Definition df-clab 2712

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 2712 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 2726). Instead, df-clab 2712 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 2712 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 2705. Therefore, df-clab 2712 can be considered a definition only in systems that can prove ax-ext 2705 (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 2715). Most often, however, 𝑥 does not occur in {𝑦 ∣ 𝜑} and 𝑦 is free in 𝜑.

3. Remark 1 stresses that df-clab 2712 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 2107). 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 2105 and cab 2711, and it can be called an "extension" of the membership predicate because of wel 2106, whose proof uses cv 1535. An a posteriori justification for cv 1535 is given by cvjust 2728, 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 3788).

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 9633 to elirr 9634.

5. Definition or axiom? The question arises with the three axiomatic statements introducing classes, df-clab 2712, df-cleq 2726, and df-clel 2813, to decide if they qualify as definitions or if they should be called axioms. Under the strict definition of "definition" (see conventions 30428), they are not definitions (see Remarks 1 and 3 above, and similarly for df-cleq 2726 and df-clel 2813). 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 2705, 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 2878 for a quick overview). For a general discussion of the theory of classes, see mmset.html#class 2878. (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 1535	. . 3 class 𝑥
3		wph	. . . 4 wff 𝜑
4		vy	. . . 4 setvar 𝑦
5	3, 4	cab 2711	. . 3 class {𝑦 ∣ 𝜑}
6	2, 5	wcel 2105	. 2 wff 𝑥 ∈ {𝑦 ∣ 𝜑}
7	3, 4, 1	wsb 2061	. 2 wff [𝑥 / 𝑦]𝜑
8	6, 7	wb 206	1 wff (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)

This definition is referenced by:  eleq1ab  2713  abid  2715  vexwt  2716  vexw  2717  nfsab1  2719  hbab1OLD  2721  hbab  2722  hbabg  2723  cvjust  2728  abbi  2804  abbib  2808  cbvabv  2809  cbvabw  2810  cbvab  2811  eqabbw  2812  eqabdv  2872  clelab  2884  nfaba1  2910  nfabdw  2924  nfabd  2925  rabrabi  3452  abv  3489  abvALT  3490  elab6g  3668  elabgw  3678  elrabi  3689  ralab  3699  dfsbcq2  3793  sbc8g  3798  sbcimdv  3864  sbcg  3869  csbied  3945  dfss2  3980  ss2abdv  4075  unabw  4312  unab  4313  inab  4314  difab  4315  notabw  4318  noel  4343  vn0  4350  eq0  4355  ab0w  4384  ab0orv  4388  eq0rdv  4412  csbab  4445  disj  4455  rzal  4514  ralf0  4519  exss  5473  iotaeq  6527  abrexex2g  7987  opabex3d  7988  opabex3rd  7989  opabex3  7990  xpab  35705  in-ax8  36206  ss-ax8  36207  cbvabdavw  36238  eliminable1  36841  eliminable-velab  36847  bj-ab0  36890  bj-elabd2ALT  36907  bj-gabima  36922  bj-snsetex  36945  wl-df-clab  37484  wl-clabv  37575  wl-clabtv  37577  wl-clabt  37578  ss2ab1  42236  scottabf  44235