Definition df-clab 2714

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 2714 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 2728). Instead, df-clab 2714 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 2714 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 2707. Therefore, df-clab 2714 can be considered a definition only in systems that can prove ax-ext 2707 (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 2717). Most often, however, 𝑥 does not occur in {𝑦 ∣ 𝜑} and 𝑦 is free in 𝜑.

3. Remark 1 stresses that df-clab 2714 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 2110). 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 2108 and cab 2713, and it can be called an "extension" of the membership predicate because of wel 2109, whose proof uses cv 1539. An a posteriori justification for cv 1539 is given by cvjust 2730, 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 3785).

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 3553 which is used, for example, to convert elirrv 9632 to elirr 9633.

5. Definition or axiom? The question arises with the three axiomatic statements introducing classes, df-clab 2714, df-cleq 2728, 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 30409), they are not definitions (see Remarks 1 and 3 above, and similarly for df-cleq 2728 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 2707, 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 eqabb 2880 for a quick overview). For a general discussion of the theory of classes, see mmset.html#class 2880. (Contributed by NM, 26-May-1993.) (Revised by BJ, 19-Aug-2023.)

Assertion

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

Detailed syntax breakdown of Definition df-clab

Step	Hyp	Ref	Expression
1		vx	. . . 4 setvar 𝑥
2	1	cv 1539	. . 3 class 𝑥
3		wph	. . . 4 wff 𝜑
4		vy	. . . 4 setvar 𝑦
5	3, 4	cab 2713	. . 3 class {𝑦 ∣ 𝜑}
6	2, 5	wcel 2108	. 2 wff 𝑥 ∈ {𝑦 ∣ 𝜑}
7	3, 4, 1	wsb 2064	. 2 wff [𝑥 / 𝑦]𝜑
8	6, 7	wb 206	1 wff (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)

This definition is referenced by:  eleq1ab  2715  abid  2717  vexwt  2718  vexw  2719  nfsab1  2721  hbab1OLD  2723  hbab  2724  hbabg  2725  cvjust  2730  abbi  2806  abbib  2810  cbvabv  2811  cbvabw  2812  cbvab  2813  eqabbw  2814  eqabdv  2874  clelab  2886  nfaba1  2912  nfabdw  2926  nfabd  2927  rabrabi  3455  abv  3491  abvALT  3492  elab6g  3668  elabgw  3676  elrabi  3686  ralab  3696  dfsbcq2  3790  sbc8g  3795  sbcimdv  3858  sbcg  3862  csbied  3934  dfss2  3968  ss2abdv  4065  unabw  4306  unab  4307  inab  4308  difab  4309  notabw  4312  noel  4337  vn0  4344  eq0  4349  ab0w  4378  ab0orv  4382  eq0rdv  4406  csbab  4439  disj  4449  rzal  4508  ralf0  4513  exss  5466  iotaeq  6524  abrexex2g  7985  opabex3d  7986  opabex3rd  7987  opabex3  7988  xpab  35719  in-ax8  36218  ss-ax8  36219  cbvabdavw  36250  eliminable1  36853  eliminable-velab  36859  bj-ab0  36902  bj-elabd2ALT  36919  bj-gabima  36934  bj-snsetex  36957  wl-df-clab  37498  wl-clabv  37589  wl-clabtv  37591  wl-clabt  37592  ss2ab1  42251  scottabf  44252