Definition df-clab 2716

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 2716 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 2729). Instead, df-clab 2716 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 2716 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 2709. Therefore, df-clab 2716 can be considered a definition only in systems that can prove ax-ext 2709 (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 2719). Most often, however, 𝑥 does not occur in {𝑦 ∣ 𝜑} and 𝑦 is free in 𝜑.

3. Remark 1 stresses that df-clab 2716 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 2116). 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 2114 and cab 2715, and it can be called an "extension" of the membership predicate because of wel 2115, whose proof uses cv 1541. An a posteriori justification for cv 1541 is given by cvjust 2731, 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 3739).

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 3512 which is used, for example, to convert elirrv 9506 to elirr 9508.

5. Definition or axiom? The question arises with the three axiomatic statements introducing classes, df-clab 2716, df-cleq 2729, and df-clel 2812, to decide if they qualify as definitions or if they should be called axioms. Under the strict definition of "definition" (see conventions 30458), they are not definitions (see Remarks 1 and 3 above, and similarly for df-cleq 2729 and df-clel 2812). 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 2709, 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 2876 for a quick overview). For a general discussion of the theory of classes, see mmset.html#class 2876. (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 1541	. . 3 class 𝑥
3		wph	. . . 4 wff 𝜑
4		vy	. . . 4 setvar 𝑦
5	3, 4	cab 2715	. . 3 class {𝑦 ∣ 𝜑}
6	2, 5	wcel 2114	. 2 wff 𝑥 ∈ {𝑦 ∣ 𝜑}
7	3, 4, 1	wsb 2068	. 2 wff [𝑥 / 𝑦]𝜑
8	6, 7	wb 206	1 wff (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)

This definition is referenced by:  eleq1ab  2717  abid  2719  vexwt  2720  vexw  2721  nfsab1  2723  hbab  2725  hbabg  2726  cvjust  2731  abbi  2802  abbib  2806  cbvabv  2807  cbvabw  2808  cbvab  2809  eqabbw  2810  eqabdv  2870  clelab  2881  nfaba1  2907  nfabdw  2921  nfabd  2922  rabrabi  3419  abv  3453  abvALT  3454  elab6g  3624  elabgw  3633  elrabi  3643  ralab  3652  dfsbcq2  3744  sbc8g  3749  sbcimdv  3810  sbcg  3814  csbied  3886  dfss2  3920  ss2abim  4013  ss2abdv  4018  unabw  4260  unab  4261  inab  4262  difab  4263  notabw  4266  noel  4291  ab0w  4332  csbab  4393  exss  5412  iotaeq  6461  abrexex2g  7910  opabex3d  7911  opabex3rd  7912  opabex3  7913  axregs  35276  xpab  35901  in-ax8  36399  ss-ax8  36400  cbvabdavw  36431  eliminable1  37035  eliminable-velab  37041  bj-ab0  37084  bj-elabd2ALT  37101  bj-gabima  37116  bj-snsetex  37139  wl-df-clab  37680  wl-clabv  37760  wl-clabtv  37762  wl-clabt  37763  scottabf  44517