Definition df-clab 2009

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 1374, which extends or "overloads" the wel 1375 definition connecting setvar variables, requires that both sides of ∈ be a class. In df-cleq 2015 and df-clel 2018, 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 1227 allows us to substitute a setvar variable x for a class variable: all sets are classes by cvjust 2017 (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 2128 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 1227	. . 3 class x
3		wph	. . . 4 wff φ
4		vy	. . . 4 setvar y
5	3, 4	cab 2008	. . 3 class {y ∣ φ}
6	2, 5	wcel 1374	. 2 wff x ∈ {y ∣ φ}
7	3, 4, 1	wsb 1627	. 2 wff [x / y]φ
8	6, 7	wb 98	1 wff (x ∈ {y ∣ φ} ↔ [x / y]φ)

This definition is referenced by:  abid  2010  hbab1  2011  hbab  2013  cvjust  2017  abbi  2133  sb8ab  2141  cbvab  2142  clelab  2144  nfabd  2178  vjust  2536  dfsbcq2  2744  sbc8g  2748  csbabg  2884  unab  3181  inab  3182  difab  3183  rabeq0  3224  abeq0  3225  oprcl  3547  exss  3937  peano1  4244  peano2  4245  iotaeq  4802  nfvres  5131  abrexex2g  5670  opabex3d  5671  opabex3  5672  abrexex2  5674  bdab  7065  bdph  7077  bdcriota  7110