MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-clab Unicode version

Definition df-clab 2240
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 2246 and df-clel 2249, 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 2248 (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 2354 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 2781 which is used, for example, to convert elirrv 7195 to elirr 7196.

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 2239 . . 3  class  { y  |  ph }
62, 5wcel 1621 . 2  wff  x  e. 
{ y  |  ph }
73, 4, 1wsb 1882 . 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  2241  hbab1  2242  hbab  2244  cvjust  2248  abbi  2359  cbvab  2367  clelab  2369  nfabd2  2403  vjust  2728  dfsbcq2  2924  sbc8g  2928  csbabg  3070  unab  3342  inab  3343  difab  3344  exss  4129  abrexex2g  5620  opabex3  5621  abrexex2  5632  iotaeq  6151  compneOLD  26810
  Copyright terms: Public domain W3C validator