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Definition df-clab 2443
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 1818, which extends or "overloads" the wel 1819 definition connecting setvar variables, requires that both sides of e. be classes. In df-cleq 2449 and df-clel 2452, 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 1394 allows us to substitute a setvar variable for a class variable: all sets are classes by cvjust 2451 (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 2581 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 3167 which is used, for example, to convert elirrv 8044 to elirr 8045.

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 2443, df-cleq 2449, and df-clel 2452 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 (Contributed by NM, 26-May-1993.)

Ref Expression

Detailed syntax breakdown of Definition df-clab
StepHypRef Expression
1 vx . . . 4
21cv 1394 . . 3
3 wph . . . 4
4 vy . . . 4
53, 4cab 2442 . . 3
62, 5wcel 1818 . 2
73, 4, 1wsb 1739 . 2
86, 7wb 184 1
Colors of variables: wff setvar class
This definition is referenced by:  abid  2444  hbab1  2445  hbab  2447  cvjust  2451  abbiOLD  2589  cbvab  2598  cbvabOLD  2599  clelab  2601  clelabOLD  2602  nfabd2  2640  vjust  3110  dfsbcq2  3330  sbc8g  3335  unab  3764  inab  3765  difab  3766  csbab  3855  csbabgOLD  3856  exss  4715  iotaeq  5564  abrexex2g  6777  opabex3d  6778  opabex3  6779  abrexex2  6781  bj-hbab1  34356  bj-abbi  34361  bj-vjust  34372  eliminable1  34415  bj-vexwt  34430  bj-vexwvt  34432  bj-abfal  34474  bj-snsetex  34521
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