Theorem wl-df.clab 37870

Description: 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 2719 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 2732). Instead, df-clab 2719 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 2719 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 2712. Therefore, df-clab 2719 can be considered a definition only in systems that can prove ax-ext 2712 (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 2722). Most often, however, 𝑥 does not occur in {𝑦 ∣ 𝜑} and 𝑦 is free in 𝜑.

3. Remark 1 stresses that df-clab 2719 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 (outside of syntax e.g. in ax-8 2121). 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 2119 and cab 2718, and it can be called an "extension" of the membership predicate because of wel 2120, whose proof uses cv 1546. An a posteriori justification for cv 1546 is given by cvjust 2734, 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 3728).

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 3502 which is used, for example, to convert elirrv 9509 to elirr 9512.

5. Definition or axiom? The question arises with the three axiomatic statements introducing classes, df-clab 2719, df-cleq 2732, and df-clel 2815, to decide if they qualify as definitions or if they should be called axioms. Under the strict definition of "definition" (see conventions 30495), they are not definitions (see Remarks 1 and 3 above, and similarly for df-cleq 2732 and df-clel 2815). 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 2712, see Appendix of [Levy] p. 357.

6. This definition (or axiom) is a class builder introducing the class {𝑥 ∣ 𝜑}, also called a class abstraction or class comprehension, i.e., it specifies a class by a condition determining its members. Another class-building operator (but no class abstraction) is cv 1546, which asserts that every set is a class. The converse need not hold: not every class is a set. A class that is not a set is called a proper class. From ru 3728, it follows that this abstraction yields proper classes, e.g. {𝑥 ∣ 𝑥 = 𝑥}.

7. 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 eqabb 2879 for a quick overview). For a general discussion of the theory of classes, see mmset.html#class 2879. (Contributed by NM, 26-May-1993.) (Revised by BJ, 19-Aug-2023.)

Assertion

Ref	Expression
wl-df.clab	⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)

Proof of Theorem wl-df.clab

Step	Hyp	Ref	Expression
1		df-clab 2719	1 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)

Syntax hints:   ↔ wb 207  [wsb 2073   ∈ wcel 2119  {cab 2718

This theorem depends on definitions:  df-clab 2719

This theorem is referenced by: (None)