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 1397 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 1397	. . 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  abbib  2349  sb8ab  2354  cbvabw  2355  cbvab  2356  clelab  2358  eqabdv  2361  nfabdw  2394  nfabd  2395  vjust  2804  dfsbcq2  3035  sbc8g  3040  csbcow  3139  csbabg  3190  unab  3476  inab  3477  difab  3478  rabeq0  3526  abeq0  3527  oprcl  3891  exss  4325  peano1  4698  peano2  4699  iotaeq  5302  nfvres  5684  abrexex2g  6291  opabex3d  6292  opabex3  6293  abrexex2  6295  modom  7037  bdab  16554  bdph  16566  bdcriota  16599