Definition df-clab 2087

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 1448, which extends or "overloads" the wel 1449 definition connecting setvar variables, requires that both sides of ∈ be a class. In df-cleq 2093 and df-clel 2096, 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 1298 allows us to substitute a setvar variable 𝑥 for a class variable: all sets are classes by cvjust 2095 (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 2208 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. (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 1298	. . 3 class 𝑥
3		wph	. . . 4 wff 𝜑
4		vy	. . . 4 setvar 𝑦
5	3, 4	cab 2086	. . 3 class {𝑦 ∣ 𝜑}
6	2, 5	wcel 1448	. 2 wff 𝑥 ∈ {𝑦 ∣ 𝜑}
7	3, 4, 1	wsb 1703	. 2 wff [𝑥 / 𝑦]𝜑
8	6, 7	wb 104	1 wff (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)

This definition is referenced by:  abid  2088  hbab1  2089  hbab  2091  cvjust  2095  abbi  2213  sb8ab  2221  cbvab  2222  clelab  2224  nfabd  2259  vjust  2642  dfsbcq2  2865  sbc8g  2869  csbabg  3011  unab  3290  inab  3291  difab  3292  rabeq0  3339  abeq0  3340  oprcl  3676  exss  4087  peano1  4446  peano2  4447  iotaeq  5032  nfvres  5386  abrexex2g  5949  opabex3d  5950  opabex3  5951  abrexex2  5953  bdab  12617  bdph  12629  bdcriota  12662