Definition df-clab 2709

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 2709 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 2722). Instead, df-clab 2709 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 2709 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 2702. Therefore, df-clab 2709 can be considered a definition only in systems that can prove ax-ext 2702 (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 2712). Most often, however, 𝑥 does not occur in {𝑦 ∣ 𝜑} and 𝑦 is free in 𝜑.

3. Remark 1 stresses that df-clab 2709 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 2112). 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 2110 and cab 2708, and it can be called an "extension" of the membership predicate because of wel 2111, whose proof uses cv 1540. An a posteriori justification for cv 1540 is given by cvjust 2724, 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 3737).

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 3507 which is used, for example, to convert elirrv 9478 to elirr 9480.

5. Definition or axiom? The question arises with the three axiomatic statements introducing classes, df-clab 2709, df-cleq 2722, and df-clel 2804, to decide if they qualify as definitions or if they should be called axioms. Under the strict definition of "definition" (see conventions 30370), they are not definitions (see Remarks 1 and 3 above, and similarly for df-cleq 2722 and df-clel 2804). 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 2702, 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 2868 for a quick overview). For a general discussion of the theory of classes, see mmset.html#class 2868. (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 1540	. . 3 class 𝑥
3		wph	. . . 4 wff 𝜑
4		vy	. . . 4 setvar 𝑦
5	3, 4	cab 2708	. . 3 class {𝑦 ∣ 𝜑}
6	2, 5	wcel 2110	. 2 wff 𝑥 ∈ {𝑦 ∣ 𝜑}
7	3, 4, 1	wsb 2066	. 2 wff [𝑥 / 𝑦]𝜑
8	6, 7	wb 206	1 wff (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)

This definition is referenced by:  eleq1ab  2710  abid  2712  vexwt  2713  vexw  2714  nfsab1  2716  hbab  2718  hbabg  2719  cvjust  2724  abbi  2795  abbib  2799  cbvabv  2800  cbvabw  2801  cbvab  2802  eqabbw  2803  eqabdv  2862  clelab  2874  nfaba1  2900  nfabdw  2914  nfabd  2915  rabrabi  3412  abv  3446  abvALT  3447  elab6g  3622  elabgw  3631  elrabi  3641  ralab  3650  dfsbcq2  3742  sbc8g  3747  sbcimdv  3808  sbcg  3812  csbied  3884  dfss2  3918  ss2abdv  4015  unabw  4255  unab  4256  inab  4257  difab  4258  notabw  4261  noel  4286  vn0  4293  eq0  4298  ab0w  4327  ab0orv  4331  eq0rdv  4355  csbab  4388  disj  4398  rzal  4457  ralf0  4462  exss  5401  iotaeq  6445  abrexex2g  7891  opabex3d  7892  opabex3rd  7893  opabex3  7894  axregs  35113  xpab  35738  in-ax8  36237  ss-ax8  36238  cbvabdavw  36269  eliminable1  36872  eliminable-velab  36878  bj-ab0  36921  bj-elabd2ALT  36938  bj-gabima  36953  bj-snsetex  36976  wl-df-clab  37517  wl-clabv  37608  wl-clabtv  37610  wl-clabt  37611  ss2ab1  42231  scottabf  44252