ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  df-clab Unicode version

Definition df-clab 2070
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 1434, which extends or "overloads" the wel 1435 definition connecting setvar variables, requires that both sides of  e. be a class. In df-cleq 2076 and df-clel 2079, 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 setvar variable. Syntax definition cv 1284 allows us to substitute a setvar variable  x for a class variable: all sets are classes by cvjust 2078 (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 2191 for a quick overview).

Because class variables can be substituted with compound expressions and setvar variables cannot, it is often useful to convert a theorem containing a free setvar variable to a more general version with a class variable.

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  setvar  x
21cv 1284 . . 3  class  x
3 wph . . . 4  wff  ph
4 vy . . . 4  setvar  y
53, 4cab 2069 . . 3  class  { y  |  ph }
62, 5wcel 1434 . 2  wff  x  e. 
{ y  |  ph }
73, 4, 1wsb 1687 . 2  wff  [ x  /  y ] ph
86, 7wb 103 1  wff  ( x  e.  { y  | 
ph }  <->  [ x  /  y ] ph )
Colors of variables: wff set class
This definition is referenced by:  abid  2071  hbab1  2072  hbab  2074  cvjust  2078  abbi  2196  sb8ab  2204  cbvab  2205  clelab  2207  nfabd  2241  vjust  2611  dfsbcq2  2828  sbc8g  2832  csbabg  2973  unab  3248  inab  3249  difab  3250  rabeq0  3291  abeq0  3292  oprcl  3615  exss  4011  peano1  4364  peano2  4365  iotaeq  4926  nfvres  5259  abrexex2g  5799  opabex3d  5800  opabex3  5801  abrexex2  5803  bdab  10880  bdph  10892  bdcriota  10925
  Copyright terms: Public domain W3C validator