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 3740).

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 3513 which is used, for example, to convert elirrv 9514 to elirr 9516.

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 30487), 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  3420  abv  3454  abvALT  3455  elab6g  3625  elabgw  3634  elrabi  3644  ralab  3653  dfsbcq2  3745  sbc8g  3750  sbcimdv  3811  sbcg  3815  csbied  3887  dfss2  3921  ss2abim  4014  ss2abdv  4019  unabw  4261  unab  4262  inab  4263  difab  4264  notabw  4267  noel  4292  ab0w  4333  csbab  4394  exss  5418  iotaeq  6468  abrexex2g  7918  opabex3d  7919  opabex3rd  7920  opabex3  7921  axregs  35314  xpab  35939  in-ax8  36437  ss-ax8  36438  cbvabdavw  36469  mh-setind  36685  regsfromunir1  36689  eliminable1  37098  eliminable-velab  37104  bj-ab0  37147  bj-elabd2ALT  37164  bj-gabima  37179  bj-snsetex  37202  wl-df-clab  37748  wl-clabv  37828  wl-clabtv  37830  wl-clabt  37831  scottabf  44585