Definition df-clab 2739

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 2739 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 2752). Instead, df-clab 2739 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 2739 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 2732. Therefore, df-clab 2739 can be considered a definition only in systems that can prove ax-ext 2732 (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 2742). Most often, however, 𝑥 does not occur in {𝑦 ∣ 𝜑} and 𝑦 is free in 𝜑.

3. Remark 1 stresses that df-clab 2739 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 2146). 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 2144 and cab 2738, and it can be called an "extension" of the membership predicate because of wel 2145, whose proof uses cv 1568. An a posteriori justification for cv 1568 is given by cvjust 2754, 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 3737).

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 3517 which is used, for example, to convert elirrv 9592 to elirr 9595.

5. Definition or axiom? The question arises with the three axiomatic statements introducing classes, df-clab 2739, df-cleq 2752, and df-clel 2835, to decide if they qualify as definitions or if they should be called axioms. Under the strict definition of "definition" (see conventions 31002), they are not definitions (see Remarks 1 and 3 above, and similarly for df-cleq 2752 and df-clel 2835). 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 2732, 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 2899 for a quick overview). For a general discussion of the theory of classes, see mmset.html#class 2899. (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 1568	. . 3 class 𝑥
3		wph	. . . 4 wff 𝜑
4		vy	. . . 4 setvar 𝑦
5	3, 4	cab 2738	. . 3 class {𝑦 ∣ 𝜑}
6	2, 5	wcel 2144	. 2 wff 𝑥 ∈ {𝑦 ∣ 𝜑}
7	3, 4, 1	wsb 2098	. 2 wff [𝑥 / 𝑦]𝜑
8	6, 7	wb 209	1 wff (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)

This definition is used by:  eleq1ab  2740  abid  2742  vexwt  2743  vexw  2744  nfsab1  2746  hbab  2748  hbabg  2749  cvjust  2754  abbi  2825  abbib  2829  cbvabv  2830  cbvabw  2831  cbvab  2832  eqabbw  2833  eqabdv  2893  clelab  2904  nfaba1  2930  nfabdw  2943  nfabd  2944  rabrabi  3430  abv  3462  abvALT  3463  elab6g  3622  elabgw  3630  elrabi  3640  ralab  3650  dfsbcq2  3741  sbc8g  3746  sbcimdv  3806  sbcg  3810  csbied  3882  ss2abim  3985  ss2abdv  3990  dfss2  4094  unabw  4252  unab  4253  inab  4254  difab  4255  notabw  4258  noel  4283  ab0w  4327  csbab  4397  exss  5430  iotaeq  6506  abrexex2g  7976  opabex3d  7977  opabex3rd  7978  opabex3  7979  scottabf  9940  axregs  35856  xpab  36566  in-ax8  37088  ss-ax8  37089  cbvabdavw  37120  mh-setind  37399  regsfromunir1  37403  bj-dfsbc  37626  eliminable1  37846  eliminable-velab  37852  bj-ab0  37895  bj-elabd2ALT  37913  bj-gabima  37928  bj-snsetex  37951  bj-vn0ALT  38062  wl-df-clab  38503  wl-df.clab  38506  wl-clabv  38592  wl-clabtv  38594  wl-clabt  38595