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Definition df-clab 2395
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 1721, which extends or "overloads" the wel 1722 definition connecting set variables, requires that both sides of  e. be a class. In df-cleq 2401 and df-clel 2404, 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 1648 allows us to substitute a set variable  x for a class variable: all sets are classes by cvjust 2403 (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 2513 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 2975 which is used, for example, to convert elirrv 7525 to elirr 7526.

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 1648 . . 3  class  x
3 wph . . . 4  wff  ph
4 vy . . . 4  set  y
53, 4cab 2394 . . 3  class  { y  |  ph }
62, 5wcel 1721 . 2  wff  x  e. 
{ y  |  ph }
73, 4, 1wsb 1655 . 2  wff  [ x  /  y ] ph
86, 7wb 177 1  wff  ( x  e.  { y  | 
ph }  <->  [ x  /  y ] ph )
Colors of variables: wff set class
This definition is referenced by:  abid  2396  hbab1  2397  hbab  2399  cvjust  2403  abbi  2518  cbvab  2526  clelab  2528  nfabd2  2562  vjust  2921  dfsbcq2  3128  sbc8g  3132  csbabg  3274  unab  3572  inab  3573  difab  3574  exss  4390  iotaeq  5389  abrexex2g  5951  opabex3d  5952  opabex3  5953  abrexex2  5964
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