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Definition df-clab 2430
Description: Define class abstraction notation (so-called by Quine), also called a "class builder" in the literature. and need not be distinct. Definition 2.1 of [Quine] p. 16. Typically, will have as a free variable, and " " is read "the class of all sets such that ( ) is true." We do not define in isolation but only as part of an expression that extends or "overloads" the e. relationship.

This is our first use of the e. symbol to connect classes instead of sets. The syntax definition wcel 1756, which extends or "overloads" the wel 1757 definition connecting setvar variables, requires that both sides of e. be classes. In df-cleq 2436 and df-clel 2439, we introduce a new kind of variable (class variable) that can be substituted with expressions such as . In the present definition, the on the left-hand side is a setvar variable. Syntax definition cv 1368 allows us to substitute a setvar variable for a class variable: all sets are classes by cvjust 2438 (but not necessarily vice-versa). For a full description of how classes are introduced and how to recover the primitive language, see the discussion in Quine (and under abeq2 2548 for a quick overview).

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 3030 which is used, for example, to convert elirrv 7812 to elirr 7813.

This is called the "axiom of class comprehension" by [Levy] p. 338, who treats the theory of classes as an extralogical extension to our logic and set theory axioms. He calls the construction a "class term".

While the three class definitions df-clab 2430, df-cleq 2436, and df-clel 2439 are eliminable and conservative and thus meet the requirements for sound definitions, they are technically axioms in that they do not satisfy the requirements for the current definition checker. The proofs of conservativity require external justification that is beyond the scope of the definition checker.

For a general discussion of the theory of classes, see http://us.metamath.org/mpeuni/mmset.html#class. (Contributed by NM, 26-May-1993.)

Assertion
Ref Expression
df-clab

Detailed syntax breakdown of Definition df-clab
StepHypRef Expression
1 vx . . . 4
21cv 1368 . . 3
3 wph . . . 4
4 vy . . . 4
53, 4cab 2429 . . 3
62, 5wcel 1756 . 2
73, 4, 1wsb 1700 . 2
86, 7wb 184 1
Colors of variables: wff setvar class
This definition is referenced by:  abid  2431  hbab1  2432  hbab  2434  cvjust  2438  abbi  2553  cbvab  2561  clelab  2563  nfabd2  2597  vjust  2973  dfsbcq2  3189  sbc8g  3194  unab  3617  inab  3618  difab  3619  csbab  3707  csbabgOLD  3708  exss  4555  iotaeq  5389  abrexex2g  6554  opabex3d  6555  opabex3  6556  abrexex2  6558  bj-hbab1  32291  bj-abbi  32296  bj-vjust  32307  eliminable1  32350  bj-vexwt  32365  bj-vexwvt  32367  bj-abfal  32409  bj-snsetex  32456
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