Definition df-clab 2225

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 2209, which extends or "overloads" the wel 2210 definition connecting setvar variables, requires that both sides of ∈ be a class. In df-cleq 2231 and df-clel 2234, 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 1401 allows us to substitute a setvar variable 𝑥 for a class variable: all sets are classes by cvjust 2233 (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 2347 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 2347. (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 1401	. . 3 class 𝑥
3		wph	. . . 4 wff 𝜑
4		vy	. . . 4 setvar 𝑦
5	3, 4	cab 2224	. . 3 class {𝑦 ∣ 𝜑}
6	2, 5	wcel 2209	. 2 wff 𝑥 ∈ {𝑦 ∣ 𝜑}
7	3, 4, 1	wsb 1815	. 2 wff [𝑥 / 𝑦]𝜑
8	6, 7	wb 105	1 wff (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)

This definition is referenced by:  abid  2226  hbab1  2227  hbab  2229  cvjust  2233  abbibcom  2352  abbib  2356  abbi  2357  sb8ab  2362  cbvabw  2363  cbvab  2364  clelab  2366  eqabdv  2369  nfabdw  2411  nfabd  2412  vjust  2822  dfsbcq2  3054  sbc8g  3059  csbcow  3158  csbabg  3209  unab  3498  inab  3499  difab  3500  ab0w  3550  rabeq0  3552  abeq0  3553  oprcl  3926  exss  4365  peano1  4739  peano2  4740  iotaeq  5344  nfvres  5729  abrexex2g  6342  opabex3d  6343  opabex3  6344  abrexex2  6346  modom  7101  bdab  16781  bdph  16793  bdcriota  16826