Definition df-clab 2744

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 2744 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 2757). Instead, df-clab 2744 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 2744 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 2737. Therefore, df-clab 2744 can be considered a definition only in systems that can prove ax-ext 2737 (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 2747). Most often, however, 𝑥 does not occur in {𝑦 ∣ 𝜑} and 𝑦 is free in 𝜑.

3. Remark 1 stresses that df-clab 2744 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 2147). 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 2145 and cab 2743, and it can be called an "extension" of the membership predicate because of wel 2146, whose proof uses cv 1562. An a posteriori justification for cv 1562 is given by cvjust 2759, 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 3746).

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 3525 which is used, for example, to convert elirrv 9547 to elirr 9550.

5. Definition or axiom? The question arises with the three axiomatic statements introducing classes, df-clab 2744, df-cleq 2757, and df-clel 2840, to decide if they qualify as definitions or if they should be called axioms. Under the strict definition of "definition" (see conventions 30660), they are not definitions (see Remarks 1 and 3 above, and similarly for df-cleq 2757 and df-clel 2840). 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 2737, 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 2904 for a quick overview). For a general discussion of the theory of classes, see mmset.html#class 2904. (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 1562	. . 3 class 𝑥
3		wph	. . . 4 wff 𝜑
4		vy	. . . 4 setvar 𝑦
5	3, 4	cab 2743	. . 3 class {𝑦 ∣ 𝜑}
6	2, 5	wcel 2145	. 2 wff 𝑥 ∈ {𝑦 ∣ 𝜑}
7	3, 4, 1	wsb 2093	. 2 wff [𝑥 / 𝑦]𝜑
8	6, 7	wb 209	1 wff (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)

This definition is referenced by:  eleq1ab  2745  abid  2747  vexwt  2748  vexw  2749  nfsab1  2751  hbab  2753  hbabg  2754  cvjust  2759  abbi  2830  abbib  2834  cbvabv  2835  cbvabw  2836  cbvab  2837  eqabbw  2838  eqabdv  2898  clelab  2909  nfaba1  2935  nfabdw  2948  nfabd  2949  rabrabi  3436  abv  3469  abvALT  3470  elab6g  3631  elabgw  3639  elrabi  3649  ralab  3659  dfsbcq2  3750  sbc8g  3755  sbcimdv  3815  sbcg  3819  csbied  3891  dfss2  3925  ss2abim  4016  ss2abdv  4021  unabw  4262  unab  4263  inab  4264  difab  4265  notabw  4268  noel  4293  ab0w  4335  csbab  4397  exss  5435  iotaeq  6493  abrexex2g  7949  opabex3d  7950  opabex3rd  7951  opabex3  7952  scottabf  9854  axregs  35447  xpab  36089  in-ax8  36597  ss-ax8  36598  cbvabdavw  36629  mh-setind  36909  regsfromunir1  36913  bj-dfsbc  37136  eliminable1  37356  eliminable-velab  37362  bj-ab0  37405  bj-elabd2ALT  37422  bj-gabima  37437  bj-snsetex  37460  wl-df-clab  38010  wl-df.clab  38013  wl-clabv  38099  wl-clabtv  38101  wl-clabt  38102