P. 21 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 2001-2100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ferio 2001	"Ferio" ("Ferioque"), one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and some 𝜒 is 𝜑, therefore some 𝜒 is not 𝜓. (In Aristotelian notation, EIO-1: MeP and SiM therefore SoP.) For example, given "No homework is fun" and "Some reading is homework", therefore "Some reading is not fun". This is essentially a logical axiom in Aristotelian logic. Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∃𝑥(𝜒 ∧ 𝜑)    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)
Theorem	barbari 2002	"Barbari", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, all 𝜒 is 𝜑, and some 𝜒 exist, therefore some 𝜒 is 𝜓. (In Aristotelian notation, AAI-1: MaP and SaM therefore SiP.) For example, given "All men are mortal", "All Greeks are men", and "Greeks exist", therefore "Some Greeks are mortal". Note the existence hypothesis (to prove the "some" in the conclusion). Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 30-Aug-2016.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜒 → 𝜑)    &   ⊢ ∃𝑥𝜒    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜓)
Theorem	celaront 2003	"Celaront", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, all 𝜒 is 𝜑, and some 𝜒 exist, therefore some 𝜒 is not 𝜓. (In Aristotelian notation, EAO-1: MeP and SaM therefore SoP.) For example, given "No reptiles have fur", "All snakes are reptiles.", and "Snakes exist.", prove "Some snakes have no fur". Note the existence hypothesis. Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∀𝑥(𝜒 → 𝜑)    &   ⊢ ∃𝑥𝜒    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)
Theorem	cesare 2004	"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 1999. (Contributed by David A. Wheeler, 27-Aug-2016.) (Revised by David A. Wheeler, 13-Nov-2016.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∀𝑥(𝜒 → 𝜓)    ⇒   ⊢ ∀𝑥(𝜒 → ¬ 𝜑)
Theorem	camestres 2005	"Camestres", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and no 𝜒 is 𝜓, therefore no 𝜒 is 𝜑. (In Aristotelian notation, AEE-2: PaM and SeM therefore SeP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜒 → ¬ 𝜓)    ⇒   ⊢ ∀𝑥(𝜒 → ¬ 𝜑)
Theorem	festino 2006	"Festino", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and some 𝜒 is 𝜓, therefore some 𝜒 is not 𝜑. (In Aristotelian notation, EIO-2: PeM and SiM therefore SoP.) (Contributed by David A. Wheeler, 25-Nov-2016.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∃𝑥(𝜒 ∧ 𝜓)    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)
Theorem	baroco 2007	"Baroco", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and some 𝜒 is not 𝜓, therefore some 𝜒 is not 𝜑. (In Aristotelian notation, AOO-2: PaM and SoM therefore SoP.) For example, "All informative things are useful", "Some websites are not useful", therefore "Some websites are not informative." (Contributed by David A. Wheeler, 28-Aug-2016.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)
Theorem	cesaro 2008	"Cesaro", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, all 𝜒 is 𝜓, and 𝜒 exist, therefore some 𝜒 is not 𝜑. (In Aristotelian notation, EAO-2: PeM and SaM therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∀𝑥(𝜒 → 𝜓)    &   ⊢ ∃𝑥𝜒    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)
Theorem	camestros 2009	"Camestros", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, no 𝜒 is 𝜓, and 𝜒 exist, therefore some 𝜒 is not 𝜑. (In Aristotelian notation, AEO-2: PaM and SeM therefore SoP.) For example, "All horses have hooves", "No humans have hooves", and humans exist, therefore "Some humans are not horses". (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜒 → ¬ 𝜓)    &   ⊢ ∃𝑥𝜒    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)
Theorem	datisi 2010	"Datisi", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and some 𝜑 is 𝜒, therefore some 𝜒 is 𝜓. (In Aristotelian notation, AII-3: MaP and MiS therefore SiP.) (Contributed by David A. Wheeler, 28-Aug-2016.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∃𝑥(𝜑 ∧ 𝜒)    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜓)
Theorem	disamis 2011	"Disamis", one of the syllogisms of Aristotelian logic. Some 𝜑 is 𝜓, and all 𝜑 is 𝜒, therefore some 𝜒 is 𝜓. (In Aristotelian notation, IAI-3: MiP and MaS therefore SiP.) (Contributed by David A. Wheeler, 28-Aug-2016.)
⊢ ∃𝑥(𝜑 ∧ 𝜓)    &   ⊢ ∀𝑥(𝜑 → 𝜒)    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜓)
Theorem	ferison 2012	"Ferison", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and some 𝜑 is 𝜒, therefore some 𝜒 is not 𝜓. (In Aristotelian notation, EIO-3: MeP and MiS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∃𝑥(𝜑 ∧ 𝜒)    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)
Theorem	bocardo 2013	"Bocardo", one of the syllogisms of Aristotelian logic. Some 𝜑 is not 𝜓, and all 𝜑 is 𝜒, therefore some 𝜒 is not 𝜓. (In Aristotelian notation, OAO-3: MoP and MaS therefore SoP.) For example, "Some cats have no tails", "All cats are mammals", therefore "Some mammals have no tails". A reorder of disamis 2011; prefer using that instead. (Contributed by David A. Wheeler, 28-Aug-2016.) (New usage is discouraged.)
⊢ ∃𝑥(𝜑 ∧ ¬ 𝜓)    &   ⊢ ∀𝑥(𝜑 → 𝜒)    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)
Theorem	felapton 2014	"Felapton", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, all 𝜑 is 𝜒, and some 𝜑 exist, therefore some 𝜒 is not 𝜓. (In Aristotelian notation, EAO-3: MeP and MaS therefore SoP.) For example, "No flowers are animals" and "All flowers are plants", therefore "Some plants are not animals". (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∀𝑥(𝜑 → 𝜒)    &   ⊢ ∃𝑥𝜑    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)
Theorem	darapti 2015	"Darapti", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, all 𝜑 is 𝜒, and some 𝜑 exist, therefore some 𝜒 is 𝜓. (In Aristotelian notation, AAI-3: MaP and MaS therefore SiP.) For example, "All squares are rectangles" and "All squares are rhombuses", therefore "Some rhombuses are rectangles". (Contributed by David A. Wheeler, 28-Aug-2016.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜑 → 𝜒)    &   ⊢ ∃𝑥𝜑    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜓)
Theorem	calemes 2016	"Calemes", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and no 𝜓 is 𝜒, therefore no 𝜒 is 𝜑. (In Aristotelian notation, AEE-4: PaM and MeS therefore SeP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜓 → ¬ 𝜒)    ⇒   ⊢ ∀𝑥(𝜒 → ¬ 𝜑)
Theorem	dimatis 2017	"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 2000 with positions interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.)
⊢ ∃𝑥(𝜑 ∧ 𝜓)    &   ⊢ ∀𝑥(𝜓 → 𝜒)    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜑)
Theorem	fresison 2018	"Fresison", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓 (PeM), and some 𝜓 is 𝜒 (MiS), therefore some 𝜒 is not 𝜑 (SoP). (In Aristotelian notation, EIO-4: PeM and MiS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∃𝑥(𝜓 ∧ 𝜒)    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)
Theorem	calemos 2019	"Calemos", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓 (PaM), no 𝜓 is 𝜒 (MeS), and 𝜒 exist, therefore some 𝜒 is not 𝜑 (SoP). (In Aristotelian notation, AEO-4: PaM and MeS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜓 → ¬ 𝜒)    &   ⊢ ∃𝑥𝜒    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)
Theorem	fesapo 2020	"Fesapo", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, all 𝜓 is 𝜒, and 𝜓 exist, therefore some 𝜒 is not 𝜑. (In Aristotelian notation, EAO-4: PeM and MaS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
⊢ ∀𝑥(𝜑 → ¬ 𝜓)    &   ⊢ ∀𝑥(𝜓 → 𝜒)    &   ⊢ ∃𝑥𝜓    ⇒   ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)
Theorem	bamalip 2021	"Bamalip", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, all 𝜓 is 𝜒, and 𝜑 exist, therefore some 𝜒 is 𝜑. (In Aristotelian notation, AAI-4: PaM and MaS therefore SiP.) Like barbari 2002. (Contributed by David A. Wheeler, 28-Aug-2016.)
⊢ ∀𝑥(𝜑 → 𝜓)    &   ⊢ ∀𝑥(𝜓 → 𝜒)    &   ⊢ ∃𝑥𝜑    ⇒   ⊢ ∃𝑥(𝜒 ∧ 𝜑)
PART 2  SET THEORY Set theory uses the formalism of propositional and predicate calculus to assert properties of arbitrary mathematical objects called "sets." A set can be an element of another set, and this relationship is indicated by the ∈ symbol. Starting with the simplest mathematical object, called the empty set, set theory builds up more and more complex structures whose existence follows from the axioms, eventually resulting in extremely complicated sets that we identify with the real numbers and other familiar mathematical objects. Here we develop set theory based on the Intuitionistic Zermelo-Fraenkel (IZF) system, mostly following the IZF axioms as laid out in [Crosilla]. Constructive Zermelo-Fraenkel (CZF), also described in Crosilla, is not as easy to formalize in metamath because the Axiom of Restricted Separation would require us to develop the ability to classify formulas as bounded formulas, similar to the machinery we have built up for asserting on whether variables are free in formulas.
2.1  IZF Set Theory - start with the Axiom of Extensionality
2.1.1  Introduce the Axiom of Extensionality
Axiom	ax-ext 2022*	Axiom of Extensionality. It states that two sets are identical if they contain the same elements. Axiom 1 of [Crosilla] p. "Axioms of CZF and IZF" (with unnecessary quantifiers removed). Set theory can also be formulated with a single primitive predicate ∈ on top of traditional predicate calculus without equality. In that case the Axiom of Extensionality becomes (∀𝑤(𝑤 ∈ 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 logical axioms, we would rewrite ax-8 1395 through ax-16 1695 with equality expanded according to the above definition. Some of those axioms could be proved from set theory and would be redundant. Not all of them are redundant, since our axioms of predicate calculus make essential use of equality for the proper substitution that is a primitive notion in traditional predicate calculus. A study of such an axiomatization would be an interesting project for someone exploring the foundations of logic. It is important to understand that strictly speaking, all of our set theory axioms are really schemes that represent an infinite number of actual axioms. This is inherent in the design of Metamath ("metavariable math"), which manipulates only metavariables. For example, the metavariable 𝑥 in ax-ext 2022 can represent any actual variable v1, v2, v3,... . Distinct variable restrictions ($d) prevent us from substituting say v1 for both 𝑥 and 𝑧. This is in contrast to typical textbook presentations that present actual axioms (except for axioms which involve wff metavariables). In practice, though, the theorems and proofs are essentially the same. The $d restrictions make each of the infinite axioms generated by the ax-ext 2022 scheme exactly logically equivalent to each other and in particular to the actual axiom of the textbook version. (Contributed by NM, 5-Aug-1993.)
⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)
Theorem	axext3 2023*	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.)
⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)
Theorem	axext4 2024*	A bidirectional version of Extensionality. Although this theorem "looks" like it is just a definition of equality, it requires the Axiom of Extensionality for its proof under our axiomatization. See the comments for ax-ext 2022. (Contributed by NM, 14-Nov-2008.)
⊢ (𝑥 = 𝑦 ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))
Theorem	bm1.1 2025*	Any set defined by a property is the only set defined by that property. Theorem 1.1 of [BellMachover] p. 462. (Contributed by NM, 30-Jun-1994.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑) → ∃!𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝜑))
2.1.2  Class abstractions (a.k.a. class builders)
Syntax	cab 2026	Introduce the class builder or class abstraction notation ("the class of sets 𝑥 such that 𝜑 is true"). Our class variables 𝐴, 𝐵, etc. range over class builders (sometimes implicitly). Note that a setvar variable can be expressed as a class builder per theorem cvjust 2035, justifying the assignment of setvar variables to class variables via the use of cv 1242.
class {𝑥 ∣ 𝜑}
Definition	df-clab 2027	Define class abstraction notation (so-called by Quine), also called a "class builder" in the literature. 𝑥 and 𝑦 need not be distinct. Definition 2.1 of [Quine] p. 16. Typically, 𝜑 will have 𝑦 as a free variable, and "{𝑦 ∣ 𝜑} " is read "the class of all sets 𝑦 such that 𝜑(𝑦) is true." We do not define {𝑦 ∣ 𝜑} in isolation but only as part of an expression that extends or "overloads" the ∈ relationship. This is our first use of the ∈ symbol to connect classes instead of sets. The syntax definition wcel 1393, which extends or "overloads" the wel 1394 definition connecting setvar variables, requires that both sides of ∈ be a class. In df-cleq 2033 and df-clel 2036, we introduce a new kind of variable (class variable) that can substituted with expressions such as {𝑦 ∣ 𝜑}. In the present definition, the 𝑥 on the left-hand side is a setvar variable. Syntax definition cv 1242 allows us to substitute a setvar variable 𝑥 for a class variable: all sets are classes by cvjust 2035 (but not necessarily vice-versa). For a full description of how classes are introduced and how to recover the primitive language, see the discussion in Quine (and under abeq2 2146 for a quick overview). Because class variables can be substituted with compound expressions and setvar variables cannot, it is often useful to convert a theorem containing a free setvar variable to a more general version with a class variable. This is called the "axiom of class comprehension" by [Levy] p. 338, who treats the theory of classes as an extralogical extension to our logic and set theory axioms. He calls the construction {𝑦 ∣ 𝜑} a "class term". For a general discussion of the theory of classes, see http://us.metamath.org/mpeuni/mmset.html#class. (Contributed by NM, 5-Aug-1993.)
⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)
Theorem	abid 2028	Simplification of class abstraction notation when the free and bound variables are identical. (Contributed by NM, 5-Aug-1993.)
⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
Theorem	hbab1 2029*	Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 5-Aug-1993.)
⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑})
Theorem	nfsab1 2030*	Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}
Theorem	hbab 2031*	Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} → ∀𝑥 𝑧 ∈ {𝑦 ∣ 𝜑})
Theorem	nfsab 2032*	Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}
Definition	df-cleq 2033*	Define the equality connective between classes. Definition 2.7 of [Quine] p. 18. Also Definition 4.5 of [TakeutiZaring] p. 13; Chapter 4 provides its justification and methods for eliminating it. Note that its elimination will not necessarily result in a single wff in the original language but possibly a "scheme" of wffs. This is an example of a somewhat "risky" definition, meaning that it has a more complex than usual soundness justification (outside of Metamath), because it "overloads" or reuses the existing equality symbol rather than introducing a new symbol. This allows us to make statements that may not hold for the original symbol. For example, it permits us to deduce 𝑦 = 𝑧 ↔ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧), which is not a theorem of logic but rather presupposes the Axiom of Extensionality (see theorem axext4 2024). We therefore include this axiom as a hypothesis, so that the use of Extensionality is properly indicated. We could avoid this complication by introducing a new symbol, say =2, in place of =. This would also have the advantage of making elimination of the definition straightforward, so that we could eliminate Extensionality as a hypothesis. We would then also have the advantage of being able to identify in various proofs exactly where Extensionality truly comes into play rather than just being an artifact of a definition. One of our theorems would then be 𝑥 =2 𝑦 ↔ 𝑥 = 𝑦 by invoking Extensionality. However, to conform to literature usage, we retain this overloaded definition. This also makes some proofs shorter and probably easier to read, without the constant switching between two kinds of equality. See also comments under df-clab 2027, df-clel 2036, and abeq2 2146. In the form of dfcleq 2034, this is called the "axiom of extensionality" by [Levy] p. 338, who treats the theory of classes as an extralogical extension to our logic and set theory axioms. For a general discussion of the theory of classes, see http://us.metamath.org/mpeuni/mmset.html#class. (Contributed by NM, 15-Sep-1993.)
⊢ (∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧) → 𝑦 = 𝑧)    ⇒   ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
Theorem	dfcleq 2034*	The same as df-cleq 2033 with the hypothesis removed using the Axiom of Extensionality ax-ext 2022. (Contributed by NM, 15-Sep-1993.)
⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
Theorem	cvjust 2035*	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 1242, which allows us to substitute a setvar variable for a class variable. See also cab 2026 and df-clab 2027. Note that this is not a rigorous justification, because cv 1242 is used as part of the proof of this theorem, but a careful argument can be made outside of the formalism of Metamath, for example as is done in Chapter 4 of Takeuti and Zaring. See also the discussion under the definition of class in [Jech] p. 4 showing that "Every set can be considered to be a class." (Contributed by NM, 7-Nov-2006.)
⊢ 𝑥 = {𝑦 ∣ 𝑦 ∈ 𝑥}
Definition	df-clel 2036*	Define the membership connective between classes. Theorem 6.3 of [Quine] p. 41, or Proposition 4.6 of [TakeutiZaring] p. 13, which we adopt as a definition. See these references for its metalogical justification. Note that like df-cleq 2033 it extends or "overloads" the use of the existing membership symbol, but unlike df-cleq 2033 it does not strengthen the set of valid wffs of logic when the class variables are replaced with setvar variables (see cleljust 1813), so we don't include any set theory axiom as a hypothesis. See also comments about the syntax under df-clab 2027. This is called the "axiom of membership" by [Levy] p. 338, who treats the theory of classes as an extralogical extension to our logic and set theory axioms. For a general discussion of the theory of classes, see http://us.metamath.org/mpeuni/mmset.html#class. (Contributed by NM, 5-Aug-1993.)
⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵))
Theorem	eqriv 2037*	Infer equality of classes from equivalence of membership. (Contributed by NM, 5-Aug-1993.)
⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)    ⇒   ⊢ 𝐴 = 𝐵
Theorem	eqrdv 2038*	Deduce equality of classes from equivalence of membership. (Contributed by NM, 17-Mar-1996.)
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	eqrdav 2039*	Deduce equality of classes from an equivalence of membership that depends on the membership variable. (Contributed by NM, 7-Nov-2008.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	eqid 2040	Law of identity (reflexivity of class equality). Theorem 6.4 of [Quine] p. 41. This law is thought to have originated with Aristotle (Metaphysics, Zeta, 17, 1041 a, 10-20). (Thanks to Stefan Allan and BJ for this information.) (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 14-Oct-2017.)
⊢ 𝐴 = 𝐴
Theorem	eqidd 2041	Class identity law with antecedent. (Contributed by NM, 21-Aug-2008.)
⊢ (𝜑 → 𝐴 = 𝐴)
Theorem	eqcom 2042	Commutative law for class equality. Theorem 6.5 of [Quine] p. 41. (Contributed by NM, 5-Aug-1993.)
⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
Theorem	eqcoms 2043	Inference applying commutative law for class equality to an antecedent. (Contributed by NM, 5-Aug-1993.)
⊢ (𝐴 = 𝐵 → 𝜑)    ⇒   ⊢ (𝐵 = 𝐴 → 𝜑)
Theorem	eqcomi 2044	Inference from commutative law for class equality. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ 𝐵 = 𝐴
Theorem	eqcomd 2045	Deduction from commutative law for class equality. (Contributed by NM, 15-Aug-1994.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐵 = 𝐴)
Theorem	eqeq1 2046	Equality implies equivalence of equalities. (Contributed by NM, 5-Aug-1993.)
⊢ (𝐴 = 𝐵 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶))
Theorem	eqeq1i 2047	Inference from equality to equivalence of equalities. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 = 𝐶 ↔ 𝐵 = 𝐶)
Theorem	eqeq1d 2048	Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶))
Theorem	eqeq2 2049	Equality implies equivalence of equalities. (Contributed by NM, 5-Aug-1993.)
⊢ (𝐴 = 𝐵 → (𝐶 = 𝐴 ↔ 𝐶 = 𝐵))
Theorem	eqeq2i 2050	Inference from equality to equivalence of equalities. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 = 𝐴 ↔ 𝐶 = 𝐵)
Theorem	eqeq2d 2051	Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 = 𝐴 ↔ 𝐶 = 𝐵))
Theorem	eqeq12 2052	Equality relationship among 4 classes. (Contributed by NM, 3-Aug-1994.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
Theorem	eqeq12i 2053	A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)
Theorem	eqeq12d 2054	A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
Theorem	eqeqan12d 2055	A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜓 → 𝐶 = 𝐷)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
Theorem	eqeqan12rd 2056	A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜓 → 𝐶 = 𝐷)    ⇒   ⊢ ((𝜓 ∧ 𝜑) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
Theorem	eqtr 2057	Transitive law for class equality. Proposition 4.7(3) of [TakeutiZaring] p. 13. (Contributed by NM, 25-Jan-2004.)
⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐶)
Theorem	eqtr2 2058	A transitive law for class equality. (Contributed by NM, 20-May-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ ((𝐴 = 𝐵 ∧ 𝐴 = 𝐶) → 𝐵 = 𝐶)
Theorem	eqtr3 2059	A transitive law for class equality. (Contributed by NM, 20-May-2005.)
⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐵)
Theorem	eqtri 2060	An equality transitivity inference. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐵 = 𝐶    ⇒   ⊢ 𝐴 = 𝐶
Theorem	eqtr2i 2061	An equality transitivity inference. (Contributed by NM, 21-Feb-1995.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐵 = 𝐶    ⇒   ⊢ 𝐶 = 𝐴
Theorem	eqtr3i 2062	An equality transitivity inference. (Contributed by NM, 6-May-1994.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐴 = 𝐶    ⇒   ⊢ 𝐵 = 𝐶
Theorem	eqtr4i 2063	An equality transitivity inference. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐵    ⇒   ⊢ 𝐴 = 𝐶
Theorem	3eqtri 2064	An inference from three chained equalities. (Contributed by NM, 29-Aug-1993.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐵 = 𝐶    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ 𝐴 = 𝐷
Theorem	3eqtrri 2065	An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐵 = 𝐶    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ 𝐷 = 𝐴
Theorem	3eqtr2i 2066	An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ 𝐴 = 𝐷
Theorem	3eqtr2ri 2067	An inference from three chained equalities. (Contributed by NM, 3-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ 𝐷 = 𝐴
Theorem	3eqtr3i 2068	An inference from three chained equalities. (Contributed by NM, 6-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐴 = 𝐶    &   ⊢ 𝐵 = 𝐷    ⇒   ⊢ 𝐶 = 𝐷
Theorem	3eqtr3ri 2069	An inference from three chained equalities. (Contributed by NM, 15-Aug-2004.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐴 = 𝐶    &   ⊢ 𝐵 = 𝐷    ⇒   ⊢ 𝐷 = 𝐶
Theorem	3eqtr4i 2070	An inference from three chained equalities. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐴    &   ⊢ 𝐷 = 𝐵    ⇒   ⊢ 𝐶 = 𝐷
Theorem	3eqtr4ri 2071	An inference from three chained equalities. (Contributed by NM, 2-Sep-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐴    &   ⊢ 𝐷 = 𝐵    ⇒   ⊢ 𝐷 = 𝐶
Theorem	eqtrd 2072	An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 = 𝐶)
Theorem	eqtr2d 2073	An equality transitivity deduction. (Contributed by NM, 18-Oct-1999.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐶 = 𝐴)
Theorem	eqtr3d 2074	An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐴 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐵 = 𝐶)
Theorem	eqtr4d 2075	An equality transitivity equality deduction. (Contributed by NM, 18-Jul-1995.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 = 𝐶)
Theorem	3eqtrd 2076	A deduction from three chained equalities. (Contributed by NM, 29-Oct-1995.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐵 = 𝐶)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → 𝐴 = 𝐷)
Theorem	3eqtrrd 2077	A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐵 = 𝐶)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → 𝐷 = 𝐴)
Theorem	3eqtr2d 2078	A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → 𝐴 = 𝐷)
Theorem	3eqtr2rd 2079	A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → 𝐷 = 𝐴)
Theorem	3eqtr3d 2080	A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐴 = 𝐶)    &   ⊢ (𝜑 → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → 𝐶 = 𝐷)
Theorem	3eqtr3rd 2081	A deduction from three chained equalities. (Contributed by NM, 14-Jan-2006.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐴 = 𝐶)    &   ⊢ (𝜑 → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → 𝐷 = 𝐶)
Theorem	3eqtr4d 2082	A deduction from three chained equalities. (Contributed by NM, 4-Aug-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐴)    &   ⊢ (𝜑 → 𝐷 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐶 = 𝐷)
Theorem	3eqtr4rd 2083	A deduction from three chained equalities. (Contributed by NM, 21-Sep-1995.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐴)    &   ⊢ (𝜑 → 𝐷 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐷 = 𝐶)
Theorem	syl5eq 2084	An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 = 𝐶)
Theorem	syl5req 2085	An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐶 = 𝐴)
Theorem	syl5eqr 2086	An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐵 = 𝐴    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 = 𝐶)
Theorem	syl5reqr 2087	An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
⊢ 𝐵 = 𝐴    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐶 = 𝐴)
Theorem	syl6eq 2088	An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ 𝐵 = 𝐶    ⇒   ⊢ (𝜑 → 𝐴 = 𝐶)
Theorem	syl6req 2089	An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ 𝐵 = 𝐶    ⇒   ⊢ (𝜑 → 𝐶 = 𝐴)
Theorem	syl6eqr 2090	An equality transitivity deduction. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ 𝐶 = 𝐵    ⇒   ⊢ (𝜑 → 𝐴 = 𝐶)
Theorem	syl6reqr 2091	An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ 𝐶 = 𝐵    ⇒   ⊢ (𝜑 → 𝐶 = 𝐴)
Theorem	sylan9eq 2092	An equality transitivity deduction. (Contributed by NM, 8-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜓 → 𝐵 = 𝐶)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶)
Theorem	sylan9req 2093	An equality transitivity deduction. (Contributed by NM, 23-Jun-2007.)
⊢ (𝜑 → 𝐵 = 𝐴)    &   ⊢ (𝜓 → 𝐵 = 𝐶)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶)
Theorem	sylan9eqr 2094	An equality transitivity deduction. (Contributed by NM, 8-May-1994.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜓 → 𝐵 = 𝐶)    ⇒   ⊢ ((𝜓 ∧ 𝜑) → 𝐴 = 𝐶)
Theorem	3eqtr3g 2095	A chained equality inference, useful for converting from definitions. (Contributed by NM, 15-Nov-1994.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ 𝐴 = 𝐶    &   ⊢ 𝐵 = 𝐷    ⇒   ⊢ (𝜑 → 𝐶 = 𝐷)
Theorem	3eqtr3a 2096	A chained equality inference, useful for converting from definitions. (Contributed by Mario Carneiro, 6-Nov-2015.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝜑 → 𝐴 = 𝐶)    &   ⊢ (𝜑 → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → 𝐶 = 𝐷)
Theorem	3eqtr4g 2097	A chained equality inference, useful for converting to definitions. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ 𝐶 = 𝐴    &   ⊢ 𝐷 = 𝐵    ⇒   ⊢ (𝜑 → 𝐶 = 𝐷)
Theorem	3eqtr4a 2098	A chained equality inference, useful for converting to definitions. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝜑 → 𝐶 = 𝐴)    &   ⊢ (𝜑 → 𝐷 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐶 = 𝐷)
Theorem	eq2tri 2099	A compound transitive inference for class equality. (Contributed by NM, 22-Jan-2004.)
⊢ (𝐴 = 𝐶 → 𝐷 = 𝐹)    &   ⊢ (𝐵 = 𝐷 → 𝐶 = 𝐺)    ⇒   ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐹) ↔ (𝐵 = 𝐷 ∧ 𝐴 = 𝐺))
Theorem	eleq1 2100	Equality implies equivalence of membership. (Contributed by NM, 5-Aug-1993.)
⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))