Definition df-clab 2704

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 2704 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 2718). Instead, df-clab 2704 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 2704 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 2697. Therefore, df-clab 2704 can be considered a definition only in systems that can prove ax-ext 2697 (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 2707). Most often, however, 𝑥 does not occur in {𝑦 ∣ 𝜑} and 𝑦 is free in 𝜑.

3. Remark 1 stresses that df-clab 2704 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 2100). 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 2098 and cab 2703, and it can be called an "extension" of the membership predicate because of wel 2099, whose proof uses cv 1532. An a posteriori justification for cv 1532 is given by cvjust 2720, 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 3771).

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 3537 which is used, for example, to convert elirrv 9590 to elirr 9591.

5. Definition or axiom? The question arises with the three axiomatic statements introducing classes, df-clab 2704, df-cleq 2718, 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 30157), they are not definitions (see Remarks 1 and 3 above, and similarly for df-cleq 2718 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 2697, 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 2867 for a quick overview). For a general discussion of the theory of classes, see mmset.html#class 2867. (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 1532	. . 3 class 𝑥
3		wph	. . . 4 wff 𝜑
4		vy	. . . 4 setvar 𝑦
5	3, 4	cab 2703	. . 3 class {𝑦 ∣ 𝜑}
6	2, 5	wcel 2098	. 2 wff 𝑥 ∈ {𝑦 ∣ 𝜑}
7	3, 4, 1	wsb 2059	. 2 wff [𝑥 / 𝑦]𝜑
8	6, 7	wb 205	1 wff (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)

This definition is referenced by:  eleq1ab  2705  abid  2707  vexwt  2708  vexw  2709  nfsab1  2711  hbab1OLD  2713  hbab  2714  hbabg  2715  cvjust  2720  abbi  2794  abbib  2798  cbvabv  2799  cbvabw  2800  cbvabwOLD  2801  cbvab  2802  eqabbw  2803  eqabdv  2861  clelab  2873  clelabOLD  2874  nfabdw  2920  nfabdwOLD  2921  nfabd  2922  rabrabi  3444  abv  3479  abvALT  3480  elab6g  3654  elrabi  3672  ralab  3682  dfsbcq2  3775  sbc8g  3780  sbcimdv  3846  sbcg  3851  csbied  3926  ss2abdv  4055  ss2abdvALT  4056  unabw  4292  unab  4293  inab  4294  difab  4295  notabw  4298  noel  4325  noelOLD  4326  vn0  4333  eq0  4338  ab0w  4368  ab0OLD  4370  ab0orv  4373  eq0rdv  4399  csbab  4432  disj  4442  rzal  4503  ralf0  4508  exss  5456  iotaeq  6501  abrexex2g  7947  opabex3d  7948  opabex3rd  7949  opabex3  7950  xpab  35228  eliminable1  36244  eliminable-velab  36250  bj-ab0  36294  bj-elabd2ALT  36311  bj-gabima  36326  bj-snsetex  36350  wl-clabv  36969  wl-clabtv  36971  wl-clabt  36972  ss2ab1  41579  elabgw  41968  scottabf  43557