Definition df-clab 2710

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 2710 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 2723). Instead, df-clab 2710 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 2710 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 2703. Therefore, df-clab 2710 can be considered a definition only in systems that can prove ax-ext 2703 (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 2713). Most often, however, 𝑥 does not occur in {𝑦 ∣ 𝜑} and 𝑦 is free in 𝜑.

3. Remark 1 stresses that df-clab 2710 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 2113). 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 2111 and cab 2709, and it can be called an "extension" of the membership predicate because of wel 2112, whose proof uses cv 1540. An a posteriori justification for cv 1540 is given by cvjust 2725, 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 3734).

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 3507 which is used, for example, to convert elirrv 9489 to elirr 9491.

5. Definition or axiom? The question arises with the three axiomatic statements introducing classes, df-clab 2710, df-cleq 2723, and df-clel 2806, to decide if they qualify as definitions or if they should be called axioms. Under the strict definition of "definition" (see conventions 30387), they are not definitions (see Remarks 1 and 3 above, and similarly for df-cleq 2723 and df-clel 2806). 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 2703, 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 2870 for a quick overview). For a general discussion of the theory of classes, see mmset.html#class 2870. (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 1540	. . 3 class 𝑥
3		wph	. . . 4 wff 𝜑
4		vy	. . . 4 setvar 𝑦
5	3, 4	cab 2709	. . 3 class {𝑦 ∣ 𝜑}
6	2, 5	wcel 2111	. 2 wff 𝑥 ∈ {𝑦 ∣ 𝜑}
7	3, 4, 1	wsb 2067	. 2 wff [𝑥 / 𝑦]𝜑
8	6, 7	wb 206	1 wff (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)

This definition is referenced by:  eleq1ab  2711  abid  2713  vexwt  2714  vexw  2715  nfsab1  2717  hbab  2719  hbabg  2720  cvjust  2725  abbi  2796  abbib  2800  cbvabv  2801  cbvabw  2802  cbvab  2803  eqabbw  2804  eqabdv  2864  clelab  2876  nfaba1  2902  nfabdw  2916  nfabd  2917  rabrabi  3414  abv  3448  abvALT  3449  elab6g  3619  elabgw  3628  elrabi  3638  ralab  3647  dfsbcq2  3739  sbc8g  3744  sbcimdv  3805  sbcg  3809  csbied  3881  dfss2  3915  ss2abim  4008  ss2abdv  4013  unabw  4256  unab  4257  inab  4258  difab  4259  notabw  4262  noel  4287  ab0w  4328  csbab  4389  ralf0  4463  exss  5406  iotaeq  6455  abrexex2g  7902  opabex3d  7903  opabex3rd  7904  opabex3  7905  axregs  35152  xpab  35777  in-ax8  36275  ss-ax8  36276  cbvabdavw  36307  eliminable1  36910  eliminable-velab  36916  bj-ab0  36959  bj-elabd2ALT  36976  bj-gabima  36991  bj-snsetex  37014  wl-df-clab  37555  wl-clabv  37635  wl-clabtv  37637  wl-clabt  37638  ss2ab1  42318  scottabf  44338