Definition df-clab 2216

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 2200, which extends or "overloads" the wel 2201 definition connecting setvar variables, requires that both sides of ∈ be a class. In df-cleq 2222 and df-clel 2225, 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 1394 allows us to substitute a setvar variable 𝑥 for a class variable: all sets are classes by cvjust 2224 (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 2338 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 2338. (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 1394	. . 3 class 𝑥
3		wph	. . . 4 wff 𝜑
4		vy	. . . 4 setvar 𝑦
5	3, 4	cab 2215	. . 3 class {𝑦 ∣ 𝜑}
6	2, 5	wcel 2200	. 2 wff 𝑥 ∈ {𝑦 ∣ 𝜑}
7	3, 4, 1	wsb 1808	. 2 wff [𝑥 / 𝑦]𝜑
8	6, 7	wb 105	1 wff (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)

This definition is referenced by:  abid  2217  hbab1  2218  hbab  2220  cvjust  2224  abbi  2343  sb8ab  2351  cbvabw  2352  cbvab  2353  clelab  2355  eqabdv  2358  nfabdw  2391  nfabd  2392  vjust  2800  dfsbcq2  3031  sbc8g  3036  csbcow  3135  csbabg  3186  unab  3471  inab  3472  difab  3473  rabeq0  3521  abeq0  3522  oprcl  3881  exss  4313  peano1  4686  peano2  4687  iotaeq  5287  nfvres  5665  abrexex2g  6271  opabex3d  6272  opabex3  6273  abrexex2  6275  bdab  16225  bdph  16237  bdcriota  16270