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

Definition df-clab 2422
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 1725, which extends or "overloads" the wel 1726 definition connecting set variables, requires that both sides of  e. be a class. In df-cleq 2428 and df-clel 2431, 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 1651 allows us to substitute a set variable  x for a class variable: all sets are classes by cvjust 2430 (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 2540 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 3003 which is used, for example, to convert elirrv 7557 to elirr 7558.

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 1651 . . 3  class  x
3 wph . . . 4  wff  ph
4 vy . . . 4  set  y
53, 4cab 2421 . . 3  class  { y  |  ph }
62, 5wcel 1725 . 2  wff  x  e. 
{ y  |  ph }
73, 4, 1wsb 1658 . 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  2423  hbab1  2424  hbab  2426  cvjust  2430  abbi  2545  cbvab  2553  clelab  2555  nfabd2  2589  vjust  2949  dfsbcq2  3156  sbc8g  3160  csbabg  3302  unab  3600  inab  3601  difab  3602  exss  4418  iotaeq  5418  abrexex2g  5980  opabex3d  5981  opabex3  5982  abrexex2  5993
  Copyright terms: Public domain W3C validator