Definition df-clab 2068

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 1433, which extends or "overloads" the wel 1434 definition connecting setvar variables, requires that both sides of ∈ be a class. In df-cleq 2074 and df-clel 2077, 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 1283 allows us to substitute a setvar variable 𝑥 for a class variable: all sets are classes by cvjust 2076 (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 2187 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 http://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 1283	. . 3 class 𝑥
3		wph	. . . 4 wff 𝜑
4		vy	. . . 4 setvar 𝑦
5	3, 4	cab 2067	. . 3 class {𝑦 ∣ 𝜑}
6	2, 5	wcel 1433	. 2 wff 𝑥 ∈ {𝑦 ∣ 𝜑}
7	3, 4, 1	wsb 1685	. 2 wff [𝑥 / 𝑦]𝜑
8	6, 7	wb 103	1 wff (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)

This definition is referenced by:  abid  2069  hbab1  2070  hbab  2072  cvjust  2076  abbi  2192  sb8ab  2200  cbvab  2201  clelab  2203  nfabd  2237  vjust  2602  dfsbcq2  2818  sbc8g  2822  csbabg  2963  unab  3231  inab  3232  difab  3233  rabeq0  3274  abeq0  3275  oprcl  3594  exss  3982  peano1  4335  peano2  4336  iotaeq  4895  nfvres  5227  abrexex2g  5767  opabex3d  5768  opabex3  5769  abrexex2  5771  bdab  10629  bdph  10641  bdcriota  10674