Definition df-clab 2340

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, φ will have y as a free variable, and "{y ∣ φ} " is read "the class of all sets y such that φ(y) is true." We do not define {y ∣ φ} in isolation but only as part of an expression that extends or "overloads" the ∈ relationship.

This is our first use of the ∈ symbol to connect classes instead of sets. The syntax definition wcel 1710, which extends or "overloads" the wel 1711 definition connecting setvar variables, requires that both sides of ∈ be a class. In df-cleq 2346 and df-clel 2349, we introduce a new kind of variable (class variable) that can substituted with expressions such as {y ∣ φ}. In the present definition, the x on the left-hand side is a setvar variable. Syntax definition cv 1641 allows us to substitute a setvar variable x for a class variable: all sets are classes by cvjust 2348 (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 2458 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 done with theorems such as vtoclg 2914 which is used, for example, to convert elirrv in set.mm to elirr in set.mm.

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 ∣ φ} a "class term".

For a general discussion of the theory of classes, see https://us.metamath.org/mpeuni/mmset.html#class 2914. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
df-clab	⊢ (x ∈ {y ∣ φ} ↔ [x / y]φ)

Detailed syntax breakdown of Definition df-clab

Step	Hyp	Ref	Expression
1		vx	. . . 4 setvar x
2	1	cv 1641	. . 3 class x
3		wph	. . . 4 wff φ
4		vy	. . . 4 setvar y
5	3, 4	cab 2339	. . 3 class {y ∣ φ}
6	2, 5	wcel 1710	. 2 wff x ∈ {y ∣ φ}
7	3, 4, 1	wsb 1648	. 2 wff [x / y]φ
8	6, 7	wb 176	1 wff (x ∈ {y ∣ φ} ↔ [x / y]φ)

This definition is referenced by:  abid  2341  hbab1  2342  hbab  2344  cvjust  2348  abbi  2463  cbvab  2471  clelab  2473  nfabd2  2507  vjust  2860  dfsbcq2  3049  sbc8g  3053  csbabg  3197  unab  3521  inab  3522  difab  3523  complab  3524  iotaeq  4347