Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > df-clab | Structured version Visualization version GIF version |
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 2801 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 2815). Instead, df-clab 2801 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 2801 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 2794. Therefore, df-clab 2801 can be considered a definition only in systems that can prove ax-ext 2794 (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 2804). Most often, however, 𝑥 does not occur in {𝑦 ∣ 𝜑} and 𝑦 is free in 𝜑. 3. Remark 1 stresses that df-clab 2801 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 2116). 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 2114 and cab 2800, and it can be called an "extension" of the membership predicate because of wel 2115, whose proof uses cv 1537. An a posteriori justification for cv 1537 is given by cvjust 2817, 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 3746). 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 3542 which is used, for example, to convert elirrv 9048 to elirr 9049. 5. Definition or axiom? The question arises with the three axiomatic statements introducing classes, df-clab 2801, df-cleq 2815, and df-clel 2894, to decide if they qualify as definitions or if they should be called axioms. Under the strict definition of "definition" (see conventions 28183), they are not definitions (see Remarks 1 and 3 above, and similarly for df-cleq 2815 and df-clel 2894). 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 2794, 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 2946 for a quick overview). For a general discussion of the theory of classes, see mmset.html#class 2946. (Contributed by NM, 26-May-1993.) (Revised by BJ, 19-Aug-2023.) |
Ref | Expression |
---|---|
df-clab | ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑) |
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 2800 | . . 3 class {𝑦 ∣ 𝜑} |
6 | 2, 5 | wcel 2114 | . 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) |
Colors of variables: wff setvar class |
This definition is referenced by: eleq1ab 2802 abid 2804 vexwt 2805 vexw 2806 hbab1 2808 nfsab1 2809 hbab 2811 hbabg 2812 cvjust 2817 abbi1 2885 abbi 2889 cbvabv 2890 cbvabw 2891 cbvabwOLD 2892 cbvab 2893 abbi2dv 2951 clelab 2957 nfabdw 3000 nfabd 3001 vjust 3470 abv 3479 elabgw 3639 dfsbcq2 3750 sbc8g 3755 unab 4244 inab 4245 difab 4246 noel 4269 csbab 4361 exss 5332 iotaeq 6305 abrexex2g 7651 opabex3d 7652 opabex3rd 7653 opabex3 7654 eliminable1 34258 eliminable-velab 34264 bj-ab0 34309 bj-snsetex 34360 bj-vjust 34431 wl-clabv 34950 wl-clabtv 34952 wl-clabt 34953 wl-dfrabv 34985 wl-dfrabf 34987 scottabf 40882 |
Copyright terms: Public domain | W3C validator |