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 2111). 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 2109 and cab 2708, and it can be called an "extension" of the membership predicate because of wel 2110, whose proof uses cv 1539. An a posteriori justification for cv 1539 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 3753).

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 3523 which is used, for example, to convert elirrv 9555 to elirr 9556.

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 30335), 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 1539	. . 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 2109	. 2 wff 𝑥 ∈ {𝑦 ∣ 𝜑}
7	3, 4, 1	wsb 2065	. 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  3428  abv  3462  abvALT  3463  elab6g  3638  elabgw  3646  elrabi  3656  ralab  3666  dfsbcq2  3758  sbc8g  3763  sbcimdv  3824  sbcg  3828  csbied  3900  dfss2  3934  ss2abdv  4031  unabw  4272  unab  4273  inab  4274  difab  4275  notabw  4278  noel  4303  vn0  4310  eq0  4315  ab0w  4344  ab0orv  4348  eq0rdv  4372  csbab  4405  disj  4415  rzal  4474  ralf0  4479  exss  5425  iotaeq  6478  abrexex2g  7945  opabex3d  7946  opabex3rd  7947  opabex3  7948  xpab  35708  in-ax8  36207  ss-ax8  36208  cbvabdavw  36239  eliminable1  36842  eliminable-velab  36848  bj-ab0  36891  bj-elabd2ALT  36908  bj-gabima  36923  bj-snsetex  36946  wl-df-clab  37487  wl-clabv  37578  wl-clabtv  37580  wl-clabt  37581  ss2ab1  42202  scottabf  44222