Definition df-clab 1441

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 1105, which extends or "overloads" the wel 1106 definition connecting set variables, requires that both sides of e. be a class. In df-cleq 1446 and df-clel 1449, 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 1098 allows us to substitute a set variable x for a class variable: all sets are classes by cvjust 1448 (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 1544 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 1822 which is used, for example, to convert elirrv 4522 to elirr 4523.

Assertion

Ref	Expression
df-clab	|- (x e. {y | ph} <-> [x / y]ph)

Detailed syntax breakdown of Definition df-clab

Step	Hyp	Ref	Expression
1		vx	. . . 4 set x
2	1	cv 1098	. . 3 class x
3		wph	. . . 4 wff ph
4		vy	. . . 4 set y
5	3, 4	cab 1440	. . 3 class {y | ph}
6	2, 5	wcel 1105	. 2 wff x e. {y | ph}
7	3, 4, 2	wsbc 1153	. 2 wff [x / y]ph
8	6, 7	wb 146	1 wff (x e. {y | ph} <-> [x / y]ph)

This definition is referenced by:  abid 1442  hbab1 1443  hbab 1444  hbabd 1445  cvjust 1448  clelab 1557  csbabg 2014  unab 2238  inab 2239  difab 2240  exss 2737  abrexex2 3810  scottexs 4642  scott0s 4643

Copyright terms: Public domain