Definition df-clab 2777

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 2777 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 2791). Instead, df-clab 2777 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 2777 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 2770. Therefore, df-clab 2777 can be considered a definition only in systems that can prove ax-ext 2770 (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 2780). Most often, however, 𝑥 does not occur in {𝑦 ∣ 𝜑} and 𝑦 is free in 𝜑.

3. Remark 1 stresses that df-clab 2777 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 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 2776, and it can be called an "extension" of the membership predicate because of wel 2112, whose proof uses cv 1537. An a posteriori justification for cv 1537 is given by cvjust 2793, 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 3719).

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 3515 which is used, for example, to convert elirrv 9044 to elirr 9045.

5. Definition or axiom? The question arises with the three axiomatic statements introducing classes, df-clab 2777, df-cleq 2791, and df-clel 2870, to decide if they qualify as definitions or if they should be called axioms. Under the strict definition of "definition" (see conventions 28185), they are not definitions (see Remarks 1 and 3 above, and similarly for df-cleq 2791 and df-clel 2870). 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 2770, 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 abeq2 2922 for a quick overview). For a general discussion of the theory of classes, see mmset.html#class 2922. (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 1537	. . 3 class 𝑥
3		wph	. . . 4 wff 𝜑
4		vy	. . . 4 setvar 𝑦
5	3, 4	cab 2776	. . 3 class {𝑦 ∣ 𝜑}
6	2, 5	wcel 2111	. 2 wff 𝑥 ∈ {𝑦 ∣ 𝜑}
7	3, 4, 1	wsb 2069	. 2 wff [𝑥 / 𝑦]𝜑
8	6, 7	wb 209	1 wff (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)

This definition is referenced by:  eleq1ab  2778  abid  2780  vexwt  2781  vexw  2782  hbab1  2784  nfsab1  2785  hbab  2787  hbabg  2788  cvjust  2793  abbi1  2861  abbi  2865  cbvabv  2866  cbvabw  2867  cbvabwOLD  2868  cbvab  2869  abbi2dv  2927  clelab  2933  nfabdw  2976  nfabd  2977  vjust  3442  abv  3451  elabgw  3612  dfsbcq2  3723  sbc8g  3728  ss2abdv  3991  ss2abdvALT  3992  unab  4222  inab  4223  difab  4224  noel  4247  csbab  4345  disj  4355  exss  5320  iotaeq  6295  abrexex2g  7647  opabex3d  7648  opabex3rd  7649  opabex3  7650  eliminable1  34297  eliminable-velab  34303  bj-ab0  34348  bj-snsetex  34399  bj-vjust  34470  wl-clabv  34992  wl-clabtv  34994  wl-clabt  34995  wl-dfrabv  35027  wl-dfrabf  35029  scottabf  40948