Definition df-clab 2010

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 1376, which extends or "overloads" the wel 1377 definition connecting setvar variables, requires that both sides of ∈ be a class. In df-cleq 2016 and df-clel 2019, 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 1373 allows us to substitute a setvar variable x for a class variable: all sets are classes by cvjust 2018 (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 2129 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 ∣ φ} 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 ∈ {y ∣ φ} ↔ [x / y]φ)

Detailed syntax breakdown of Definition df-clab

Step	Hyp	Ref	Expression
1		vx	. . . 4 setvar x
2	1	cv 1373	. . 3 class x
3		wph	. . . 4 wff φ
4		vy	. . . 4 setvar y
5	3, 4	cab 2009	. . 3 class {y ∣ φ}
6	2, 5	wcel 1376	. 2 wff x ∈ {y ∣ φ}
7	3, 4, 1	wsb 1628	. 2 wff [x / y]φ
8	6, 7	wb 98	1 wff (x ∈ {y ∣ φ} ↔ [x / y]φ)

This definition is referenced by:  abid  2011  hbab1  2012  hbab  2014  cvjust  2018  abbi  2134  sb8ab  2142  cbvab  2143  clelab  2145  nfabd  2179  vjust  2535  dfsbcq2  2743  sbc8g  2747  csbabg  2886  unab  3183  inab  3184  difab  3185  rabeq0  3226  abeq0  3227  oprcl  3526  exss  3916  peano1  4220  peano2  4221  iotaeq  4779  nfvres  5108  abrexex2g  5646  opabex3d  5647  opabex3  5648  abrexex2  5650