Definition df-clab 2716

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 2716 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 2730). Instead, df-clab 2716 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 2716 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 2709. Therefore, df-clab 2716 can be considered a definition only in systems that can prove ax-ext 2709 (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 2719). Most often, however, 𝑥 does not occur in {𝑦 ∣ 𝜑} and 𝑦 is free in 𝜑.

3. Remark 1 stresses that df-clab 2716 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 2108). 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 2106 and cab 2715, and it can be called an "extension" of the membership predicate because of wel 2107, whose proof uses cv 1538. An a posteriori justification for cv 1538 is given by cvjust 2732, 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 3715).

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 3505 which is used, for example, to convert elirrv 9355 to elirr 9356.

5. Definition or axiom? The question arises with the three axiomatic statements introducing classes, df-clab 2716, df-cleq 2730, and df-clel 2816, to decide if they qualify as definitions or if they should be called axioms. Under the strict definition of "definition" (see conventions 28764), they are not definitions (see Remarks 1 and 3 above, and similarly for df-cleq 2730 and df-clel 2816). 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 2709, 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 2872 for a quick overview). For a general discussion of the theory of classes, see mmset.html#class 2872. (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 1538	. . 3 class 𝑥
3		wph	. . . 4 wff 𝜑
4		vy	. . . 4 setvar 𝑦
5	3, 4	cab 2715	. . 3 class {𝑦 ∣ 𝜑}
6	2, 5	wcel 2106	. 2 wff 𝑥 ∈ {𝑦 ∣ 𝜑}
7	3, 4, 1	wsb 2067	. 2 wff [𝑥 / 𝑦]𝜑
8	6, 7	wb 205	1 wff (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)

This definition is referenced by:  eleq1ab  2717  abid  2719  vexwt  2720  vexw  2721  nfsab1  2723  hbab1OLD  2725  hbab  2726  hbabg  2727  cvjust  2732  abbi1  2806  abbi  2810  cbvabv  2811  cbvabw  2812  cbvabwOLD  2813  cbvab  2814  abeq2w  2815  abbi2dv  2877  clelab  2883  clelabOLD  2884  nfabdw  2930  nfabdwOLD  2931  nfabd  2932  rabrabi  3427  abv  3443  abvALT  3444  elab6g  3600  elrabi  3618  ralab  3628  dfsbcq2  3719  sbc8g  3724  sbcimdv  3790  sbcg  3795  csbied  3870  ss2abdv  3997  ss2abdvALT  3998  unabw  4231  unab  4232  inab  4233  difab  4234  notabw  4237  noel  4264  noelOLD  4265  vn0  4272  eq0  4277  ab0w  4307  ab0OLD  4309  ab0orv  4312  eq0rdv  4338  csbab  4371  disj  4381  rzal  4439  ralf0  4444  exss  5378  iotaeq  6404  abrexex2g  7807  opabex3d  7808  opabex3rd  7809  opabex3  7810  xpab  33677  eliminable1  35043  eliminable-velab  35049  bj-ab0  35093  bj-elabd2ALT  35113  bj-gabima  35128  bj-snsetex  35153  wl-clabv  35746  wl-clabtv  35748  wl-clabt  35749  elabgw  40165  scottabf  41858