Definition df-clab 2609

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 1990, which extends or "overloads" the wel 1991 definition connecting setvar variables, requires that both sides of ∈ be classes. In df-cleq 2615 and df-clel 2618, we introduce a new kind of variable (class variable) that can be 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 1482 allows us to substitute a setvar variable 𝑥 for a class variable: all sets are classes by cvjust 2617 (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 2732 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 done with theorems such as vtoclg 3266 which is used, for example, to convert elirrv 8504 to elirr 8505.

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".

While the three class definitions df-clab 2609, df-cleq 2615, and df-clel 2618 are eliminable and conservative and thus meet the requirements for sound definitions, they are technically axioms in that they do not satisfy the requirements for the current definition checker. The proofs of conservativity require external justification that is beyond the scope of the definition checker.

For a general discussion of the theory of classes, see mmset.html#class. (Contributed by NM, 26-May-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 1482	. . 3 class 𝑥
3		wph	. . . 4 wff 𝜑
4		vy	. . . 4 setvar 𝑦
5	3, 4	cab 2608	. . 3 class {𝑦 ∣ 𝜑}
6	2, 5	wcel 1990	. 2 wff 𝑥 ∈ {𝑦 ∣ 𝜑}
7	3, 4, 1	wsb 1880	. 2 wff [𝑥 / 𝑦]𝜑
8	6, 7	wb 196	1 wff (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)

This definition is referenced by:  abid  2610  hbab1  2611  hbab  2613  cvjust  2617  cbvab  2746  clelab  2748  nfabd2  2784  vjust  3201  abv  3206  dfsbcq2  3438  sbc8g  3443  unab  3894  inab  3895  difab  3896  csbab  4008  exss  4931  iotaeq  5859  abrexex2g  7144  opabex3d  7145  opabex3  7146  abrexex2OLD  7150  bj-hbab1  32771  bj-abbi  32775  bj-vjust  32786  eliminable1  32840  bj-cleljustab  32847  bj-vexwt  32854  bj-vexwvt  32856  bj-ab0  32902  bj-snsetex  32951  bj-vjust2  33015  csbabgOLD  39050