Definition df-clab 2193

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 2177, which extends or "overloads" the wel 2178 definition connecting setvar variables, requires that both sides of ∈ be a class. In df-cleq 2199 and df-clel 2202, 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 1372 allows us to substitute a setvar variable 𝑥 for a class variable: all sets are classes by cvjust 2201 (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 2315 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 2315. (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 1372	. . 3 class 𝑥
3		wph	. . . 4 wff 𝜑
4		vy	. . . 4 setvar 𝑦
5	3, 4	cab 2192	. . 3 class {𝑦 ∣ 𝜑}
6	2, 5	wcel 2177	. 2 wff 𝑥 ∈ {𝑦 ∣ 𝜑}
7	3, 4, 1	wsb 1786	. 2 wff [𝑥 / 𝑦]𝜑
8	6, 7	wb 105	1 wff (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)

This definition is referenced by:  abid  2194  hbab1  2195  hbab  2197  cvjust  2201  abbi  2320  sb8ab  2328  cbvabw  2329  cbvab  2330  clelab  2332  eqabdv  2335  nfabdw  2368  nfabd  2369  vjust  2774  dfsbcq2  3002  sbc8g  3007  csbcow  3105  csbabg  3156  unab  3441  inab  3442  difab  3443  rabeq0  3491  abeq0  3492  oprcl  3845  exss  4275  peano1  4646  peano2  4647  iotaeq  5245  nfvres  5617  abrexex2g  6212  opabex3d  6213  opabex3  6214  abrexex2  6216  bdab  15848  bdph  15860  bdcriota  15893