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Definition df-clab 2440
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 (y) 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 1758, which extends or "overloads" the wel 1759 definition connecting setvar variables, requires that both sides of e. be classes. In df-cleq 2446 and df-clel 2449, 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 1369 allows us to substitute a setvar variable for a class variable: all sets are classes by cvjust 2448 (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 2578 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 3139 which is used, for example, to convert elirrv 7949 to elirr 7950.

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 2440, df-cleq 2446, and df-clel 2449 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 1369 . . 3
3 wph . . . 4
4 vy . . . 4
53, 4cab 2439 . . 3
62, 5wcel 1758 . 2
73, 4, 1wsb 1702 . 2
86, 7wb 184 1
Colors of variables: wff setvar class
This definition is referenced by:  abid  2441  hbab1  2442  hbab  2444  cvjust  2448  abbiOLD  2586  cbvab  2595  cbvabOLD  2596  clelab  2598  clelabOLD  2599  nfabd2  2637  vjust  3082  dfsbcq2  3300  sbc8g  3305  unab  3731  inab  3732  difab  3733  csbab  3821  csbabgOLD  3822  exss  4672  iotaeq  5508  abrexex2g  6687  opabex3d  6688  opabex3  6689  abrexex2  6691  bj-hbab1  33136  bj-abbi  33141  bj-vjust  33152  eliminable1  33195  bj-vexwt  33210  bj-vexwvt  33212  bj-abfal  33254  bj-snsetex  33301
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