Definition df-clab 2714

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 2714 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 2727). Instead, df-clab 2714 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 2714 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 2707. Therefore, df-clab 2714 can be considered a definition only in systems that can prove ax-ext 2707 (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 2717). Most often, however, 𝑥 does not occur in {𝑦 ∣ 𝜑} and 𝑦 is free in 𝜑.

3. Remark 1 stresses that df-clab 2714 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 2110). 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 2108 and cab 2713, and it can be called an "extension" of the membership predicate because of wel 2109, whose proof uses cv 1539. An a posteriori justification for cv 1539 is given by cvjust 2729, 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 3763).

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 3533 which is used, for example, to convert elirrv 9608 to elirr 9609.

5. Definition or axiom? The question arises with the three axiomatic statements introducing classes, df-clab 2714, df-cleq 2727, and df-clel 2809, to decide if they qualify as definitions or if they should be called axioms. Under the strict definition of "definition" (see conventions 30327), they are not definitions (see Remarks 1 and 3 above, and similarly for df-cleq 2727 and df-clel 2809). 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 2707, 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 1539	. . 3 class 𝑥
3		wph	. . . 4 wff 𝜑
4		vy	. . . 4 setvar 𝑦
5	3, 4	cab 2713	. . 3 class {𝑦 ∣ 𝜑}
6	2, 5	wcel 2108	. 2 wff 𝑥 ∈ {𝑦 ∣ 𝜑}
7	3, 4, 1	wsb 2064	. 2 wff [𝑥 / 𝑦]𝜑
8	6, 7	wb 206	1 wff (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)

This definition is referenced by:  eleq1ab  2715  abid  2717  vexwt  2718  vexw  2719  nfsab1  2721  hbab  2723  hbabg  2724  cvjust  2729  abbi  2800  abbib  2804  cbvabv  2805  cbvabw  2806  cbvab  2807  eqabbw  2808  eqabdv  2868  clelab  2880  nfaba1  2906  nfabdw  2920  nfabd  2921  rabrabi  3435  abv  3471  abvALT  3472  elab6g  3648  elabgw  3656  elrabi  3666  ralab  3676  dfsbcq2  3768  sbc8g  3773  sbcimdv  3834  sbcg  3838  csbied  3910  dfss2  3944  ss2abdv  4041  unabw  4282  unab  4283  inab  4284  difab  4285  notabw  4288  noel  4313  vn0  4320  eq0  4325  ab0w  4354  ab0orv  4358  eq0rdv  4382  csbab  4415  disj  4425  rzal  4484  ralf0  4489  exss  5438  iotaeq  6495  abrexex2g  7961  opabex3d  7962  opabex3rd  7963  opabex3  7964  xpab  35689  in-ax8  36188  ss-ax8  36189  cbvabdavw  36220  eliminable1  36823  eliminable-velab  36829  bj-ab0  36872  bj-elabd2ALT  36889  bj-gabima  36904  bj-snsetex  36927  wl-df-clab  37468  wl-clabv  37559  wl-clabtv  37561  wl-clabt  37562  ss2ab1  42216  scottabf  44212