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Definition df-clab 2283
Description: Define class abstraction notation (so-called by Quine), also called a "class builder" in the literature.  x and  y need not be distinct. Definition 2.1 of [Quine] p. 16. Typically,  ph will have  y as a free variable, and " { y  |  ph } " is read "the class of all sets  y such that  ph ( y ) is true." We do not define  { y  |  ph } 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 1696, which extends or "overloads" the wel 1697 definition connecting set variables, requires that both sides of  e. be a class. In df-cleq 2289 and df-clel 2292, we introduce a new kind of variable (class variable) that can substituted with expressions such as  { y  | 
ph }. In the present definition, the  x on the left-hand side is a set variable. Syntax definition cv 1631 allows us to substitute a set variable  x for a class variable: all sets are classes by cvjust 2291 (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 2401 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 2856 which is used, for example, to convert elirrv 7327 to elirr 7328.

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  {
y  |  ph } 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  |-  ( x  e.  { y  | 
ph }  <->  [ x  /  y ] ph )

Detailed syntax breakdown of Definition df-clab
StepHypRef Expression
1 vx . . . 4  set  x
21cv 1631 . . 3  class  x
3 wph . . . 4  wff  ph
4 vy . . . 4  set  y
53, 4cab 2282 . . 3  class  { y  |  ph }
62, 5wcel 1696 . 2  wff  x  e. 
{ y  |  ph }
73, 4, 1wsb 1638 . 2  wff  [ x  /  y ] ph
86, 7wb 176 1  wff  ( x  e.  { y  | 
ph }  <->  [ x  /  y ] ph )
Colors of variables: wff set class
This definition is referenced by:  abid  2284  hbab1  2285  hbab  2287  cvjust  2291  abbi  2406  cbvab  2414  clelab  2416  nfabd2  2450  vjust  2802  dfsbcq2  3007  sbc8g  3011  csbabg  3155  unab  3448  inab  3449  difab  3450  exss  4252  iotaeq  5243  abrexex2g  5784  opabex3  5785  abrexex2  5796  compneOLD  27746  opabex3d  28190
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