed by NM, 17-Mar-1996.) |
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | eqrdav 2797* | Deduce equality of classes from an equivalence of membership that depends on the membership variable. (Contributed by NM, 7-Nov-2008.) (Proof shortened by Wolf Lammen, 19-Nov-2019.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐶) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | eqid 2798 |
Law of identity (reflexivity of class equality). Theorem 6.4 of [Quine]
p. 41.
This is part of Frege's eighth axiom per Proposition 54 of [Frege1879] p. 50; see also biid 264. An early mention of this law can be found in Aristotle, Metaphysics, Z.17, 1041a10-20. (Thanks to Stefan Allan and BJ for this information.) (Contributed by NM, 21-Jun-1993.) (Revised by BJ, 14-Oct-2017.) |
⊢ 𝐴 = 𝐴 | ||
Theorem | eqidd 2799 | Class identity law with antecedent. (Contributed by NM, 21-Aug-2008.) |
⊢ (𝜑 → 𝐴 = 𝐴) | ||
Theorem | eqeq1d 2800 | Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 5-Dec-2019.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶)) |
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