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Definition df-clab 2425
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 1726, which extends or "overloads" the wel 1727 definition connecting set variables, requires that both sides of  e. be a class. In df-cleq 2431 and df-clel 2434, 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 1652 allows us to substitute a set variable  x for a class variable: all sets are classes by cvjust 2433 (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 2543 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 3013 which is used, for example, to convert elirrv 7567 to elirr 7568.

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 1652 . . 3  class  x
3 wph . . . 4  wff  ph
4 vy . . . 4  set  y
53, 4cab 2424 . . 3  class  { y  |  ph }
62, 5wcel 1726 . 2  wff  x  e. 
{ y  |  ph }
73, 4, 1wsb 1659 . 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  2426  hbab1  2427  hbab  2429  cvjust  2433  abbi  2548  cbvab  2556  clelab  2558  nfabd2  2592  vjust  2959  dfsbcq2  3166  sbc8g  3170  csbabg  3312  unab  3610  inab  3611  difab  3612  exss  4428  iotaeq  5428  abrexex2g  5990  opabex3d  5991  opabex3  5992  abrexex2  6003
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