Definition df-clab 2164

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 2148, which extends or "overloads" the wel 2149 definition connecting setvar variables, requires that both sides of ∈ be a class. In df-cleq 2170 and df-clel 2173, 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 1352 allows us to substitute a setvar variable 𝑥 for a class variable: all sets are classes by cvjust 2172 (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 2286 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 2286. (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 1352	. . 3 class 𝑥
3		wph	. . . 4 wff 𝜑
4		vy	. . . 4 setvar 𝑦
5	3, 4	cab 2163	. . 3 class {𝑦 ∣ 𝜑}
6	2, 5	wcel 2148	. 2 wff 𝑥 ∈ {𝑦 ∣ 𝜑}
7	3, 4, 1	wsb 1762	. 2 wff [𝑥 / 𝑦]𝜑
8	6, 7	wb 105	1 wff (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)

This definition is referenced by:  abid  2165  hbab1  2166  hbab  2168  cvjust  2172  abbi  2291  sb8ab  2299  cbvabw  2300  cbvab  2301  clelab  2303  nfabdw  2338  nfabd  2339  vjust  2738  dfsbcq2  2965  sbc8g  2970  csbcow  3068  csbabg  3118  unab  3402  inab  3403  difab  3404  rabeq0  3452  abeq0  3453  oprcl  3802  exss  4226  peano1  4592  peano2  4593  iotaeq  5185  nfvres  5547  abrexex2g  6118  opabex3d  6119  opabex3  6120  abrexex2  6122  bdab  14450  bdph  14462  bdcriota  14495