Definition df-clab 2739

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 2739 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 2752). Instead, df-clab 2739 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 2739 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 2732. Therefore, df-clab 2739 can be considered a definition only in systems that can prove ax-ext 2732 (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 2742). Most often, however, 𝑥 does not occur in {𝑦 ∣ 𝜑} and 𝑦 is free in 𝜑.

3. Remark 1 stresses that df-clab 2739 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 2147). 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 2145 and cab 2738, and it can be called an "extension" of the membership predicate because of wel 2146, whose proof uses cv 1569. An a posteriori justification for cv 1569 is given by cvjust 2754, 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 3738).

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 3517 which is used, for example, to convert elirrv 9569 to elirr 9572.

5. Definition or axiom? The question arises with the three axiomatic statements introducing classes, df-clab 2739, df-cleq 2752, and df-clel 2835, to decide if they qualify as definitions or if they should be called axioms. Under the strict definition of "definition" (see conventions 30880), they are not definitions (see Remarks 1 and 3 above, and similarly for df-cleq 2752 and df-clel 2835). 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 2732, 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 2899 for a quick overview). For a general discussion of the theory of classes, see mmset.html#class 2899. (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 1569	. . 3 class 𝑥
3		wph	. . . 4 wff 𝜑
4		vy	. . . 4 setvar 𝑦
5	3, 4	cab 2738	. . 3 class {𝑦 ∣ 𝜑}
6	2, 5	wcel 2145	. 2 wff 𝑥 ∈ {𝑦 ∣ 𝜑}
7	3, 4, 1	wsb 2099	. 2 wff [𝑥 / 𝑦]𝜑
8	6, 7	wb 209	1 wff (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)

This definition is used by:  eleq1ab  2740  abid  2742  vexwt  2743  vexw  2744  nfsab1  2746  hbab  2748  hbabg  2749  cvjust  2754  abbi  2825  abbib  2829  cbvabv  2830  cbvabw  2831  cbvab  2832  eqabbw  2833  eqabdv  2893  clelab  2904  nfaba1  2930  nfabdw  2943  nfabd  2944  rabrabi  3430  abv  3462  abvALT  3463  elab6g  3623  elabgw  3631  elrabi  3641  ralab  3651  dfsbcq2  3742  sbc8g  3747  sbcimdv  3807  sbcg  3811  csbied  3883  dfss2  3917  ss2abim  4008  ss2abdv  4013  unabw  4253  unab  4254  inab  4255  difab  4256  notabw  4259  noel  4284  ab0w  4328  csbab  4398  exss  5438  iotaeq  6501  abrexex2g  7961  opabex3d  7962  opabex3rd  7963  opabex3  7964  scottabf  9878  axregs  35665  xpab  36305  in-ax8  36844  ss-ax8  36845  cbvabdavw  36876  mh-setind  37155  regsfromunir1  37159  bj-dfsbc  37382  eliminable1  37602  eliminable-velab  37608  bj-ab0  37651  bj-elabd2ALT  37669  bj-gabima  37684  bj-snsetex  37707  bj-vn0ALT  37816  wl-df-clab  38258  wl-df.clab  38261  wl-clabv  38347  wl-clabtv  38349  wl-clabt  38350