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Definition df-clab 2243
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 1621, which extends or "overloads" the wel 1622 definition connecting set variables, requires that both sides of  e. be a class. In df-cleq 2249 and df-clel 2252, 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 1618 allows us to substitute a set variable  x for a class variable: all sets are classes by cvjust 2251 (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 2361 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 2811 which is used, for example, to convert elirrv 7265 to elirr 7266.

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/mpegif/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 1618 . . 3  class  x
3 wph . . . 4  wff  ph
4 vy . . . 4  set  y
53, 4cab 2242 . . 3  class  { y  |  ph }
62, 5wcel 1621 . 2  wff  x  e. 
{ y  |  ph }
73, 4, 1wsb 1883 . 2  wff  [ x  /  y ] ph
86, 7wb 178 1  wff  ( x  e.  { y  | 
ph }  <->  [ x  /  y ] ph )
Colors of variables: wff set class
This definition is referenced by:  abid  2244  hbab1  2245  hbab  2247  cvjust  2251  abbi  2366  cbvab  2374  clelab  2376  nfabd2  2410  vjust  2758  dfsbcq2  2955  sbc8g  2959  csbabg  3103  unab  3396  inab  3397  difab  3398  exss  4194  abrexex2g  5688  opabex3  5689  abrexex2  5700  iotaeq  6219  compneOLD  26997
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