Definition df-clab 2221

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 2205, which extends or "overloads" the wel 2206 definition connecting setvar variables, requires that both sides of ∈ be a class. In df-cleq 2227 and df-clel 2230, 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 2229 (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 2343 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 2343. (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 2220	. . 3 class {𝑦 ∣ 𝜑}
6	2, 5	wcel 2205	. 2 wff 𝑥 ∈ {𝑦 ∣ 𝜑}
7	3, 4, 1	wsb 1811	. 2 wff [𝑥 / 𝑦]𝜑
8	6, 7	wb 105	1 wff (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)

This definition is referenced by:  abid  2222  hbab1  2223  hbab  2225  cvjust  2229  abbibcom  2348  abbib  2352  abbi  2353  sb8ab  2358  cbvabw  2359  cbvab  2360  clelab  2362  eqabdv  2365  nfabdw  2405  nfabd  2406  vjust  2816  dfsbcq2  3047  sbc8g  3052  csbcow  3151  csbabg  3202  unab  3490  inab  3491  difab  3492  rabeq0  3540  abeq0  3541  oprcl  3909  exss  4345  peano1  4718  peano2  4719  iotaeq  5323  nfvres  5708  abrexex2g  6315  opabex3d  6316  opabex3  6317  abrexex2  6319  modom  7063  bdab  16625  bdph  16637  bdcriota  16670