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Definition df-clab 2043
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 1409, which extends or "overloads" the wel 1410 definition connecting setvar variables, requires that both sides of  e. be a class. In df-cleq 2049 and df-clel 2052, 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 1258 allows us to substitute a setvar variable  x for a class variable: all sets are classes by cvjust 2051 (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 2162 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 1258 . . 3  class  x
3 wph . . . 4  wff  ph
4 vy . . . 4  setvar  y
53, 4cab 2042 . . 3  class  { y  |  ph }
62, 5wcel 1409 . 2  wff  x  e. 
{ y  |  ph }
73, 4, 1wsb 1661 . 2  wff  [ x  /  y ] ph
86, 7wb 102 1  wff  ( x  e.  { y  | 
ph }  <->  [ x  /  y ] ph )
Colors of variables: wff set class
This definition is referenced by:  abid  2044  hbab1  2045  hbab  2047  cvjust  2051  abbi  2167  sb8ab  2175  cbvab  2176  clelab  2178  nfabd  2212  vjust  2575  dfsbcq2  2789  sbc8g  2793  csbabg  2934  unab  3231  inab  3232  difab  3233  rabeq0  3274  abeq0  3275  oprcl  3600  exss  3990  peano1  4344  peano2  4345  iotaeq  4902  nfvres  5233  abrexex2g  5774  opabex3d  5775  opabex3  5776  abrexex2  5778  bdab  10324  bdph  10336  bdcriota  10369
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