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Definition df-clab 2367
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 1717, which extends or "overloads" the wel 1718 definition connecting set variables, requires that both sides of  e. be a class. In df-cleq 2373 and df-clel 2376, 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 2375 (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 2485 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 2947 which is used, for example, to convert elirrv 7491 to elirr 7492.

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 2366 . . 3  class  { y  |  ph }
62, 5wcel 1717 . 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  2368  hbab1  2369  hbab  2371  cvjust  2375  abbi  2490  cbvab  2498  clelab  2500  nfabd2  2534  vjust  2893  dfsbcq2  3100  sbc8g  3104  csbabg  3246  unab  3544  inab  3545  difab  3546  exss  4360  iotaeq  5359  abrexex2g  5920  opabex3d  5921  opabex3  5922  abrexex2  5933
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