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Definition df-clab 2801
Description: 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 2801 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 2815). Instead, df-clab 2801 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 2801 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 2794. Therefore, df-clab 2801 can be considered a definition only in systems that can prove ax-ext 2794 (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 ∈ {𝑥𝜑} which are sometimes useful to shorten proofs (because of abid 2804). Most often, however, 𝑥 does not occur in {𝑦𝜑} and 𝑦 is free in 𝜑.

3. Remark 1 stresses that df-clab 2801 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 2116). 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 2114 and cab 2800, and it can be called an "extension" of the membership predicate because of wel 2115, whose proof uses cv 1537. An a posteriori justification for cv 1537 is given by cvjust 2817, 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 3746).

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 3542 which is used, for example, to convert elirrv 9048 to elirr 9049.

5. Definition or axiom? The question arises with the three axiomatic statements introducing classes, df-clab 2801, df-cleq 2815, and df-clel 2894, to decide if they qualify as definitions or if they should be called axioms. Under the strict definition of "definition" (see conventions 28183), they are not definitions (see Remarks 1 and 3 above, and similarly for df-cleq 2815 and df-clel 2894). 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, due to the difference between the "scheme level" of 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 2794, 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 2946 for a quick overview). For a general discussion of the theory of classes, see mmset.html#class 2946. (Contributed by NM, 26-May-1993.) (Revised by BJ, 19-Aug-2023.)

Ref Expression
df-clab (𝑥 ∈ {𝑦𝜑} ↔ [𝑥 / 𝑦]𝜑)

Detailed syntax breakdown of Definition df-clab
StepHypRef Expression
1 vx . . . 4 setvar 𝑥
21cv 1537 . . 3 class 𝑥
3 wph . . . 4 wff 𝜑
4 vy . . . 4 setvar 𝑦
53, 4cab 2800 . . 3 class {𝑦𝜑}
62, 5wcel 2114 . 2 wff 𝑥 ∈ {𝑦𝜑}
73, 4, 1wsb 2069 . 2 wff [𝑥 / 𝑦]𝜑
86, 7wb 209 1 wff (𝑥 ∈ {𝑦𝜑} ↔ [𝑥 / 𝑦]𝜑)
Colors of variables: wff setvar class
This definition is referenced by:  eleq1ab  2802  abid  2804  vexwt  2805  vexw  2806  hbab1  2808  nfsab1  2809  hbab  2811  hbabg  2812  cvjust  2817  abbi1  2885  abbi  2889  cbvabv  2890  cbvabw  2891  cbvabwOLD  2892  cbvab  2893  abbi2dv  2951  clelab  2957  nfabdw  3000  nfabd  3001  vjust  3470  abv  3479  elabgw  3639  dfsbcq2  3750  sbc8g  3755  unab  4244  inab  4245  difab  4246  noel  4269  csbab  4361  exss  5332  iotaeq  6305  abrexex2g  7651  opabex3d  7652  opabex3rd  7653  opabex3  7654  eliminable1  34258  eliminable-velab  34264  bj-ab0  34309  bj-snsetex  34360  bj-vjust  34431  wl-clabv  34950  wl-clabtv  34952  wl-clabt  34953  wl-dfrabv  34985  wl-dfrabf  34987  scottabf  40882
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