Definition df-clab 2180

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 2164, which extends or "overloads" the wel 2165 definition connecting setvar variables, requires that both sides of ∈ be a class. In df-cleq 2186 and df-clel 2189, 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 1363 allows us to substitute a setvar variable 𝑥 for a class variable: all sets are classes by cvjust 2188 (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 2302 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 2302. (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 1363	. . 3 class 𝑥
3		wph	. . . 4 wff 𝜑
4		vy	. . . 4 setvar 𝑦
5	3, 4	cab 2179	. . 3 class {𝑦 ∣ 𝜑}
6	2, 5	wcel 2164	. 2 wff 𝑥 ∈ {𝑦 ∣ 𝜑}
7	3, 4, 1	wsb 1773	. 2 wff [𝑥 / 𝑦]𝜑
8	6, 7	wb 105	1 wff (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)

This definition is referenced by:  abid  2181  hbab1  2182  hbab  2184  cvjust  2188  abbi  2307  sb8ab  2315  cbvabw  2316  cbvab  2317  clelab  2319  eqabdv  2322  nfabdw  2355  nfabd  2356  vjust  2761  dfsbcq2  2988  sbc8g  2993  csbcow  3091  csbabg  3142  unab  3426  inab  3427  difab  3428  rabeq0  3476  abeq0  3477  oprcl  3828  exss  4256  peano1  4626  peano2  4627  iotaeq  5223  nfvres  5588  abrexex2g  6172  opabex3d  6173  opabex3  6174  abrexex2  6176  bdab  15330  bdph  15342  bdcriota  15375