MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-clab Unicode version

Definition df-clab 2409
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 1749, which extends or "overloads" the wel 1750 definition connecting setvar variables, requires that both sides of e. be a class. In df-cleq 2415 and df-clel 2418, 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 setvar variable. Syntax definition cv 1686 allows us to substitute a setvar variable for a class variable: all sets are classes by cvjust 2417 (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 2527 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 3008 which is used, for example, to convert elirrv 7759 to elirr 7760.

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 3 class definitions df-clab 2409, df-cleq 2415, and df-clel 2418 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, 5-Aug-1993.)

Assertion
Ref Expression
df-clab

Detailed syntax breakdown of Definition df-clab
StepHypRef Expression
1 vx . . . 4
21cv 1686 . . 3
3 wph . . . 4
4 vy . . . 4
53, 4cab 2408 . . 3
62, 5wcel 1749 . 2
73, 4, 1wsb 1693 . 2
86, 7wb 178 1
Colors of variables: wff setvar class
This definition is referenced by:  abid  2410  hbab1  2411  hbab  2413  cvjust  2417  abbi  2532  cbvab  2540  clelab  2542  nfabd2  2576  vjust  2952  dfsbcq2  3167  sbc8g  3171  unab  3594  inab  3595  difab  3596  csbab  3684  csbabgOLD  3685  exss  4527  iotaeq  5361  abrexex2g  6523  opabex3d  6524  opabex3  6525  abrexex2  6527  bj-hbab1  31787  bj-abbi  31792  bj-vjust  31802  bj-vexwt  31842  bj-vexwv  31844  bj-snsetex  31903
  Copyright terms: Public domain W3C validator