Definition df-clab 2218

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 2202, which extends or "overloads" the wel 2203 definition connecting setvar variables, requires that both sides of ∈ be a class. In df-cleq 2224 and df-clel 2227, 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 1396 allows us to substitute a setvar variable 𝑥 for a class variable: all sets are classes by cvjust 2226 (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 2340 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 https://us.metamath.org/mpeuni/mmset.html#class 2340. (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 1396	. . 3 class 𝑥
3		wph	. . . 4 wff 𝜑
4		vy	. . . 4 setvar 𝑦
5	3, 4	cab 2217	. . 3 class {𝑦 ∣ 𝜑}
6	2, 5	wcel 2202	. 2 wff 𝑥 ∈ {𝑦 ∣ 𝜑}
7	3, 4, 1	wsb 1810	. 2 wff [𝑥 / 𝑦]𝜑
8	6, 7	wb 105	1 wff (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)

This definition is referenced by:  abid  2219  hbab1  2220  hbab  2222  cvjust  2226  abbi  2345  sb8ab  2353  cbvabw  2354  cbvab  2355  clelab  2357  eqabdv  2360  nfabdw  2393  nfabd  2394  vjust  2803  dfsbcq2  3034  sbc8g  3039  csbcow  3138  csbabg  3189  unab  3474  inab  3475  difab  3476  rabeq0  3524  abeq0  3525  oprcl  3886  exss  4319  peano1  4692  peano2  4693  iotaeq  5295  nfvres  5675  abrexex2g  6281  opabex3d  6282  opabex3  6283  abrexex2  6285  modom  6993  bdab  16433  bdph  16445  bdcriota  16478