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Definition df-clab 2476
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 1732, which extends or "overloads" the wel 1733 definition connecting set variables, requires that both sides of e. be a class. In df-cleq 2482 and df-clel 2485, we introduce a new kind of variable (class variable) that can substituted with expressions such as . In the present definition, the on the left-hand side is a set variable. Syntax definition cv 1669 allows us to substitute a set variable for a class variable: all sets are classes by cvjust 2484 (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 2594 for a quick overview).

Because class variables can be substituted with compound expressions and set variables cannot, it is often useful to convert a theorem containing a free set variable to a more general version with a class variable. This is done with theorems such as vtoclg 3070 which is used, for example, to convert elirrv 7732 to elirr 7733.

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".

For a general discussion of the theory of classes, see http://us.metamath.org/mpeuni/mmset.html#class. (Contributed by NM, 5-Aug-1993.)

Assertion
Ref Expression
df-clab

Detailed syntax breakdown of Definition df-clab
StepHypRef Expression
1 vx . . . 4
21cv 1669 . . 3
3 wph . . . 4
4 vy . . . 4
53, 4cab 2475 . . 3
62, 5wcel 1732 . 2
73, 4, 1wsb 1676 . 2
86, 7wb 178 1
Colors of variables: wff set class
This definition is referenced by:  abid  2477  hbab1  2478  hbab  2480  cvjust  2484  abbi  2599  cbvab  2607  clelab  2609  nfabd2  2643  vjust  3016  dfsbcq2  3227  sbc8g  3231  unab  3653  inab  3654  difab  3655  csbab  3742  csbabgOLD  3743  exss  4578  iotaeq  5409  abrexex2g  6560  opabex3d  6561  opabex3  6562  abrexex2  6564  bj-snsetex  30947  bj-pr1val  30980  bj-pr2val  30981
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