Definition df-clab 2157

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 2141, which extends or "overloads" the wel 2142 definition connecting setvar variables, requires that both sides of ∈ be a class. In df-cleq 2163 and df-clel 2166, 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 1347 allows us to substitute a setvar variable 𝑥 for a class variable: all sets are classes by cvjust 2165 (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 2279 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 2279. (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 1347	. . 3 class 𝑥
3		wph	. . . 4 wff 𝜑
4		vy	. . . 4 setvar 𝑦
5	3, 4	cab 2156	. . 3 class {𝑦 ∣ 𝜑}
6	2, 5	wcel 2141	. 2 wff 𝑥 ∈ {𝑦 ∣ 𝜑}
7	3, 4, 1	wsb 1755	. 2 wff [𝑥 / 𝑦]𝜑
8	6, 7	wb 104	1 wff (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)

This definition is referenced by:  abid  2158  hbab1  2159  hbab  2161  cvjust  2165  abbi  2284  sb8ab  2292  cbvabw  2293  cbvab  2294  clelab  2296  nfabdw  2331  nfabd  2332  vjust  2731  dfsbcq2  2958  sbc8g  2962  csbcow  3060  csbabg  3110  unab  3394  inab  3395  difab  3396  rabeq0  3444  abeq0  3445  oprcl  3789  exss  4212  peano1  4578  peano2  4579  iotaeq  5168  nfvres  5529  abrexex2g  6099  opabex3d  6100  opabex3  6101  abrexex2  6103  bdab  13873  bdph  13885  bdcriota  13918