Definition df-clab 2758

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 2145, which extends or "overloads" the wel 2146 definition connecting setvar variables, requires that both sides of ∈ be classes. In df-cleq 2764 and df-clel 2767, 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 1630 allows us to substitute a setvar variable 𝑥 for a class variable: all sets are classes by cvjust 2766 (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 2881 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 3417 which is used, for example, to convert elirrv 8660 to elirr 8661.

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 2758, df-cleq 2764, and df-clel 2767 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 1630	. . 3 class 𝑥
3		wph	. . . 4 wff 𝜑
4		vy	. . . 4 setvar 𝑦
5	3, 4	cab 2757	. . 3 class {𝑦 ∣ 𝜑}
6	2, 5	wcel 2145	. 2 wff 𝑥 ∈ {𝑦 ∣ 𝜑}
7	3, 4, 1	wsb 2049	. 2 wff [𝑥 / 𝑦]𝜑
8	6, 7	wb 196	1 wff (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑)

This definition is referenced by:  abid  2759  hbab1  2760  hbab  2762  cvjust  2766  cbvab  2895  clelab  2897  nfabd2  2933  vjust  3352  abv  3358  dfsbcq2  3590  sbc8g  3595  unab  4042  inab  4043  difab  4044  csbab  4153  exss  5060  iotaeq  6001  abrexex2g  7294  opabex3d  7295  opabex3  7296  abrexex2OLD  7300  bj-hbab1  33106  bj-abbi  33110  bj-vjust  33121  eliminable1  33173  bj-cleljustab  33180  bj-vexwt  33182  bj-vexwvt  33184  bj-ab0  33230  bj-snsetex  33281  bj-vjust2  33345  csbabgOLD  39575