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

Definition df-clab 2391
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 2397 and df-clel 2400, 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 2399 (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 2509 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 2971 which is used, for example, to convert elirrv 7521 to elirr 7522.

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 2390 . . 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  2392  hbab1  2393  hbab  2395  cvjust  2399  abbi  2514  cbvab  2522  clelab  2524  nfabd2  2558  vjust  2917  dfsbcq2  3124  sbc8g  3128  csbabg  3270  unab  3568  inab  3569  difab  3570  exss  4386  iotaeq  5385  abrexex2g  5947  opabex3d  5948  opabex3  5949  abrexex2  5960
  Copyright terms: Public domain W3C validator