Definition df-clab 2711

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 2711 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 2725). Instead, df-clab 2711 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 2711 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 2704. Therefore, df-clab 2711 can be considered a definition only in systems that can prove ax-ext 2704 (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 2714). Most often, however, 𝑥 does not occur in {𝑦 ∣ 𝜑} and 𝑦 is free in 𝜑.

3. Remark 1 stresses that df-clab 2711 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 (e.g., in the non-syntactic statement ax-8 2109). 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 2107 and cab 2710, and it can be called an "extension" of the membership predicate because of wel 2108, whose proof uses cv 1541. An a posteriori justification for cv 1541 is given by cvjust 2727, 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 3775).

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 3556 which is used, for example, to convert elirrv 9587 to elirr 9588.

5. Definition or axiom? The question arises with the three axiomatic statements introducing classes, df-clab 2711, df-cleq 2725, and df-clel 2811, to decide if they qualify as definitions or if they should be called axioms. Under the strict definition of "definition" (see conventions 29633), they are not definitions (see Remarks 1 and 3 above, and similarly for df-cleq 2725 and df-clel 2811). 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 2704, 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 2874 for a quick overview). For a general discussion of the theory of classes, see mmset.html#class 2874. (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 2710	. . 3 class {𝑦 ∣ 𝜑}
6	2, 5	wcel 2107	. 2 wff 𝑥 ∈ {𝑦 ∣ 𝜑}
7	3, 4, 1	wsb 2068	. 2 wff [𝑥 / 𝑦]𝜑
8	6, 7	wb 205	1 wff (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)

This definition is referenced by:  eleq1ab  2712  abid  2714  vexwt  2715  vexw  2716  nfsab1  2718  hbab1OLD  2720  hbab  2721  hbabg  2722  cvjust  2727  abbi  2801  abbib  2805  cbvabv  2806  cbvabw  2807  cbvabwOLD  2808  cbvab  2809  eqabbw  2810  eqabdv  2868  clelab  2880  clelabOLD  2881  nfabdw  2927  nfabdwOLD  2928  nfabd  2929  rabrabi  3451  abv  3486  abvALT  3487  elab6g  3658  elrabi  3676  ralab  3686  dfsbcq2  3779  sbc8g  3784  sbcimdv  3850  sbcg  3855  csbied  3930  ss2abdv  4059  ss2abdvALT  4060  unabw  4296  unab  4297  inab  4298  difab  4299  notabw  4302  noel  4329  noelOLD  4330  vn0  4337  eq0  4342  ab0w  4372  ab0OLD  4374  ab0orv  4377  eq0rdv  4403  csbab  4436  disj  4446  rzal  4507  ralf0  4512  exss  5462  iotaeq  6505  abrexex2g  7946  opabex3d  7947  opabex3rd  7948  opabex3  7949  xpab  34633  eliminable1  35676  eliminable-velab  35682  bj-ab0  35726  bj-elabd2ALT  35743  bj-gabima  35758  bj-snsetex  35782  wl-clabv  36395  wl-clabtv  36397  wl-clabt  36398  elabgw  40963  ss2ab1  40984  scottabf  42932