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  2740  dfsbcq2  2967  sbc8g  2972  csbcow  3070  csbabg  3120  unab  3404  inab  3405  difab  3406  rabeq0  3454  abeq0  3455  oprcl  3804  exss  4229  peano1  4595  peano2  4596  iotaeq  5188  nfvres  5550  abrexex2g  6123  opabex3d  6124  opabex3  6125  abrexex2  6127  bdab  14629  bdph  14641  bdcriota  14674